Linear Discrete Parabolic Problems

LINEAR DISCRETE PARABOLIC PARABOLIC PROBLEMS PROBLEMS

NORTH-HOLLAND MATHEMATICS MATHEMATICS STUDIES STUDIES 203 203

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LINEAR DISCRETE PARABOLIC PARABOLIC PROBLEMS

NIKOLAI YU. BAKAEV BAKAEV Moscow, Russia

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Preface In this book we give a systematic systematic approach approach to to the the study study ofof linear linear discrete discrete parabolic problems for which which we we examine examine the the question question of of well-posedness, well-posedness, in particular, of stability. stability. The The developed developed techniques techniques can can also also be be applied applied in the analysis of convergence convergence of of the the corresponding corresponding discrete discrete solutions, solutions, asas demonstrated in Chapter 7. 7. We We remark remark that that the the use use of of the the concept concept ofofevoevolution equation in discrete time time and and of of some some related related concepts, concepts, which which will will be be explained in the sequel, is is of of great great importance importance for for our our investigation. investigation. This This approach appears natural natural in in view view of of the the fact fact that that our our subsequent subsequent consideraconsiderations are restricted to studying studying the the parabolic parabolic case case only. only. We Weexpect expect that that the the reader will have gotten the the impression impression that that the the developed developed discrete discrete theory theory by no means is a direct analogue analogue of of the the modern modern theory theory ofof parabolic parabolic differdifferential equations in Banach Banach space. space. ItIt will will therefore therefore be be seen seen that that the the below below approach possesses its own own distinctive distinctive features features for for which which one one can can find find no no analogue in the continuous continuous case. case. On On the the other other hand, hand, in in what what follows follows we we emphasize frequently the parallels parallels between between the the present present discrete discrete theory theory and and the classical continuous one. one. Our Our assertions assertions of of general general character character are are further further used for examining certain certain discretization-in-time discretization-in-time methods methods as as applied applied toto ababstract parabolic equations, equations, and and as as such such we we consider consider Runge-Kutta Runge-Kutta methods methods as well as other widespread widespread methods. methods. Note Note that that the the achieved achieved results results can can be used for the study of fully fully discrete discrete approximations approximations to to parabolic parabolic partial partial differential problems that are are constructed constructed on on the the basis basis of offinite finitedifference difference oror finite element discretization discretization in in space. space. However However such such an an investigation investigation would would to describe describe the the necessary necessary details, details, and and we we give give up up require much more room to idea at at least least in in the the present present book; book; perhaps perhaps itit will will be be the realization of this idea one. the subject for another one.

It is well known that any any modern modern discretization discretization method method used used inin the the nununon-stationary problems problems can can be be equivalently equivalently rewritten rewritten merical treatment of non-stationary step method method with with respect respect to to the the time time variable, variable, under under in the form of a single step restrictions. In In fact, fact, this this circumstance circumstance isis an an essential essential inincertain reasonable restrictions. subsequent considerations. considerations. Moreover, Moreover, itit allows allowsus usto toexamine examine gredient of our subsequent

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discrete problems at the the first first stage stage of of analysis analysis without without any any connection connection with with the starting parabolic problem problem that that has has been been discretized. discretized. It is generally acknowledged acknowledged that that the the questions questions of of stability stability and and converconvergence are the most essential essential for for numerical numerical analysis analysis of of discretizations. discretizations. In In parparticular, if the method is is stable, stable, any any numerical numerical errors errors which which arise arise during during the the calculating process do not not grow grow in in the the sequel. sequel. The The analysis analysis of of convergence convergence is closely related to that that of of stability. stability. Moreover, Moreover, the the so-called so-called equivalence equivalence theorems (see, e.g., Richtmyer Richtmyer & & Morton Morton [135]) [135]) allow allow one one to to derive derive error error estimates for both stable stable and and consistent consistent discrete discrete methods, methods, assuming assuming that that the data are smooth enough. enough. In In dealing dealing with with the the parabolic parabolic case case we wehave have the the possibility of working with with non-smooth non-smooth data data as as well. well. The The question question ofof stabilstability/convergence is then then settled settled by by using using aa priori priori inequalities inequalities related related toto the the property of strong stability stability of of holomorphic holomorphic semigroups semigroups (cf. (cf. Larsson Larsson [99]). [99]). When working with concrete concrete classes classes of of discretization discretization methods, methods, the the questions questions of existence and uniqueness uniqueness of of aa solution solution in in the the necessary necessary sense sense also also need need toto be settled, and then we we have have to to discuss discuss well-posedness well-posedness instead instead ofof stability. stability. Much work has been devoted devoted to to the the verification verification of of stability stability and and by by now now many different tools for for its its analysis analysis are are known. known. We We can can mention, mention, among among others, the Fourier transform transform method method (see, (see, e.g., e.g., Lax Lax & & Wendroff Wendroff [101] [101] and and Richtmyer & Morton [135]); [135]); the the method method of of separation separation of of variables variables (see, (see, e.g., e.g., Douglas & Gunn [66]); the the energy energy method method (see, (see, e.g., e.g., Kreiss Kreiss [98]); [98]);the the method method based on estimating the the Green's Green's functions functions (Brenner (Brenner & & Thomee Thomee [51], [51],Schatz, Schatz, 145]); and and the the method method Thomee, and Wahlbin [141], and Serdyukova [143, 145]); based on the maximum maximum principle principle (see, (see, e.g., e.g., Samarskii Samarskii [139]). [139]). Recently, many authors authors have have successfully successfully dealt dealt with with discrete discrete initialinitialboundary value problems problems thinking thinking of of them them as as initial initial value value problems problems for for difference-operator equations equations in in Banach Banach (in (in particular, particular, in in Hilbert) Hilbert) spaces. spaces. As As a consequence, the stability stability theory theory of of linear linear discrete discrete non-stationary non-stationary problems problems in Hilbert spaces is, at present, present, well well advanced advanced and and fairly fairly effective effective inin applicaapplications connected with the the finite finite element element and and finite finite difference difference methods methods as aswell well as with other approaches approaches (see, (see, e.g., e.g., Bramble Bramble et et al. al. [49], [49], Karakashian Karakashian [90], [90], and Keeling [92], Oden k Reddy Reddy [121], Samarskii [139], Thomee [161, 162], and references therein). However However related related analysis analysis within within Banach Banach space space settings settings is much more complicated, complicated, which which results results from from complexity complexity of of the the geometry geometryofof Banach spaces as compared compared with with that that of of Hilbert Hilbert spaces. spaces. Most Most ofof this this work work has been devoted to the the case case of of linear linear differential differential problems problems with with aa constant constant operator (see, e.g., Bakaev Bakaev [10, [10, 16, 16, 17, 17, 18, 18, 20, 20, 28, 28, 30, 30, 31, 31, 32, 32, 37], 37], Brenner Brenner & Thomee [51, 52], Crouzeix Crouzeix [60], [60], Crouzeix Crouzeix et et al. al. [61], [61], Crouzeix, Crouzeix, Larsson, Larsson, and Thomee [62], Lubich Lubich & & Nevanlinna Nevanlinna [104], Nitsche & Wheeler [119], Palencia [127, 128], Schatz, Schatz, Thomee, Thomee, and and Wahlbin Wahlbin [141], and Thomee [162]).

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In applications such a situation situation corresponds corresponds to to partial partial differential differential initialinitialboundary value problems problems with with time-independent time-independent coefficients. coefficients. We We also also reremark that certain results results for for discretized discretized linear linear integro-differential integro-differential equations equations have appeared (see, e.g., e.g., Bakaev, Bakaev, Larsson, Larsson, and and Thomee Thomee [40, [40, 41], 41], Sloan Sloan && Thomee [147], and Zhang [168]). Less is known for the case case of of non-stationary non-stationary problems problems with with aa variable variable (that is, time-dependent) time-dependent) operator operator in in Banach Banach space. space. The The first first achievements achievements in this direction have dealt dealt with with situations situations with with one one spatial spatial dimension dimension (see, (see, e.g., Thomee and Wahlbin Wahlbin [163]) [163]) or or with with concrete concrete families families of of discrete discrete probproblems (see, e.g., Gudovich Gudovich hh Terteryan Terteryan [78] [78] and and Sobolevskii Sobolevskii [151, [151, 152]). 152]). For For results of general character, character, we we mention mention our our work work [10, [10, 16, 16, 25, 25,26, 26, 27, 27,28] 28] carcarried out in the course of of Banach Banach space space settings settings and and covering covering aalarge largevariety varietyofof applications. 1 Also, problems with a splitting splitting operator operator have have been been considered considered [15, 34, 35], as well as problems problems with with generalized generalized input input data data [20, [20, 34, 34, 35] 35] and and problems with a variable variable stepsize stepsize [44, [44, 30, 30, 36]. 36]. Such Such results results allow allow one one to to exexamine, in particular, wide wide classes classes of of parabolic parabolic semidiscrete semidiscrete and and fully fully discrete discrete problems with variable coefficients, coefficients, constructed constructed on on the the basis basis of of discretization discretization in time by single step or or multistep multistep methods methods and and of of finite finite difference difference or or finite finite element discretization in in space. space. This This research research has has become become possible possible due due toto certain resolvent estimates estimates that that have have been been established established for for elliptic elliptic differential, differential, finite difference, and finite finite element element operators operators (see (see [156, [156, 14, 14, 29, 29, 38, 38, 43, 43,39]). 39]).

The present book actually actually gives gives an an unfolded unfolded and and considerably considerably extended extended exposition of our results [13], [28], results collected collected in in the the papers papers [10] [10] [13], [15] [15] [28], and [30] [36], of which which some some are are quite quite old old and and some some more more or or less less recent. recent. Most of the above cited cited works works in in that that direction direction are are either either announcements announcements of results or have been been published published in in editions editions almost almost unattainable unattainable for for the the readers outside of Russia. Russia. Although in this book we we deal deal only only with with linear linear problems, problems, our our achieveachievements are significant for for studying studying numerical numerical methods methods for for nonlinear nonlinear parabolic parabolic equations, and we intend intend to to demonstrate demonstrate in in our our future future work work how how the the asserassertions in this volume can can be be applied applied to to the the analysis analysis of of nonlinear nonlinear problems problems as well. Recent results results devoted devoted to to examining examining nonlinear nonlinear discrete discrete parabolic parabolic problems in Hilbert and and Banach Banach space space settings settings can can be be found, found, inin particparticular, in Akrivis & Crouzeix Crouzeix [4], [4], Akrivis, Akrivis, Crouzeix, Crouzeix, and and Makridakis Makridakis [5], [5], Akrivis, Karakashian, and and Karakatsani Karakatsani [6], [6], Akrivis Akrivis & & Makridakis Makridakis [7], [7], Beyn Beyn & Garay [47], Gonzalez Gonzalez et et al. al. [72], [72], Ostermann Ostermann k.k. Thalhammer Thalhammer [125], and *Not so long ago Gonzalez Gonzalez & & Palencia Palencia [73, [73, 74], 74], using using different different techniques, techniques, showed showed certain stability assertions assertions that that are are similar similar to to some some of of those those established established within withinour ourearlier earlier work.

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Ostermann, Thalhammer, Thalhammer,and andKirlinger Kirlinger[126]. [126]. Ostermann, considerations start start bybyexamining examininggeneral generalquestions questions stability Our considerations of of stability within the the framework frameworkofofBanach Banachspaces spaces (see Part This investigation within (see Part I). I). This investigation is, inis, in the core coreof ofthe thebook. book.The Themain mainworking working concept in Part is that of evofact, the concept in Part I isIthat of evolution equation equation inindiscrete discretetime. time.On Onthe theone one hand, modern (even implicit lution hand, anyany modern (even implicit and/or multistep) multistep)discretization discretizationmethod method which is applied a non-stationary and/or which is applied to atonon-stationary problem necessarily necessarilyreduces reducestotothe theCauchy Cauchyproblem problem evolution equaproblem forfor an an evolution equain discrete discrete time time (under (underreasonable reasonablerestrictions). restrictions).OnOn other hand, tion in thethe other hand, rewriting the the starting startingdiscrete discreteproblem problemin in form evolution equation rewriting thethe form of of an an evolution equation in discrete discrete time time allows allowsone onetotoobserve observeitsits most important features a more most important features in ainmore transparent way. way. As Asititwill willbebeseen seenbelow, below, discrete theory presented transparent thethe discrete theory presented in in looks very veryoften oftenrather rathersimilar similartoto modern analysis of abstract Part II looks thethe modern analysis of abstract dif- differential equations equationsininBanach Banachspace space(see, (see, e.g., Amann or Lunardi [108]). ferential e.g., Amann [9] [9] or Lunardi [108]). the same same time time itit possesses possessesitsitsown owndistinctive distinctive features which At the features which havehave no no analogues in in continuous continuoustheory. theory.Apart Apartfrom fromeverything everything else, hope analogues else, we we hope thatthat our investigation investigation makes makesa avaluable valuablecontribution contribution theory of discrete to to thethe theory of discrete semigroups, whichisisatatthe themoment momentonly only a development stage semigroups, which at ata development stage (see,(see, e.g.,e.g., Lyubich [110], Nagy Nagy & & Zemanek Zemanek [114], [114], and Nevanlinna Nevanlinna [116, [116,117, 117,118]). 118]). Lyubich [110], main problem problem totobebeconsidered consideredis isposed posedin in Chapter 1, where The main Chapter 1, where we we introduce the the necessary necessaryworking workingconcepts concepts subsequent in Chapters introduce forfor ourour subsequent useuse in Chapters 2, 3, and and 4. 4. Furthermore, Chapter Chapter 22isisdevoted devotedtotothetheanalysis analysis stability of the Furthermore, of of stability of the Cauchy problem problemfor foran anevolution evolutionequation equation discrete time. These results Cauchy in in discrete time. These results are are most crucial crucial for for the thewhole wholesubsequent subsequentexposition. exposition.WeWe distinguish between most distinguish between autonomous and and non-autonomous non-autonomouscases casesforforwhich which essentially the autonomous we we essentially use use different techniques. techniques. InInthe theformer formercase casethethe main tools of analysis different main tools of analysis are are the the 2 operator calculus calculus methods methods while the the study study ofofnon-autonomous non-autonomousequations equations operator while based on on the the use useofofcertain certaindiscrete discreteversions versions Gronwall's lemma. is based of of Gronwall's lemma. TheThe corresponding stability stabilityestimates estimatesare are then valid finite time intervals corresponding then valid forfor finite time intervals only.only. The considerations considerations ininChapter Chapter3 3are areof ofparticular particular interest concern interest andand concern discrete problemswith witha asplitting splittingoperator; operator; ideas employed in the discrete problems thethe ideas employed in the pre-preceding chapter are are further further developed developedforforexamining examining stability of such ceding chapter thethe stability of such problems. significantthat thatour ouranalysis analysis based assumption problems. ItIt isissignificant is is notnot based on on thethe assumption that the the separate separatecomponents componentsofofthe thesplitting splitting necessarily commute; avoiding necessarily commute; avoiding this assumption assumptionisisessential essentialforformany manyimportant important applications. In this chapter applications. In this chapter we also also consider consider aamore moreinvolved involvedsplitting splittingconstruction construction which appears, which appears, for for 2 such methods methodsusually usuallydeals dealswith withintegration integration over suitable contours applicationof such The application over suitable contours complex plane. plane. Throughout Throughout the book book the orientation orientation of any any contour contour is taken taken in the complex what is accepted accepted in Appendix AppendixA.2. according according to what

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example, when handling second second order order discretizations. discretizations. In Chapter 4 we study from from the the viewpoint viewpoint of of stability stability more more general general equations, that is, discrete discrete evolution evolution equations equations with with aa memory memory term. term. The The techniques used here allow allow us us to to show show stability stability on on finite finite intervals intervals only. only. The general results of Part Part II can can further further be be applied applied to to the the study study ofofdisdiscretizations in time of abstract abstract linear linear parabolic parabolic problems problems inin Banach Banach space. space. This program of action is is realized realized in in Parts Parts IIII and and III. III. The The investigation investigation inin these parts is based on reducing reducing the the starting starting discrete discrete problem problem to tothe the Cauchy Cauchy problem for an evolution equation equation in in discrete discrete time. time. We Wetherefore therefore discuss discusshow how to perform such a reduction reduction and and how how to to verify verify the the meaning meaning ofof not not clear clear rerestrictions on the operator operator in in question question ifif we we proceed proceed from from aa discretization discretization in time of the original differential differential problem. problem. In In Part Part IIII we we concentrate concentrate on on the study of single step methods, methods, and and as as such such we we consider, consider, in in the the concrete, concrete, the Runge-Kutta methods. methods. However However the the below below techniques techniques can can be be extended extended in order to examine stability stability for for other other families families of of modern modern single single step step methmethods (see, e.g., Butcher [53], [53], Hairer, Hairer, N0rsett, N0rsett, and and Wanner Wanner [80], [80], Hairer Hairer && Wanner [81], or Stetter [155]). [155]). Next, Next, in in Part Part III, III, in in order order to to reveal reveal further further possibilities of our approach, approach, we we discuss discuss some some other other ways ways ofof discretization, discretization, more precisely the ^-method, ^-method, some some operator operator splitting splitting methods, methods, and and the the linlinear multistep methods. In In this this investigation investigation we we stay stay along along the the same samelines linesofof analysis as above; however however our our exposition exposition in in Part Part III III isis not not as as detailed detailed as as that carried out in Part II. II. In Chapter 5 we consider consider Runge-Kutta Runge-Kutta discretization discretization for for the the abstract abstract parabolic Cauchy problem. problem. We We introduce introduce some some working working concepts concepts and and hyhypotheses intended for subsequent subsequent use, use, among among others, others, certain certain conditions conditions ofof Holder-continuity; meanwhile meanwhile some some auxiliary auxiliary assertions assertions are are established established here here as well. Also, we give a suitable suitable classification classification of of the the Runge-Kutta Runge-Kutta methods. methods. However the most important important point point isis that that we we show, show, under under the the sectorialness sectorialness hypothesis and under one one of of the the Holder-continuity Holder-continuity type type conditions, conditions, that that the starting discrete problem problem can can be be equivalently equivalently represented represented as asthe the Cauchy Cauchy problem for an evolution equation equation in in discrete discrete time time and and is, is,therefore, therefore, uniquely uniquely solvable. Next, in Chapter 6, we derive derive stability stability estimates estimates of ofthe the discrete discrete solutions solutions for different classes of Runge-Kutta Runge-Kutta methods. methods. These These results results are are achieved achieved by by showing certain resolvent estimates estimates and and Holder-continuity Holder-continuity type type results results and and by subsequently using the the general general stability stability results results established established inin Part Part I.I. As concerns convergence estimates, estimates, these these are are presented presented in in Chapter Chapter 77 under various restrictions restrictions on on the the Runge-Kutta Runge-Kutta method method in in question, question, distindistinguishing between the autonomous autonomous and and non-autonomous non-autonomous cases. cases. Also, Also, we wedo do not pass over the phenomenon phenomenon of of order order reduction reduction which which may may be be observed observed for for

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approximations of equations equations with with an an unbounded unbounded operator operator and and which which shows shows itself by the fact that the the classical classical order order of of accuracy accuracy isis not not really really achieved. achieved. Completing Part II in Chapter Chapter 8, 8, we we study study variable variable stepsize stepsize Runge-Kutta Runge-Kutta discretizations and give stability stability estimates estimates for for such such discrete discrete problems problems under under some (rather general) restrictions restrictions on on the the families families of of time time partitions partitions used. used. In the next three Chapters Chapters 99 — —1111,, constituting constituting Part Part III, III, we we focus focus on on showing stability only and and carry carry out out the the investigation investigation under under further further restricrestrictions and simplifications. This This narrowing narrowing of of scope scope allows allows us us to to concentrate concentrate on the most essential aspects aspects of of the the considered considered methods. methods. Our analysis in Chapter 99 concerns concerns the the (9-method, (9-method, and and we we show show that that certain refined stability estimates estimates are are possible possible ifif the the operator operator inin question question isis under stronger restrictions restrictions than than just just the the sectorialness sectorialness hypothesis. hypothesis. In Chapter 10, under the the assumption assumption that that the the operator operator in in question question can can be decomposed into the sum sum of of two two generally generally non-commutative non-commutative operators, operators, we study certain methods methods with with aa splitting splitting operator. operator. In Chapter 11 we deal with with some some classes classes of of problems problems constructed constructed by by means of linear multistep multistep discretization. discretization. Finally, in Part IV which which consists consists of of only only one one chapter, chapter, we we discuss discuss the the stability of certain approximations approximations to to linear linear integro-differential integro-differential equations equationsinin Banach space by applying applying the the results results of of Chapter Chapter 4. 4. Our results below will mainly mainly be be expressed expressed in in terms terms ofof stability stability and and convergence estimates in the the norm norm of of an an abstract abstract Banach Banach space. space. Throughout Throughout the whole exposition there there occur occur constants constants in in assumed, assumed, intermediate, intermediate, and and final estimates, whose sizes sizes are are unessential unessential for for our our analysis. analysis. They They will will be be denoted by C and c when subject subject to to C C> > 00 and and cc >> 0,0, respectively. respectively. Both Boththe the C's and c's may implicitly implicitly depend depend on on other other parameters parameters which which are are fixed fixed inin traced. At At the the same same time time we wetake takecare care the context; 3 this dependence is never traced. of giving an explicit mention mention whenever whenever there there may may be be aa danger danger ofof confusion. confusion. As already mentioned above, above, our our considerations considerations will willbe becarried carried out outwithin within the framework of Banach Banach spaces. spaces. Throughout Throughout the the book, book, given given an an abstract abstract Banach space X, \\ || stands for for both both the the vector vector and and operator operator norm norm inin X, X, denote the the space space of of all all linear linear bounded bounded operators operators on on while B{X) is used to denote denotes the the dual dual space space for for X, X, \\\\ ||* ||* the the corresponding corresponding X. Furthermore, X* denotes norm in X*, and , ) the duality pairing pairing between between X* X*and and X. X. By By II we wedenote denote the identity operator both both in in XX and and in in X*. X*. In In what what follows follows we we also also deal deal 3

A parameter £ is thought of of as as fixed fixed without without explicit explicit mention mention if,if, for forexample, example,ititappears appears with the words 'there exists exists aa number number ££ ...' ...' or or 'for 'for some some££ ...'. ...'. The The same sameapplies appliestotosubsets subsets of the complex plane. On On the the contrary, contrary, ifif ££ appears appears with with the the words words 'for 'for all all ££...' ...' oror 'for 'for any £...', it is by no means means fixed. fixed. In In doubtful doubtful cases cases itit isis explicitly explicitly specified specified ifif££isisfixed fixedoror not.

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with some concrete function function Banach Banach spaces; spaces; the the corresponding corresponding notation notation for for their norms will be introduced introduced immediately immediately as as required. required. Furthermore, Furthermore, given given a linear (generally unbounded) unbounded) operator operator £, £, we we denote denote by by Dom£ Dom£ and and RRaann££ its domain and range, respectively, respectively, while while £* £* denotes denotes the the adjoint adjoint for for ££ (if (if defined). Next, if | | is a seminorm, seminorm, the the symbol symbol Dom Dom || || isis used used to to denote denote its its domain which is always always assumed assumed to to be be aa linear linear manifold. manifold. For For future future needs needs we also define some useful useful classes classes of of unbounded unbounded operators. operators. In In order order to to do do this, we first denote, given <
Definition 0.1. Let £ :: Dom£ >>XX be Dom£ C C XX — — be aa closed closed linear linear operator. operator. We We write £ € S(ip,a), with some some <
for all X e - ^ - ^

(1) (1)

resolvent estimate estimate The operator £ itself is then then called called ((p, cr)-sectorial and the resolvent (1) the sectorial estimate. estimate. It is easily seen that the the fact fact that that ££ 66 S(tp, S(tp, a) a) implies implies k£ k£ ££ S((p, S((p,ka) ka) for for any k > 0. In our subsequent considerations considerations we we deal, deal, in in fact, fact, with with (ip, (ip,cr)-sectorial cr)-sectorial equations with with such such operators operators operators for some 0 <

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Nirenberg [3]) for elliptic elliptic partial partial differential differential operators operators and and their their discrete discrete analogues. We suppose that the reader reader is is familiar familiar with with the the fundamental fundamental concepts concepts and results of the theory theory of of functions functions of of aa complex complex variable variable and and possesses possesses a basic knowledge of functional functional analysis. analysis. Nevertheless, Nevertheless, in in order order to to raise raise the level of self-containedness self-containedness of of the the book, book, we we collect collect some some important important facts, facts, needed for our analysis, in in an an appendix. appendix. In conclusion, the author would would like like to to thank thank the the many many colleagues colleagues from from abroad who kindly sent him him reprints reprints of of their their works works that that are are unavailable unavailable inin Moscow.

Contents Preface

I

EVOLUTION EQUATIONS EQUATIONS IN DISCRETE TIME

1

1 Preliminaries 1.1 Preliminaries 1.2 Comments and bibliographical bibliographical remarks remarks

33 33 19 19

2 Main Results on Stability Stability 2.1 Stability of discrete discrete semigroups semigroups 2.2 Stability of discrete discrete semigroups: semigroups: some some special special points points 2.3 Stability of discrete discrete evolution evolution operators operators 2.4 Stability of discrete discrete Cauchy Cauchy problems problems 2.5 Comments and bibliographical bibliographical remarks remarks

21 21 21 21 25 25 33 33 48 48 51 51

3 Operator Splitting Splitting Problems Problems 3.1 Estimates for discrete discrete semigroups semigroups 3.2 Stability of operator operator splitting splitting problems problems 3.3 A symmetric operator operator splitting splitting method method 3.4 Comments and bibliographical bibliographical remarks remarks

53 53 55 55 63 63 66 66 69 69

4 Equations with Memory Memory 4.1 The general case 4.2 A particular case 4.3 Comments and bibliographical bibliographical remarks remarks

71 71 74 74 76 76 79 79

xm

xiv II

CONTENTS CONTENTS RUNGE-KUTTA RUNGE-KUTTA METHODS METHODS

81 81

5 Discretization by Runge-Kutta Runge-Kutta Methods Methods 5.1 Introductory light 5.2 Discretization 5.3 Connection with discrete discrete evolution evolution equations equations 5.4 Comments and bibliographical bibliographical remarks remarks

83 83 83 83 94 94 102 102 110 110

6 Analysis of Stability Stability 6.1 Some resolvent estimates estimates 6.2 Holder-continuity type type results results 6.3 Stability estimates 6.4 Comments and bibliographical bibliographical remarks remarks

111 111 IIllll 125 125 131 131 145 145

7 Convergence Estimates Estimates 7.1 Preliminaries 7.2 Autonomous equations equations 7.3 Non-autonomous equations equations 7.4 Working against order order reduction reduction 7.5 Comments and bibliographical bibliographical remarks remarks

147 147 147 147 152 152 164 164 173 173 177 177

8 Variable Stepsize Approximations Approximations 8.1 The commutative case case 8.2 The non-commutative non-commutative case case 8.3 Error estimates 8.4 Comments and bibliographical bibliographical remarks remarks

179 179 181 181 190 190 199 199 204 204

III

207 207

OTHER DISCRETIZATION DISCRETIZATION METHODS METHODS

9 The (9-Method 9.1 Stability results 9.2 Comments and bibliographical bibliographical remarks remarks

211 211 212 212 217 217

10 Methods with Splitting Splitting Operator Operator

219 219

11 Linear Multistep Methods Methods 11.1 Linear multistep discretization discretization 11.2 Auxiliary results 11.3 Stability 11.4 Comments and bibliographical bibliographical remarks remarks

227 227 227 227 233 233 238 238 242 242

CONTENTS CONTENTS

xv

xv

IV INTEGRO-DIFFERENTIAL INTEGRO-DIFFERENTIALEQUATIONS EQUATIONS UNDER UNDER DISCRETIZATION DISCRETIZATION 243 243 Integro-DifferentialEquations Equations 12 Integro-Differential 12.1 Stabilityof discretizations discretizations 12.1 Stability 12.2 Quadraturerules rules 12.2 Quadrature

247 247 247247 252252

Functions of Linear Linear Operators Operators A Functions A.I Analytic Analyticcontinuation continuationof the the resolvent resolvent A.2 Functions Functionsof bounded bounded and and sectorial sectorial operators operators A.3 Fractional Fractionalpowers powersof operators operators A.4 Functions Functionsof ordered ordered non-commutative non-commutativeoperators operators

255 255 255 255 258 258 260 260 262 262

B Linear Linear Cauchy CauchyProblems Problemsin Banach Banach Space Space

265 265

Bibliography Bibliography

269269

Index Index

284 284

This Page is Intentionally Left Blank

Part II Part

EVOLUTIONEQUATIONS EQUATIONS EVOLUTION IN DISCRETE DISCRETETIME TIME IN

This Page is Intentionally Left Blank

Chapter 1

Preliminaries 1.1

Preliminaries

Let there be given a complex Banach space X and an interval [0,T], T > 0. In addition, let £1^ = {tn : tn = nk for n = 0 , . . . , N} be a uniform grid in the interval [0,T], where k > 0 and N e N are are connected by the relation Nk = T. We allow for the possibility that T = oo; in this case nk = {tn : tn — nk for n = 0,1,...} and k > 0 is set arbitrarily. The last case is formally arrived at if we put T — oo,N oo,N = = oo. oo. We We also use the notation &k — {tn

tn — nk for n — 0 , . . . , N — 1}.

In this part of the book all working objects (operators, (operators, seminorms, seminorms, etc.), etc.), except the space X itself, generally depend on k;1 this dependence will not mentioned explicitly. At the same time, given k, the dependence on tn, if generally available, will always always be shown explicitly by adding n as a subscript functions of discrete to the corresponding object. 2 In particular, (X)-valued functions time tn are used. Given such a function Zn : 0,^ —+ X, we denote dZn = k~1(Zn+\ — Zn), £n € fife. (This notation will will be employed below for scalar discrete functions as well.) The main object of consideration in the present part of the book will be the discrete Cauchy problem problem 8Yn + %nYn = QnFn

(ortnenk,

Y 0 = VKy°,

(1.1)

for some given %l n,£ln,9{ G B(X). The vector y° G X and the discrete function Fn : &k —> X are considered as the input data of problem (1.1). 1

In principle, we could let the space X depend on k as well, but this is not needed for our subsequent aims. 2 This means that every object object that that depends depends on tn will have n as a subscript, separated by a semicolon from any additional subscripts.

4

CHAPTER CHAPTER 1. 1. PRELIMINARIES PRELIMINARIES

The discrete function YYn : £)& —> 36 is to be found. found. YYn is therefore thought of as a solution of (1.1). (1.1). In the case T < oo, we we employ employ the the condition condition 21 N — 21JV-I- This is not an important restriction, restriction, since since only only the the values values of of 2t 2tn, n = 0 , . . . , N — 1, are essential, but it will be convenient convenient in in our our subsequent subsequent statements. statements. Clearly, problem (1.1) can can be be equivalently equivalently rewritten rewritten as as follows follows

which is an explicit single single step step equation equation with with the the transition transition operator operator

il n = / - f c 0 n .

(1.3) (1.3)

It is therefore easily seen seen that that problem problem (1.1) (1.1) isis uniquely uniquely solvable solvable and and its its solution Y n is given by 3 n-l 0

Yn = Un-lflXy + kYsUn-hj+l&jFj,

(1.4)

3=0

where, for 0 < j , n < N, lj

if if

jn.

In spite of the fact that that the the unique unique solvability solvability of of problem problem (1.1) (1.1) isis evident, evident, itit is important to establish establish stability stability estimates estimates for for the the solution solution of of (1.1), (1.1), which which is the main objective in in this this part part of of the the book. book. As far as possible applications applications are are concerned, concerned, note note that that all all modern modern disdiscretization methods (including (including implicit implicit and/or and/or multistep multistep ones) ones) can, can, under under certain reasonable restrictions, restrictions, be be equivalently equivalently written written in in the the form form (1.1) (1.1) oror (1.2). In particular, for Runge-Kutta Runge-Kutta methods methods this this fact fact isis shown shown inin Section Section 5.3 while for linear multistep multistep methods methods this this observation observation isis pointed pointed out out inin Section 11.1. Now we give a series of of working working definitions. definitions. Definition 1.1. A difference difference equation equation of of the theform form (1.1) (1.1) isis called calledan anevolution evolution equation in discrete time. time. The The operator operator iinj iinj given given by by (1.5) (1.5) isis called called the the discrete evolution operator operator for for problem problem (1.1) (1.1) or, or, equivalently, equivalently, for for problem problem (1.2). In the particular case when when 2t 2tn = 21 and On = il do not not depend depend on on ttn, semigroup generated generated by by the the operator operator21. 21. iinfl = il n is called the discrete semigroup Throughout the book we we use use the the convention convention 52?=i 52?=i

== 00 ifif nn <
1.1.

PRELIMINARIES

Figure 1.1: A set of A3-conflguration A3-conflguration When dealing with subsets subsets of of the the complex complex plane plane C, C, given given VV CC C, C, we wededenote by Int V, Cl V, and and dV, dV, its its interior, interior, closure, closure, and and boundary, boundary, respectively. respectively. Also, we use the notation T>(z; r) — {A : |A — z\ < r} and T>(r) = T>(0; r) for z G C and r > 0. Definition 1.2. A set TT C CC C is is called called aa set set of of Aj-configuration Aj-configuration with with some some integer J>lifTis

the the union union of of aa disk disk V(l\ V(l\ a) a) and and of of JJ sets sets ofof the the form form

a3 = {A : | arg(A - Q - a r g ( l - 0 ) 1 < P} D 0 ( 0 ; d),j = l,...,

J,

s o m e a € ( 0 , 1 ) a n d /3 G (0, a r c s i n a ) , where d = cos/3 — ( a 2 — sin 2 /3) 1 / / 2 and ^i = 0, ^2, , O are different points lying on the circle circle dV(l; dV(l; 1). 1). The The constants a and (5 are called called the the parameters parameters of of the the set set TT and and the the points points Q, Q, j — 1,. .., J, the singular singular points points of of T. T. For instance, when J — — 1, 1, we we deal deal with with aa set set of of Ai-configuration Ai-configuration which which has only one singular point: point: £1 £1 == 0. 0. Such Such sets sets will will be be of of particular particular interest interest in our subsequent considerations. considerations. Figure Figure 1.1 1.1 gives gives an an example example ofof aa set set ofof A3-configuration. Definition 1.3. An operator operator ££ n G B(X) is said to satisfy satisfy the the Aj-condition Aj-condition with some integer J > 11 ifif there there exist exist numbers numbers KK >> 0, 0, 88 >> 0,0, and and aa set set TT of A j-configuration such such that that (i) for all k G (0, K], the the operator operator k£ k£n has its spectrum contained contained in in the the set set IntT f c U{0} ; where fc / 5 / c ) u T , (1.6) (1.6)

6

CHAPTER 1. PRELIMINARIES CHAPTER 1. PRELIMINARIES

and and satisfiesthe theresolvent resolvent estimate, that is, for G (0,K], Gtflk, flk, (ii) ££n satisfies estimate, that is, for all kallGk(0,K], tn G (ii) and \glntrk, and \glntr jA — O ll~~ ,,

(1-7) (1-7)

3=1 3=1

where Q, jj — —1 , ......,, J, are arethe thesingular singular points points of of T. T. //// (1.7) (1.7)holds holds for for all all where Q, the operator operator £ n is is said said to to satisfy satisfy the the Aj-condition. Aj-condition. Any Any set set TT X fi I nnttTT,,4 the of A j-configuration, j-configuration, which whichmay may be beemployed employed in in the the above above context, context, is is called called Aj-proper fAj-proper,) fAj-proper,)for for the the operator operator £„. £„. IfIf TT isisAkj-proper j-proper (Aj-proper) (Aj-proper) Aj-proper the singular singular points points of of T T are arealso also called calledthe the singular singular points points of of the the for £ n ; the operator operator £ n . Re m maarrkk 1.1. 1.1. ItIt follows follows by by duality duality that that if an an operator operator £ n satisfies satisfies the theA Ajj-condition (the AAjj-condition) -condition) in in X, X, the thesame same is is true true for for its its adjoint adjoint £^ £^ in in condition (the X*. X*. The above aboveconcepts conceptsare areof of great importance for our subsequent considThe great importance for our subsequent considin2t(1.1) (1.1) erations. One Oneofofthe themain mainrestrictions restrictions below on the operator erations. below on the operator 2tn in will be be that that 2t2t satisfies the theAjAj-ororAj-condition Aj-condition with some integer will with some integer J > J1, > 1, n satisfies and the the situation situationwhen whenJ J—— \ will of particular interest. Atsame the same and \ will be be of particular interest. At the time some some seminorms, seminorms,whose whose properties related to above the above Aj- and time properties are are related to the Aj- and Ajconditions,are areused. used. Aj- conditions, Definition 1.4. Let Letthere there be begiven given an an ££n G B(X) B(X) and andfixed fixed numbers numbersJ G G N, N, Definition 1.4. m G GN, N, ££>> 0,0,with with m > > max{l,£}, max{l,£}, and and let let |||||| ||||n and and |||||| |*|*; nn be be seminorms seminorms andX*, X*,respectively. respectively. The Thepairs pairs (||| (||| ||| n ;£n) ;£n) and and (||| (||| ||*; ||*;n ;£n) ;£n) are arecalled called on X and and Aj(*;^|m)-concordant, Aj(*;^|m)-concordant, respectively, respectively, if there there exist exist Aj(£|m)-concordant if Aj(£|m)-concordant and numbers K > > 00 and and55>>0 0and anda a ofj A -configuration j suchthat thatthe the numbers K setset T T of A -configuration such fe spectrum of the the operator operator k£, k£,n is contained contained in in the the set setInt IntTT {0} whenever whenever spectrum of U {0} (0,K], where whereTk is is defined defined by by (1-6), (1-6), and and the therespective respective inequalities inequalitieshold, hold, k G (0,K], fc for all all kk G G(0,K], (0,K], tn G GU Uk, X X ££ IntT IntTfc , uG GX, X, and andxxGGX*, X*, j

|(AJ k£n)-mu\in <
(1.8) (1.8)

3=1 3=1

and and

llxll*, llxll*,

(1-9) (1-9)

3 = 11

where Q, where Q, jj — —1 , ......,, J, are are the thesingular singular points points of of T. T.55 If If in in addition addition the the 4

This means meansthat thatone onecan cantake take 5 = = 0. 0. This By z" z" we wedenote denote the the conjugate conjugate comlex comlexnumber numberfor for z. z.

5

1.1. PRELIMINARIES

77

estimates (1.8) and (1.9) (1.9) hold hold for for all all A A ££ IntT, IntT, then then the the pairs pairs (§ (§ |||| n ;£n) and (I ||* ;n ;£n) are called Aj(£|m)-concordant Aj(£|m)-concordant and and Aj(*;£|m)-concordant, Aj(*;£|m)-concordant, respectively. Any set T of of Kj-configuration, Kj-configuration, which which may may be be employed employed inin the the corresponding above context, is called Aj(.. Aj(.. .)-proper .)-proper (Aj(.. .)-properj for the corresponding pair. If the size of m is is not not important important in in the the context, context, mm does does not not appear appear inin the above notation? Apart Apart from from everything everything else, else, ifif the the resolvent resolvent estimates estimates (1.8) and (1.9) hold for all all ttn,t\ G Ofc with ||| |||; and ||| |* ;; in place of ||| |||n and I |* ;n , respectively, then the pairs pairs (||| (||| || n ; £ n ) and (||| |* ;n ; £ n ) are called strictly Aj(.. .)-concordant .)-concordant (Strictly (Strictly Aj(.. Aj(.. .)-concordant,). .)-concordant,). We also need to give some some generalizations generalizations of of the the above above notions, notions, which which are based on the use of certain certain weighted weighted resolvent resolvent estimates. estimates. numDefinition 1.5. Let there there be be given given operators operators ££n,%3n G B(X) and fixed numbers J £ N, m G N, £ >> 0, 0, with with m m> > max{l,£}. In addition, let ||| || n and I |*;n be two seminorms defined defined on on XX and and X*, X*, respectively. respectively. The The triplets triplets k (III ' Iln;£n;23n) and (1 |*;n;£n;2Jn) |*;n;£n;2Jn) are are called called A j(£\m)- and A^(*;£; \m)concordant, respectively, respectively, ifif there there exist exist numbers numbers KK >> 00 and and 55 >> 00 and and aaset set TT of A j-configuration such such that that the the spectrum spectrum of of the the operator operator k£ k£n is contained in the set lntTk U {0} whenever k £ (0, (0, K], K], where where TTfc is defined by (1.6), and the respective inequalities inequalities hold, for all k G (0,-JsT], t n G fifc, A ^ IntT f e ; u G X, andx& X*, J

k£ n)-mxnu\u < ck-t J2I A - 0 l ^ m \H

(i-io)

\i(\*i - fc£;rmQ?;x«l*;n < cfc-« J2IA - 0 l c " m llxll*.

(i-ii)

I( and

where Q, j = 1 , . . . , J, are are the the singular singular points points of of T. T. IfIf (1.10) (1.10) and and (1.11) (1.11) hold for all A £ I n t T , tfie tfie tripiets tripiets (|(| !„;£„; !„;£„; 9J 9Jn) and (||| !*;„;£„; 2?n) are called Aj(£\m)- and Aj(*\^\m)-concordant, Aj(*\^\m)-concordant, respectively. respectively. Any Any set set TT ofof AjAjconfiguration, which may may be be employed employed in in the the above abovecontext, context, isis called calledAj(.. Aj(.. .).)proper fAj(.. .)-properj for for the the corresponding corresponding triplet. triplet. IfIf the the size size ofof mm isis not not important in the context, context, m m does does not not appear appear in in the the above above notation. notation. Furthermore, we give a collection collection of of some some assertions assertions involving involving the the aboveaboveintroduced concepts and and intended intended for for subsequent subsequent reference. reference. 6

For example, we then write write simply simply Aj(*;£) Aj(*;£) instead instead of of Aj(*;£,\m). Aj(*;£,\m).

8

CHAPTER CHAPTER 1. 1.

PRELIMINARIES PRELIMINARIES

Lemma 1.1. If an operator operator ££ n G B(X) satisfies the Aj-condition, Aj-condition, itit then then follows that ||fc£ n|| < C for all k G (0, K] K] and and ttn € fifc. Proof. This fact can easily be shown shown by by using using the the operator operator calculus calculus formula formula (A.2) with /(A) = A and and with with k£ k£n substituted for £, where where the the contour contour H is taken independent independent of of kk and and bounded bounded away away from from the the disk disk T>(1; 1). We therefore have

f \(\I = J_ 2m J E

and a trivial estimation estimation shows shows the the claim. claim. (0,K] Lemma 1.2. Suppose that that ££ n € B(X) while ||fc£ n|| < C for all k G (0,K] andtn G fife. Lei a/so seminorms seminorms ||-| ||-| n and |-|||*;n /OTTTI respectively Alj(iy\m)and Aj(*;7*|m)-concordant Aj(*;7*|m)-concordant pairs pairs w£/i w£/i £„ £„ /or /or some some J,J, m m GG NN and and 7,7* 7,7*GG [0, m]. Then, with any fixed fixed dd G G [0,7] [0,7] andd* G [0,7*], the seminorms k"9 | - | n and fc''*! |*. n m/Z /orm respectively Aj(7 Aj(7 — i9|m)- and Aj(*;7* — i?*|m)concordant pairs with ££ n . Furthermore, for all kk G G (0, (0, if] if] and and ttn G fifc, F sup ffluln + A:7* sup |xi*;n < C.

(1.12) (1.12)

Proof. We will show the claim claim in in the the case case of of the the seminorm seminorm k^\l k^\l |||| n only, noting that the needed result result for for fc^ fc^1 |* ;n can be established in a similar similar manner. Accordingly, we we will will prove prove only only that that the the first first term term on on the the left-hand left-hand side of (1.12) can be estimated estimated as as stated. stated. Let R > 0 be denned by by R=2

sup

ke(o,K],tnenk

||A;£n||. ||A;£n||.

Using a standard argument, argument, itit follows follows that that for for |A| |A| >> R, R, since since lAI^Ufc^H lAI^Ufc^H << 1/2, 1 1 11 11 11 (1.13) . Furthermore, let T be aa Aj(7|m)-proper Aj(7|m)-proper set set for for the the pair pair (||| (||| |||| n ;£n), whose singular points are Cii >O> s o * n a t with some 5 > 0 and with T fe given by (1.6), for all A £ Int TTk and ueX, j

|A - or m IMI-

(i.i4)

1.1. PRELIMINARIES

99

The last inequality yields, yields, under under the the additional additional restriction restriction |A| |A| << R, R, j

|A - Q^-m\\u\\.

(1.15)

On the other hand, assuming assuming without without loss loss of of generality generality that that K5 K5 << 1,1, by by fc (1.13) and (1.14) we have, have, for for kk G G (0,K\, A £ IntT , |A| > R, and « e l , since - 1 ^ IntT fc and ||fc£ n|| < C, that

j

|A - Cjl 7~*~m||«ll-

(1-16)

Now the Aj(7 — i9|m)-concordance of the the pair pair (fc (fciJ|fl || n ; £ n ) follows by combining (1.15) and (1.16). (1.16). Using the above argument argument we we also also find, find, for for all all kk €€ (0,K] (0,K] and and uu GG3t, 3t, that

< C\\{I + k£ n)mu\\

< C\\u\\,

which shows the desired desired estimate estimate for for the the first first term term on on the the left-hand left-hand side sideofof (1.12).

L e m m a 1.3. Suppose that that £„ £„ ee B(X) B(X) and and \\k£n\\ < C for all k G (0,K] (0,K] seminorms |||||| |||| n and ||| |* ;n , defined respectively on and tn & flk- Let also seminorms X and X*, satisfy the conditions, conditions, with with some some 7,7* 7,7* €€ [0,1], ip € (0, TT/2), and 8>0, for all k G {0,K}, tn E Clk, A ^ Int (£,/, UV(5)), uu G G X, X,

and

Then, for any fixed d G [0,7] [0,7] and and ** G G [0,7*], /or aH k G (0, -ff], i n

and

10

CHAPTER 1. PRELIMINARIES CHAPTER 1. PRELIMINARIES

The proofis is similar similar to to that that of of Lemma Lemma1.2. 1.2. The proof Lemma 1.4. Let Letan anoperator operator£„ £„ satisfy satisfythe theA^-condition A^-conditionwith with some >1. 1. Lemma 1.4. some J> T/ien i/iere ezisi ezisia a ip ipG G(0,7r/2) (0,7r/2) and and aa/ii /ii>>0 0suc/i suc/i i/iai i/iai/or /or aZ ZZZk G G(0, (0,if], if], T/ien i/iere Gf2fc, M M> > Mi, Mi,a™d a™dAA Int(E^,U tn G g gInt(E^,

|A| ||(XI -- Z^W Z^W ++ !!(£„ +J) (A/ (A/-- ££n )- 1 || < < C. C. |A| ||(XI !!(£„ +M M 7/£ satisfiesthe theA j -condition, -condition,then then the thelast last estimate estimateholds holds with [i\ = =0. 0. 7/£ n satisfies with [i\ Proof. isobvious obviousthat thatthe thefirst first term termon onthe theleft-hand left-handside sideof of the theinequality inequality Proof. It is can be estimated estimatedas as needed, needed,with withsome some G (0, TT/2) and and[i\ [i\>>0.0.Using Usingthis this ip ip G (0, TT/2) can be the identity identity fact together togetherwith withthe fact x + fil) fil)(XI (XI- -££ +(A (A+ /x) /x)(XI (XI- - Sin)Sin)-1 (£ n + n ) - = -- II +

leads to the thedesired desired estimate estimatefor for the thesecond second term termas aswell. well. leads to thecase case where where£ n satisfies satisfiesthe theAj-condition, Aj-condition,it it is isnot nothard hard to to see seethat that In the for ji\ ji\==0.0. the sameargument argumentalso also works the same works for Lemma 1.5. Let Letan anoperator operator£ n satisfy satisfythe thekkj-condition j-conditionwith withsome some integer Lemma 1.5. integer J> > 1. 1. Then Then there thereexists existsa pb\ pb\ > > 00 such such that thatwith withany anyfixed fixed/J,Q /J,Q G GR, R,/or /orall all fj, > > Mi, Mi,& G (0, (0,iiff]], ,t n G Gfifc; fifc; u u ££ X, X, and andx xGG3£*, 3£*,

||«||<
(1.17) (1.17)

and and Proof. Since £ £n satisfies satisfies the the Aj-condition, Aj-condition,by by Lemma Lemma 1.4 1.4 itit follows follows that that Proof. Since /xi >>0, 0, with some some/xi with 1


for forall/x> all/x> MI, MI,

(1.18) (1.18)

thefirst firstterm termon onthe theleft-hand left-hand side sideof of (1.17) (1.17) can canbe beestimated estimatedas as stated stated thus the thus viewof in view of the theidentity identity

1

(Sin ) (Sin

It is is also also easily easilyseen seenthat that(1.18) (1.18) leads leads to to the thedesired desiredestimate estimatefor for the thesecond second on the theleft-hand left-hand side sideof of (1.17). (1.17). term term on now nowfollows follows in in view view of of Remark Remark 1.1. 1.1. The The latter latterstated statedestimate estimate

1.1. PRELIMINARIES

11 11

L e m m a 1.6. Let, for some some JJ G GN N and and £,£* £,£* G G [0,1], seminorms § || n and Aj(£|l)- and and Aj(*;£*|l)-concordant Aj(*;£*|l)-concordant pairs pairs with with an an I |* ;n form respectively Aj(£|l)exist aa ip ip G G (0,TT/2) and a S > 0 such that operator £ n G B(X). Then there exist for all k G (0,K), tn G Qk, A g Int(S^ UP(<5)), uu G G ll ,, and and xx ee X*, X*,

and The proof is immediate. immediate. and let let aa set set TT be be Lemma 1.7. Lei an operator operator £, £,n satisfy the h\-condition and h.\-proper for £ n . Then there exists a 6 >> 00 such such that that with with any any fixed fixed /xo /xo GGM, M, for all k G (0, K], t n G fifc, and A £ Int (T U V(5k)), V(5k)),

Proof. By the given conditions, conditions, the the operator operator ££ n satisfies (1.7) for k G (0, (0, K] K] and A ^ Int T, |A| > 6k, 6k, with with JJ == 11 and and some some 66 >> 0.0. With With this this fact fact inin mind mind and using the identity + /i 0 /) (A/ - fc^)-1 = - 7 + (A + /iOfc) (AI - fc-C)- 1, as well as taking into account account that that |A| |A| >> 6k 6k we we get get the the stated stated result. result.

Lemma 1.8. Suppose that that seminorms seminorms |||||| ||||||n and |]j |* ;n form respectively Akj(£\m)- and k kj{*;£\m)-concordant pairs pairs with with an an operator operator ££ n G B(JE) for some J G N, m G N, and and ^^ G G [0, [0,m]. m]. T/ien T/ien7 wii/i any fixed £i G [^,?n], [^,?n], the pairs (||| | n ; £ n ) and (||| l* ;n ;£n) wi^ be Aj(^i|m)- and miconcordant, respectively. respectively. Proof It suffices to note note that that one one can can take take the the 66 in in Definition Definition 1.4 1.4 to to be be strictly positive. Observe that if an operator operator ££ n fulfils the Aj-condition, Aj-condition, itit follows follows from from Lemma 1.4 that with some some ip ip G G (0,TT/2) and with any fixed IJ,O > 0 sufficiently large, for all k G G (0, (0, K], K], ttn G ftfc, and A

Therefore, if ^o > 0 is is sufficiently sufficiently large, large, the the operator operator ££ n + [IQI is sectorial, it has a bounded bounded inverse, inverse, and and its its fractional fractional powers powers (£ (£ n + —oo < £ < oo, are defined defined and and bounded bounded on on ££ (see (see Appendix Appendix A.3). A.3).

12

CHAPTER CHAPTER 1. 1.

PRELIMINARIES PRELIMINARIES

Lemma 1.9. Let an operator operator ££ n satisfy the Aj-condition Aj-condition and and let let seminorms seminorms and A^(*;7*|l)-concordant A^(*;7*|l)-concordant pairs pairs I || n and I |* ;n form respectively Aj(7|l)- and with £ n , for some J G N and 7,7* 7,7* G G [0,1). [0,1). Then Then there there exists exists aa/xi /xi >> 00 such such that with any fixed 71 >> 77 and and 7*1 7*1 >> 7*, 7*, for for all all kk GG (0, (0,K], K], ttn G fifc; u G X, and x G 3-*; (1.19) «|| (1.19)

Proof. We will show only (1.19), (1.19), noting noting that that the the second second estimate estimate can can be be proved in a similar way. way. Also, Also, by by the the convexity convexity inequality inequality (A. (A.18) 18) itit follows follows that ||(£ n + H\I)~^\\ < C for any fixed [i\ > > 00 sufficiently sufficiently large large and and for for any any fixed £ > 0. Therefore Therefore without without loss loss of of generality generality itit can can be be assumed assumed that that 7i < 1Let i\) G (0, TT/2) and 6 > 0 be the same as as in in the the assertion assertion of of Lemma Lemma 1.6 1.6 stated for £ = 7, and let let [i\ [i\ > > 00 be be aa fixed fixed number number such such that that |A |A — pL\\ > 5 for A <£j Int £,/,. Let furthermore furthermore the the contour contour FF be be given given by by FF == dT,^. dT,^. By By the operator calculus formula formula (A.15), (A.15), we we get, get, for for all all uu GG X, X,

Since the pair (||| ||| n ;£ n ) is A^(7)-concordant and |A |A — /J,Ij > c(|A| + 1) for A G F, taking next the seminorm seminorm |||||| |||| n of both sides and using Lemma Lemma 1.6 1.6 yields + ^iiyilu\L
/ |A|- 71 (|A| + 1) 7 ~1 |dA|
which shows (1.19).

DD

Lemma 1.10. Let £ n G B{X), let seminorms |[|[|| |||| n and ||| ||*;n form respecpairs with with ££ n , andlet9Jl n G #(3C) tively Aj(7|l)- and A j(*; 7*| 1) -concordant pairs 6e a "small" perturbation perturbation to to ££ n in t/ie sense that for all all kk G G (0, (0, K], K], ttn G fife, X G X*, and ii G £, £,

/or some J G N, 7,7* G G [0,1), [0,1), and and G [0,1] such that 7 + 7* < 11 ++ Then /or aH aH A A;;G G (0, (0, K], K], ttn G fife, there exist a tp G (0, TT/2) and a 5 > 0 suc/i i/iai /or A ^ Int (S^ U T>(5)), uGX, uGX, and and xx €€ X*, X*,

KAI-CCn + Srtn))-1?*!,, ^ C I A ^ I H I ,

(1.21) (1.21)

1.1. PRELIMINARIES

13 13

ffl(AJ - (£*n + m*n)rlX\Un

< ClAr-^Hxll*,

(1-22) (1-22)

and

IKAI-OCn + J O T O r ^ C CllA Arr 1 .

(1.23)

Proof. We will show (1.21) only, noting noting that that the other estimates (1.22) and (1.23) can be proved in a similar manner. Let 7' and 7* be fixed numbers such that that 7' G (7,1], 7I G ( T * , 1 ] , and 27' + 7* — 7 < 1 + d. By (1.20) and Lemma 1.9, there exists a Mi > 0 such that for all t n G fife, u G X, and x G X*, ||

(1.24) (1.24)

and

|||xllU;n
(1-25) (1-25)

Further, by the convexity inequality (A. (A.18) 18) and Lemma 1.4, we have, with some i\) G (0, TT/2) and <50 > 0, for all A £ Int (S^ (S^ U U V(60)), - £ n ) " 1 | | < C||(£ n + MI/) (A/ - £ n ) - 1 | | 7 '

||(£ n + inl)*(XI

'W1-1'
(1.26)

and likewise, with i9' = min{'i9,7 / + 7^,}, taking into account account (A.14), (A.14), Lemma Lemma 1.1, and the fact that 7' + 7^ < 1 + 7

'(AJ - £n)~ 1 (£n + Mi/) 71 II ^ £

n

+ M I / / | | ||(£n + M i / ) 7 ' + 7 l ^ ' ( A / - fin)"1!!

^"*'-1.

(1.27) (1.27)

Let now <5 be a fixed number such that that 6 > 5Q. Multiplying then both sides sides of the identity ( £ n + Tin))'1 (A/ - (£

= (XI - Zn)- 1 + (XI - £ n )- 1 9Jt n (AI - (£ (£n + 2tt n))-\ (1.28) from the left by ( £ n + Ml/) 7 a n d using furthermore a simple estimation by (1.20), (1.24), (1.25), (1.26), and (1.27), we get, for all A £ Int (S,f, (S,f, UX>(<5)), 7 1 With W = || (£ ( £ n + Ml/) ' (XI - (£ (£n + ^n))- !!, w

'- 1 + C|A| y+7*-1?'-1u; < CIA!7'"1 +

Note that in this estimate no constant C depends on 5. Taking therefore 5 > 0 sufficiently large we obtain '-1.

(1.29)

14

CHAPTER PRELIMINARIES CHAPTER 1. 1.PRELIMINARIES

Applying then the above above argument argument as well as (1.20), (1.20), (1.24), (1.24), (1.25), (1.25),(1-27), (1-27), Applying then (1.29), it follows follows that that with with the same same #' as as above, above, for all all A A^^ Int Int(E^, (E^,UU and (1.29), l V(S)) and and uu ee X, X, since since 27' + + 7^ 7^-- 77 << 11++d'd'and and|A| |A|>>8~8~ , sup 1 (XI -- £ n )- 1 9Jt n (A/ - ( ££n + IMI=i IMI=i

miy'(\I - (£ (£n ||(£n + miy'(\I With this result result in mind, mind, the claim claim now now follows follows from from (1.28) (1.28) with with both both With this right by u, u, using using a simple simple estimation estimation and and taking taking multiplied from from the right sides multiplied account Lemma Lemma 1.6 and and the thefact that that the pair (||| || n ;£n) ;£n) is Aj(7|l)Aj(7|l)into account concordant. concordant. D D future needs needs we examine examine here here also also the behaviour behaviour of fractional fractional For our future operator in question question is under under perturbation. perturbation. powers when the operator powers when Le m m maa 1.11. 1.11. Suppose Suppose that that seminorms seminorms ||| ||||n and and |||||| |* |*;ri form respectively respectively ;ri form Aj(7|l)andA\{*\^*\l)-concordant A\{*\^*\l)-concordant pairs pairswith with£ n G B(X), B(X), ££n itself itself is under under Aj(7|l)- and )-condition, and and DJln € B(X) B(X) satisfies satisfies condition condition (1.20), (1.20), with withsome some the A A1)-condition, N, i9 i9 G G [0,1], [0,1], and and 7,7* 7,7* G G [0,1) [0,1)such such that that 7 + 7* 7* << 11++ d. d. Then, Then, J € N, any fixed fixed n\ > > 00 sufficiently sufficiently large largeand any any fixed fixed £ G [0,1 [0,1++ ## — 7*), 7*), with any £, € [0,1 [0,1++ 00 -- 7), 7),for forall allkkGG(0, (0,K), K),tn t G Q Qk, u G GX, X, and andxx GG 36*, 36*, || (£n + + Win + Hil)*u\\ Hil)*u\\ < C C II (£„ + + W J)*u|| J)*u||

(1.30) (1.30)

and Proof, demonstrate how how to to prove prove (1.30) (1.30) since since the second second estimate estimatecan Proof, We demonstrate derived in aa similar similar manner. manner. Also, Also,note notethat thatthe case £ = 00 is is trivial, trivial, so be derived it can be be assumed assumed that that £ > 0. 0. Let if) if) and SS be be chosen chosen as in in Lemma Lemma 1.10 1.107 and let let 7' 7' be beaa fixed fixed number number G (7,1) (7,1)and and7'7'++7* 7*<<11++$.$.Note Note that that in the the below below proof proofwe such that that 7' G can use the the estimates estimates (1.22) (1.22) and and (1.23), (1.23), by Lemma Lemma 1.10. 1.10. Also, Also, in the the case 7, by by Lemma Lemma 1.9 and andanalogy analogy with with (1.27) (1.27)we have, have, since since 7' —£ < 1, 1, with £ < 7, fixed //1 > > 00 sufficiently sufficiently large, large, for all all A A^^ Int Int(E^, (E^,UUT>(8)) T>(8))and uu G GX, X, any fixed

< C||(£n+ C||(£n+ M1-0 M1-07 '(A/-£nrM '(A/-£nrM 7 + UlrfuW UlrfuW < C||(£ C||(£n + /Xl/) '^(A/ - fin)"1!! ||(£n + 1 Kf M M (1-31) (1-31) 7 assume without without loss lossof generality generality that that tp and and 55 are areare arechosen chosen as in in Lemmas Lemmas We can assume 1.3 and and 1.4 1.4which which will will be applied applied together together with withLemma Lemma1.10 in in the thepresent present proof. proof.

1.1. PRELIMINARIES 1.1. PRELIMINARIES

15

At thesame sametime, time, if££G G(7,1 (7,1+ + — —7*), 7*),since since £nn satisfies satisfiesthe theAj-condition, Aj-condition, At the if £ usingagain again Lemma 1.9yields, yields,for forall all A^ ^ Int Int(E^, (E^,UUV(5)) V(5)) anduue e 1.9 A and X,X, using Lemma Zn^uln < < \I(\I- -Zn^uln \I(\I ^^

zii/^uH. ++zii/^uH.

(1-32)(1-32)

byLemma Lemma1.1 1.1we wehave have Next note that Next note that by \\k£n\\
(1.33)(1.33)

inturn turngives gives that which that which in SUp ||M|nn + +V* V* SS U ||x||*;n<
c, * <<--^^ << c,

which, combined (1.33), obviously which, combined withwith (1.33), obviously implies implies

With thisestimate estimate inmind mindand andusing usingLemma Lemma 1.3and and (1.22),aasimple simplecalcalin 1.3 (1.22), With this culation gives, d'==min{7*, min{7*, , ,for forall all A^ ^ Int Int(£,/, (£,/,UX>(5)) UX>(5)) x£X*, culation gives, withwith d' A andand x£X*, 11 fc^'iKA/- -(-c;(-c; +im;))im;))xil*;n<
(1.34) (1.34)

Fromnow now onwe we assume assume that n\>>00 chosensufficiently sufficiently sothat that From on that n\ isis chosen largelarge so — /ii /ii| |> >55for for A^ ^ Int IntE^. E^.We We then apply theidentity identity A— IA A then apply the 11 (£nn + +m mnn + +M MII/ /) ) ")11" -- ((A A/ -/ (£n (£n+ + Mi-Or Mi-Or {xi -- (£ {xi /))-11OT OTnn(A7 (A7-- (£ = (A/ (A/-- (£ (£nn+ +m mnn + +/xx11/))(£nn+ +M MII/ /) ) ")11", = 88 and the theestimates estimates(1.20), (1.20), (1.31), and(1.34) (1.34) toobtain, obtain,for forall all A£ £ and (1.31), (1.32),(1.32), and to A IntE,/, IntE,/,and anduu€ € X,X, since sincekk < >c(|A| c(|A|+ + 1), 1), 11 11 I -- (£ (£nn + +m mnn + +Mi/))" Mi/))" -- (AI (AI-- (£ (£nn+ + Mi/))" Mi/))" ) «|| «||

1 ? 22

Cfc^'|A|«(|A| l ) > l' )' >| '||((££ Cfc^'|A|«(|A| s

ifif ^^G [ oo, ,7 ] ,,

(7,1+ +^^-7*)-7*)/ /HH l lif if £G £G (7,1 (1.35) (1.35)

weuse use the the last last three threeestimates estimates with with A— — /n /n A in inplace placeof ofA, A,for forA A^ ^ Int Int£^. £^. In fact, fact,we In

15

16

CHAPTER CHAPTER 1. 1. PRELIMINARIES PRELIMINARIES

Next we denote R = 2sup fce(0tK ],t n en k (\\k(£>n+ViI)\\ + \\k9Jln\\). By Lemma 1.4, (1-23), and a standard argument, it follows that for all A ^ Int (S^, (~l | | ( A / - ( £ n + 2Jl n + M!/))- 1 1|+ | | ( A / - ( £ n + /i 1 /))- 1 1|
If £ < 1, using with d(Y.^r\V(R/k)). d(Y.^r\V(R/k)). Let now F ^ be the contour coinciding with the operator calculus formula formula (A.13) (A.13) with with m — 1, recalling that £n,9Jln G B(X) and taking (1.36) into account, account, we get, after a suitable deformation of the path of integration in (A.13), for all u G 3C,

(£n + Tln + frrfu - (£ n - (£„ (1-37) Furthermore, with the aid of the identity

= A((AJ - ( £ n + mn + Mx/))" 1 - (XI -

we obtain, instead of (1.37), - (£n

J A«((AJ - (£ n + mn + Mi/))- 1 - (XI - (£ n + Ml /))

= ~ r

(*0

(1.38)

A slightly more delicate delicate calculation, calculation, based based on applying (A. 13) with m = 2 and using Cauchy's Theorem, Theorem, shows shows that that (1.38) (1.38) remains remains valid valid for £ £ (1,1 + # — 7*) as well. Also, in the case £ — 1, (1.38) follows directly from from (A.2) with /(A) = A and E = F F{k). Since i + 7» < 1 + $', £ + + 7* 7* < < 11 ++ i?, i?, and jfc^'lAI* < C|A| max ^- 1?+ ' 9 '' 0 > for A G r (fe) , the norm of the integral on the right-hand side of (1.38) can be estimated with the aid of (1.35) by

C||(£ n + Mi-OHl w h i c h

shows

(1-30).

R e m a r k 1.2. The reader will have no difficulty in showing that in the particular case where ||| || n = | | ( £ n + M i - 0 7 II and 1 ||*;n = \\(£n + Mi-^) 7 * " II; with some [i\ > 0 sufficiently large, the assertions of Lemmas 1.10 and 1.11 are still valid if one requires that 7,7* belong to [0,1] instead of [0,1).

1.1. PRELIMINARIES

17

In the remainder of the the section section we we want want to to show show that that under under reasonable reasonable conditions on the operator operator in in question question there there do do exist exist seminorms seminorms with with the the properties stated in Definitions Definitions 1.4 1.4 and and 1.5. 1.5. L e m m a 1.12. Let £ n satisfy the Aj-condition Aj-condition (Aj-condition) (Aj-condition) with with some some integer J > 1 and let Q, Q, jj = = 11,,......,, J, J, be be the the singular singular points points ofof ££ n . Then, (with fio fio == 00 in in the the case case of of the the AjAjwith any fixed JIQ > 0 sufficiently large (with condition), with any fixed m £ N, N, rj rj 66 N NU U {0}, {0}, and and ££ ££ [0,min{m,?7}] [0,min{m,?7}]; with J

(1.39) and with the seminorms |||||| |||| n and § |* ;n given by

H\n = and

the pairs (||| | n ; £ n ) and concordant (Aj(£\m)- and and

^ ) will be A^(£|m)- and and )-concordant), respectively. respectively.

Proof. For the sake of brevity brevity we we denote denote 5J 5J J;n = / — £ 1k£n, j = 2,. .., . . , J, so that we have QJn = TL-oOJ 7?,,. Our argument is based based on on using using certain certain smoothing smoothing properties properties ofof the the operators 2X,;n, j = 2 , . . . , J, and 5J n . Note first that the operators operators QJj;n are uniformly bounded ||QJj;n||
j = 2 , . . . , J,

(1.40) (1.40)

because the operator k£ k£n is uniformly bounded bounded by by Lemma Lemma 1.1. 1.1. Let Let furfurthermore T be a Aj-proper Aj-proper (Aj-proper) (Aj-proper) set set for for £„. £„. With With the the aid aid ofof the the identity

(A/ - feSn)- 1*^ = ^ - ^ (A7 - k2 nyl + g- 1 /, and of the given condition condition on on ££ n , it follows that with some some 5i 5i > > 00 sufficiently sufficiently small and some 6 > 00 (8 (8 == 00 in in the the case case of of the the A ./-condition), for all j = 2 , . . . , J and A £ Int T, <5i > |A |A -- Cj| Cj| >> 5fc, 5fc, ||(A/-fc£ n )- 1 QJ j;Tl ||


18

CHAPTER CHAPTER 1. 1. PRELIMINARIES PRELIMINARIES

Using this result, (1.40) (1.40) and the fact that £ n satisfies the Aj-condition (Ajcondition) yields, for all A £ Int ( U^ =2 V(Q;8k) U T ) , A G U^=2V(Cj; <5I), J m

||9Jn(AJ - k£n)- \\

< CJ2 \ X ~ Cj| m i n { O l I 7 - m } .

(1.41)

3=2

Also, by (1.40) it follows that for all A £ Int ( u / = 1 V(Q; Si) U T ) , j

m

-

(1-42)

3=2

Recalling that ||fc£ n|| < C, by the convexity inequality (A. (A.18) 18) we have ^|
(1.43) (1.43)

which, combined with (1.41) (1.41) and (1.42), yields, since 0 < £ < rj, for A

Int {U^2V(CJ;5k) UT), 6^1X1, l*l*~m-

(1.44)

i=2

Furthermore, without loss loss of generality it can be assumed that T/> in the statement of Lemma 1.4 coincides with the parameter j3 of the set T (see Definition 1.2). Therefore, by (A.18), (1.40), and Lemma 1.4 with \i — HQ we get, for all A ^ IntT, 5\ > |A| > 6k, ||(£ n + fxolf (XI - fc£ n)-m5Jn|| <

Ck-*\\\t- m.

Together with (1-44), this this inequality inequality shows shows that that the seminorm ||| || n forms a Aj(^|m)-concordant (Aj(^|m)-concordant) (Aj(^|m)-concordant) pair pair with with £ n . Similarly, with the aid of Remark 1.1 it can be shown that the seminorm (A,/(*;£|m)-concordant) pair pair with with I |* ; n forms a Aj(*;£|m)-concordant (A,/(*;£|m)-concordant) £n- ' L e m m a 1.13. Let £ n satisfy the Aj-condition (Kj-condition) (Kj-condition) for some integer J > 1. Let £j, j = 1 , . . . , J, be the singular points of £ n and let 5J n be given by (1.39) for some ij 6 N U {0}. Then, with any fixed /Uo > 0 sufficiently large (with /io = 0 in the case that £ n satisfies the A j -condition), with any fixed m G N and £ G [0, min{m, 77}], and with the seminorms ||| |||n and I |* ; n given by

\\u\ln — || (£ n + fiorfuW

forueX

1.2. COMMENTS AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL REMARKS REMARKS

19 19

and the triplets (||| || n ;£n; 2Jn) and (||| ||*; n ;£n; 53n) a r e respectively Aj(£|m)anrf Aj(*;£|m)-concordant Aj(*;£|m)-concordant (Aj(£\m)(Aj(£\m)- and and Aj(*;£\m)-concordant). Aj(*;£\m)-concordant). The proof actually reduces reduces to to that that of of Lemma Lemma 1.12 1.12 because because the the operators operators -1 (A/ — fe£n) and QJn are permutable.

1.2

Comments and bibliographical bibliographical remarks remarks

Discrete semigroups are used used in in Kato Kato [91] [91] and and Pazy Pazy [131] [131] as as aa discrete discrete anaanalogue of semigroups of linear linear bounded bounded operators operators in in operator operator theory. theory. Evolution Evolution equations in discrete time time actually actually appear appear in in [10, [10, 11, 11, 12], 12],but but the the correspondcorresponding terminology is introduced introduced later, later, in in [34]. [34]. The The A^-condition A^-condition (in (in particular, particular, the Aj-condition) is actually actually introduced introduced already already in in [10, [10, 11, 11, 12]. 12]. Note Note that that the A j-condition may be regarded regarded as as aa discrete discrete analogue analogue of of the the sectorialness sectorialness property in the theory of of holomorphic holomorphic semigroups. semigroups. The The Aj(.. Aj(.. .)-concordant .)-concordant pairs are introduced in this this book book and and are are further further developed developed herein herein for for pracpractical purposes.

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Chapter 2

Main Results on Stability Stability This chapter deals with with the stability of problem (1.1). In fact, the solution Yn of this problem is estimated in seminorms forming concordant concordant pairs pairs with with the operator 2l n in (1.1), in the sense of Definitions 1.4 or 1.5. We also show Throughout we dual estimates which are often used in numerical analysis. Throughout employ the notation Un and ilnj introduced by (1.3) and (1.5) without explicit reference to the corresponding formulas. formulas.

2.1

Stability of discrete semigroups

that arise arise Our first step is examining the stability of discrete semigroups that from problem (1.1) considered with frozen frozen operator. operator. In this section we therefore concentrate on the problem of estimating the powers of linear bounded operators. As pointed out above, our analysis will be carried out by using special seminorms. seminorms. The results of the present section allow us to get stability estimates for problem (1.1) in the autonomous case, that that is, where the operator 2l n does not depend on tn (see Section 2.4). At the same time it turns out that these estimates are also important for the analysis of the non-autonomous case. case. Theorem 2.1. Let a seminorm ||| || n form a Aj(£)-concordant pair with 2ln for some J £ N and £ > 0. Then, ifT< oo ; we have

for all k £ (0, K], ti,tn £ fife, and u £ X. Furthermore, if the pair (||| || n ; 2t n ) also for T = oo ; that is, for all is Aj(£)-concordant, the last estimate holds also l,n — 0 , 1 , . ... 21

22

CHAPTER 2.2. MAIN MAINRESULTS RESULTSONONSTABILITY STABILITY CHAPTER

Proof. present the the proof proof only onlyfor forthe thecase casewhere wherethe thepair pair(||| (|||||n;2ln) || is Proof. We present is ./(^-concordant.1 Therefore, Therefore, with with this thisassumed assumedcondition, condition,there thereexists exists A ./(^-concordant. anan integer m m> > max{l, max{l,£} £}such suchthat thatthe thepair pair(|||(||||| n ; 2ln) is Aj(£|m)-concordant. Aj(£|m)-concordant. integer set TT be be Aj-proper Aj-proper for forthis thispair, pair,letleta aand and(3(3 parameters of T, Let a set bebe thethe parameters of T, let Q, Q, jj == 1,...,J, 1,...,J, be its its singular singular points points (see (seeDefinition Definition1.2). 1.2). Let Let and let further be expressed expressed via viaaaand and/3/3asasspecified specifiedin inDefinition Definition1.21.2 further dd be andand let let Tl T be the the contour contour coinciding coincidingwith withthe theboundary boundaryofof thesetset the U-j=1D((3-a(l

l)-l)UT, + l)-

11==

0,1,.... 0,1,....

By the the operator operator calculus calculus formula formula (A.2) (A.2)wewehave, have,forforallall I — 0,1,. .. and I — 0,1,. .. and

u<=3Z, that u<=3Z, (2m)-1 / ( I -- X)l{XI iilnu = (2m)rl

mn)-lud\. m

2 Thus integrating integrating by byparts, parts,we wefind find

(2.1)

seminorm and andthe thepair pair(|||(||| || n||; 2ln) is Aj(£\m)Aj(£\m)the functional functional |||||| ||||n is a seminorm Since the concordant, itit follows follows from from (2.1) (2.1)that thatforfor/ =/ =0 , 1 , ......,, concordant,

Jr' where we used used the the notation notation where we j

Clearly, TTl can be be decomposed decomposed asasfollows follows Clearly,

r' = u/ u/= 1 (rlhj u r'2J) u r 3 , x

ii == o, o,1,...,

(2.3) (2.3)

from what what follows followsthat thatour argument argument can easily easily be extended extended to the the will be seen from It will general case case corresponding correspondingto the the first part part of the the statement statement of Theorem Theorem 2.1 (see (see also general Remark 2.1). Remark 2 It is assumed assumed that that UT^i UT^i1 + g) g) == 11ifif m m ==1.1.

2.1. STABILITY

OF DISCRETE DISCRETE SEMIGROUPS SEMIGROUPS

23 23

where T[d

=

Tl2j =

{A 6 C : | arg(O -- A) A) -- arg(O arg(O -- 1)| 1)| << ftft |A |A-- 01 01 == d(l d(l ++ I ) " 1 } , {A G C : arg(A - 00 )) == arg(l arg(l -- 00))

ftft d/(l d/(l ++ 1) 1) << |A |A-- 00| | << d}, d},

and F3 is the part of V 1 whose points lie on the the circle circle dV{\\ dV{\\ a). a). Further, Further, for for all j = 1 , . . . , J, I = 0 , 1 , . . . , and A € F^-, we have have |1 |1 -- A A^^™ ™-- 1 < C and G(A) < C(l + l ) m ~ c , consequently,

f /

ff

|1 — A| + m ~ G(A) IdAI < C(l + l) m ~^ /

\d\\ < C(l + l ) m ~ ~^. (2.4)

Next, in view of the restriction restriction j3 j3 < < TT/2, we find that for all j — — 11,,......,, J, J, I = 0 , 1 , . . . , and A G F;2j-,

- 01). Thus it follows that X

C I J

exp(-(Z exp(-(Z ++ ll))ccxx))^^"" m dx (2.5)

Finally, it is easily seen seen that that G(A) G(A) << CC ifif A A€€ F3, F3, which which gives, gives, since since aa << 1,1, for a l l / = 0 , 1 , . . . ,
< C ( / + l)771"1^.

So the claim follows by combining combining (2.2), (2.2), (2.3), (2.3), (2.4), (2.4), (2.5), (2.5), and and (2.6). (2.6).

(2.6) HI HI

R e m a r k 2.1. The general general case case in in Theorem Theorem 2.1 2.1 where where the the pair pair (|||(||| ||||||n;2ln) is Aj(0~concordant can be be settled settled similarly similarly by by using using aa slightly slightly different different path path ofof integration. The path of of integration integration should should deviate deviate from from each each singular singular point point x of the set T, appearing appearing in in the the proof proof of of the the theorem, theorem, by by O((l O((l ++ l)~ l)~ + k), while the path follows the the boundary boundary of of TT as as long long as as the the deviation deviation exceeds exceeds this value. Note that the the assumption assumption TT << oo oo isis essential essential in in the the general general case case 1 order since we then have to use use the the fact fact that that (1 (1 ++ O(k)) O(k)) < C for t\ G £lk in order to bound suitably the quantity quantity |1 |1 — A|' ++mm ~ 1 for A £ T[ . We state as well a dual dual stability stability estimate. estimate.

24

CHAPTER CHAPTER 2. MAIN RESULTS ON STABILITY

T h e o r e m 2.2. Let a seminorm ||| |* ;n form a Aj(*;£)-concordant pair with with 2ln for some J G N and £ > 0. T/ien, if T < oo, for all k G G (0,K],

Furthermore, if the pair (|||-||*. n ;2l n ) is Aj(*; £)-concordant, holds as well for T = oo, that is, for all l,n — 0,1,....

the last

estimate

The proof is similar to that of Theorem 2.1. Considerations in subsequent parts of this book will use the application of the above Theorems 2.1 and 2.2. We make this statement now in view of its importance. T h e o r e m 2.3. 2 . 3 . Let the operator 2l n satisfy the Aj-condition and let QJn e B(X) be given by (1.39), with some J € N, r] G Nu{0} and with some HQ > 0 sufficiently large. Then, Then, ifT< ifT< oo, we have, with any fixed £ G [0,77], for all k G (0,K] andti,tn G Qk> that Ct^

K)l\\*
(2.7) (2.7)

(2.8)

If in addition the operator 2l n satisfies the Aj-condition, both estimates estimates are in force for T = oo, that is, for l,n — 0 , 1 , . . . , provided that [IQ = 0. In particular, if 2l n satisfies the k\-condition (A\-condition), (A\-condition), the above holds with VOn — I. Theorem 2.3 is a direct consequence of Lemma 1.12 combined with Theorems 2.1 and 2.2. In what follows we also use the following generalizations generalizations of the above results. T h e o r e m 2.4. Let there be given numbers J G N and £ > 0, an an operator 53n G B(X), and a seminorm ||| || n such that the triplet (||| || n ; 2ln;QJn) is Aj(£)-concordant. Then, Then, ifT < oo, we have, for all k G (0,K], ti,t n G flk, and u £ l , that

Furthermore, if the triplet (||| | | n ; 2 l n ; ^ n ) is Aj(£)-concordant, timate is in force for T — oo, that is, for all I,n — 0,1,....

the last es-

2.2. DISCRETE SEMIGROUPS: SEMIGROUPS: SOME SOME SPECIAL SPECIAL POINTS POINTS

25

T h e o r e m 2 . 5 . Let there be given numbers J e f f and £ > 0, an operator 2J n G B(X), and a seminorm ||| | * ; n such that the triplet (||| |*; n ;2ln;5?n) is Aj(*;O-concordant. Then, Then, ifT < oo, we have, for all k G (0,K], ti,t n G fJfc, and x G X*

Furthermore, if the triplet (||| ||* ; n;2ln;^n) is Aj(*; ^-concordant, 0,1,.... estimate is in force for T = oo, that is, for all I,n =

the last

The proofs of Theorems 2.4 and 2.5 are similar to that of Theorem 2.1.

Remark 2.2. Note that Theorem 2.3 can be stated with the order of V3n and illn reversed in (2.7) and with the one of QJ^ and (il£)' reversed in (2.8) because the operators iin and 2J n commute. Clearly, in this new form Theorem 2.3 may be thought of as a particular case of both Theorems 2.4 and 2.5 collected in one statement. Remark 2.3. The assertions of Theorems 2.1, 2.2, 2.4, and 2.5 2.5 remain valid for all tn, ti,tq G ttk, with ||| |||g and ||| |* ;g in place of ||| || n and ||| |* ;ri ;ri, respectively, if instead of the Aj(...)-concordance (Aj(...)-concordance) (Aj(...)-concordance) of the corresponding pairs/triplets pairs/triplets one requires their strict concordance. concordance.

2.2

Stability of discrete discrete semigroups: semigroups: some some special special points

In this section we present certain certain aspects aspects of the stability analysis of discrete semigroups, which are connected connected with with modifications modifications of our starting hypotheses. Concentrating on special questions of stability, we therefore introduce and make make our our certain assumptions that facilitate facilitate the following presentation and ourselves essentially essentially to analysis more transparent. transparent. In particular, we restrict ourselves the autonomous case, 3 and we consider only the case T = oo. Also, we deal (not of the Aj-condition). with analogues of the Aj-condition only (not We begin by posing a slight modification of the above Aj-condition. Definition 2.1. We say that an operator £ € B(3L) satisfies the Aj(M)condition with some M > 1 and J G N if it satisfies the Aj-condition in which the constant C (see (1-7)) is replaced by M. Aj-proper sets for the operator £ are then defined in the same way as in Definition 1.3. 3

We then write simply 21 and and il instead of 2ln and il n , respectively.

26

CHAPTER 2. MAIN RESULTS ON STABILITY

In fact, employing the Aj(M)-condition means that our results may be used for situations in which the size of the constant in the corresponding resolvent estimate varies considerably. Now we state an analogue of Theorem 2.3 with £ = 0.4 Theorem 2.6. Let the operator 21 satisfy the Aj(M)-condition with some J G N and M > 1. Then, for all k G (0, K) and 1 = 0,1,..., ||il'||
This estimate follows from (2.1) with m = 1 in a manner similar to that used in the proof of Theorem 2.1. For simplicity we present an analogue of Theorem 2.3 with £ > 0 in the case J = 1 only. Theorem 2.7. Let the operator 21 satisfy the Ai(M)-condition with some Then, with any fixed £ > 0, for all k G (0, K) and I = 0 , 1 , . . . , M>\.

Proof. Unfortunately, we cannot act on the complete analogy of the proof of Lemma 1.12 which further implies Theorem 2.3 since in that proof we have used the convexity inequality which does not control the size of the stability constant. We will therefore use instead a slightly different argument which is helpful in proving directly Theorem 2.3 as well. As in Section 1.1, it follows from Lemma 1.4 that the fractional powers 2ln, £ > 0, are denned and bounded on X. Further, let a set T be Ai-proper for 21 and let F be the contour coinciding with dT. Using the operator calculus formula (A.17) with /(A) = (1 — A)', with A;2t in place of £, and with F in place of H^,5 we have, for I = 0 , 1 , . . . ,

Now it is not hard to see that the claim follows because of the reasonings already used in the proof of Theorem 2.1. Note also that Theorems 2.1, 2.2, 2.4, and 2.5 can be similarly developed. We leave the details for the reader. In the remainder of the section we intend to get stability estimates based on a slightly different resolvent condition. 4

In this section we present no estimates involving the adjoint operators. Clearly, the mentioned deformation of the original contour is tolerable by Cauchy's Theorem. 5

2.2. DISCRETE SEMIGROUPS: SEMIGROUPS: SOME SOME SPECIAL SPECIAL POINTS POINTS

27 27

Definition 2.2. We say say that that an an operator operator ££ G GB(X) B(X) satisfies satisfies the the STj(M)STj(M)condition® with some M M >> 11 and and JJ G GN N ifif its its spectrum spectrum isis contained contained inin the the shifted unit disk X>(1; 1) 1) whenever whenever kk G G (0, (0,K] K] and and

\\{\I - S,)- 1]] < M ^ l A - C j T 1

with Ci = 0, <^2,

for alike

(0,K) and and A Agg IntX>(l; IntX>(l; 1), 1),

, CJ given fixed fixed points points lying lying on on the the circle circle dT>{\\ 1).

On the one hand, if £ satisfies satisfies the the Aj(M)-condition, Aj(M)-condition, then then the the operator operator fc£ clearly has satisfied the the ST/(M)-condition ST/(M)-condition with with the the same same constant constant M. M. On the other hand, it can can be be shown shown by by analytic analytic continuation continuation (see (see Theorems Theorems A.I and A.2) that the STj(M)-condition STj(M)-condition for for the the operator operator fc£ fc£ has has implied implied the Aj(Mi)-condition for for the the operator operator ££ with with some some M\ M\ >> M. M. Therefore, Therefore, ifif we want to use the above above techniques techniques in in order order to to get get stability stability estimates, estimates, we we have to trace the dependence dependence M\ M\ on on M. M.

Theorem 2.8. Let the operator operator fc2t fc2tsatisfy satisfy the the STj(M)-condition STj(M)-condition J G N and M > 1. Then, Then, for for all all kk G G {0,K] {0,K] and and 11== 0,1,..., 0,1,...,

with with some some

||H'|| < C M l o g ( l + M).

Proof. For the sake of simplicity, simplicity, we we show show the the result result in in the the particular particular case case J = I. 7 Assuming therefore this this condition, condition, let let (p (p== arcsinl/(2M). arcsinl/(2M). Assume first that (I + + I)" I)"1 < 2sinM = fl{M U ff,'M U f f , where = = =

{AGC: {AGC: { A G C : | A - 1 | = 1, 1, ||aarrggA A||<<77rr//22--<
Since, by the accepted conditions, conditions, ||(A/ - fcSl)- 1!! < M|A|- X 6

for all A with R\ == 0, 0,

The abbreviature ST comes comes of of two two words: words: "shifted" "shifted" and and "Tadmor" "Tadmor" ofof which which the the second is the name of the the mathematician mathematician who who introduced introduced aa resolvent resolvent condition condition related related toto the STj(M)-condition with with JJ == 11 (see (see Tadmor Tadmor [158]). [158]). In In fact, fact, both both conditions conditionsdiffer differ from from each other by a shift of the the spectrum spectrum of of the the operator operator in in question. question. Note Notethat that Ritt Ritt [136] [136]has has worked with a resolvent condition condition which which actually actually isis very very close closeto to Tadmor's Tadmor's condition. condition. 7 It is seen from the present present proof proof that that our our argument argument can can easily easily be be extended extended toto the the general case J > 1.

28

CHAPTER CHAPTER 2. MAIN MAIN RESULTS RESULTS ON ON STABILITY STABILITY

applying applying Theorem Theorem A.2 A.2 yields, yields, for for || arg(—A)| arg(—A)| < TT/2 TT/2 + ip,

which which shows shows in particular particular that that ||(AJ - fcSt)fcSt)-1!! < 2M(l 2M(l+ +1) 1) ||(AJ

for for A e T Tl{M.

(2.10) (2.10)

Also for for further further reference, reference,observe observethat that the theST\ ST\ (M)-condition (M)-condition directly directly imimAlso plies plies

Using Using now nowthe the operator operator calculus calculusformula formula(A.2), (A.2), we we have, have, for for II — 0,1,..., 0,1,..., (l-A)'(A/-^2l)(l-A)'(A/-^2l)- 1 dA

it' = = -^r -^r // 27H 27H Jfl.,M Jfl.,M

—(f - —

++ f

+ + f ))

Since |1 -- A|' Since A|' < C C for for A G G f i' M , with with the the aid aid of (2.10) (2.10) it easily easily follows follows that that ||/i|| < 2M(l ||/i|| 2M(l++l)l) [[l M ||ll--AA| |' '| d| dAA| | < CM, CM, Jf {

11 = 0,1,.... 0,1,....

(2.13) (2.13)

Next, using Next, using the the estimates, estimates, for for II== 0 , 11,,......,, 1-xexp(i(7r/2-( 1 - x e x p ( i ( 7 r / 2 - ( / 9))| 9))|/

< < (1 (1-- xx sin
iiffOO<<xx<<s isni (n^( ^ (2.14) (2.14)

and and exp (i{ir/2 (i{ir/2 - ip)) ip)) | < < 11 ifif sin sin ip < xx < < 2 sin sin tp, tp, 1 - xx exp a simple simple calculation calculationyields yields

x

// zz++ 11 X Xdx fl^'P fl^'P dx dx exp[-x——smip) exp[-x——smip) 1 / 1 / — — 22 V JJ xx Jsintp J 1/(1+1) 1/(1+1) V Jsintp XX < C lloogg((ll
,sinV

2.2. DISCRETE DISCRETE SEMIGROUPS: SOME POINTS SPECIAL POINTS 2.2. SEMIGROUPS: SOME SPECIAL 29 29 Applying inequality and(2.9), (2.9),we weget, get, forI I— — 0,1,..., 0,1,..., Applying thisthis inequality and for ^ ^

r2sintp r2sintp

\\h\\ < 4M 4M \\h\\ < / /

J

J

l \l-xex.p(i(ir/2-
< CMlog(l C M l o g+ + ( lM). M). <

(2.15)

(2.15)

=22and and|1 |1--A| A|= = 11for for AGGff ff, , with with Finally, since |d(2cosi?exp(i0))/di9| Finally, since |d(2cosi?exp(i0))/di9| = A the aid aidofof (2.11)we wefind, find,for forI = I= the (2.11) 00, ,11, .,. . ,. . , M M

CC

rl\ rl\

rir/2-tp rir/2-tp

\\h\\ << -z--z<2M , \\h\\ , U U<2M Jfucos(argA) cos(argA) JooJ 22 Jfu < CMlog(lH-Af). CMlog(lH-Af). <

( c o s t(fc) "o11s^^t f ) " (2.16)

1 1 Nowit itfollows followsfrom from (2.12), and (2.15), (2.16)that that if(I(I < Now (2.12), (2.13), (2.13), (2.15), and (2.16) if ++ I)'I)' < 2 sinif, if, 2 sin |||il'||< | i l ' | | > 2sin<^ 2sin<^gives, gives, for A argument in case (I for 1 = 0,1,..., 0,1,..., 1=
The claim follows bycombining combining thelast lasttwo twoestimates estimates for||il'||. ||il'||. The claim thusthus follows by the for Theorem 2.9.Let Let theoperator operator k%satisfy satisfythe theST\(M)-condition ST\(M)-condition with some Theorem 2.9. the k% with some M> >1.1.Then, Then,with with any fixed£ £>>0,0, for all11= = 0,1,..., 0,1,..., any fixed for all M

rfvv (2.17) (2.17) rf where® where®

=L€J, L€J, ifif CC= 11 22 K{M) =I I M M -L€J+ K{M) = -L€J+ ««

I

[ej L £ + <5, 5, (2.18) if if [ej < £ <
M M Proof. Asjibovejet a r c s i n land and / ( 2 let M let)TT bethe thecontour contourspecified specified Proof. Asjibovejet tpj= tpj= arcsinl/(2M) be

M byffM =Pf PU f Uffff U Ufff, f,where where = by

= = = = 88

{A {A GG 99EETT/ 22--vv :: siny; siny;< <|A| |A|< < (2 (2- {A {A GG E Eww//22__vv :: A A= = (2 (2 - -1/M) 1/M) cos(arg cos(arg A) A) exp(i exp(i arg argA)}. A)}.

In this In thiscase case the thepath pathof ofintegration integration contains contains no noline linesegments. segments. [£J [£J denotes denotes the thelargest largestinteger integer not notexceeding exceeding £. £.

9 9

(2.16)

30

CHAPTER CHAPTER 2. MAIN RESULTS ON STABILITY

Using now (A.17) with /(A) = (1 - A)' and with fM in place of S L , 1 0 we have, for Z = 0 , 1 , . . . ,

1

ff

Jf JfM

\JrM =

Jv™

J

Jvf

(2Ttik^- 1(l1 + I2 + h).

(2.19) (2.19)

estimates for I\, I 2, and I3. It thus remains to show suitable estimates A simple estimation which which is similar to the one that leads to (2.15) gives, with the aid of (2.14), for 1 = 0,1,..., ~ ||JiH

f sin v < 4M / x^" 1 l-xexp(i(ir/2-
,,

Jo

<

X x ? ~ 1 e x p ( - - (I+ 1) sin
CM Jo

<

dx

CM 1+{(! + l)- { .

(2.20) (2.20)

Likewise, with the aid of the estimate 1 — x exp (i{ir/2 —
and since x~ x (l — e'x) < x~^ for x > 0, we we obtain in the case 0 < £ < 1, for 1 = 0 , 1 , . . . , \\h\\

< 4Af Jsiny) y

1

e x p ( - ( / + 1) simp (sirup - x/2)) dx Jsintp

<

CM (sin
(2.21) (2.21)

Next, with (2.11), using using the analytic continuation of the resolvent of A;2l from the set dV(\; 1) \ {0} towards the origin within the sector S 7r//2_¥, and 10

By Cauchy's Theorem the mentioned deformation of the original path of integration is admissible.

2.2. DISCRETE SEMIGROUPS: SEMIGROUPS: SOME SOME SPECIAL SPECIAL POINTS POINTS

31 31

applying Theorem A.I, A.I, we we come come to to the the following following estimate, estimate, for for all all AA££ II33 , , > l-M(2cos(argA))

1

M

1

Af- cos(argA)

cos(argA)' cos(argA)'

At the same time we have, have, for for A AG G FF^^,, |1 — A|* = (1 — (2 — 1/M)/M cos2 argA)' /2 < Cexp ( - ^ - t i cos 2 arg A).

Therefore, observing that l/M)cosi9exp(M?))/di?| << 2, 2, applying applying the the that |d((2 |d((2 — l/M)cosi9exp(M?))/di?| last two estimates together together with with T

Lf°°

e-i

o

;;

11 11

;;

11 i i

and using the above argument, argument, we we obtain, obtain, for for II == 0 , 1 , . . . ,

ll/sll < M I

< CM ! i0

|A|«(cos(argA))" |A|«(cos(argA))"1|l-A|'|dA| (cos df- 1 expv( - - ^ 1 cos21?); ^ 2M l)^.

(2.22) (2.22)

Combining (2.19), (2.20), (2.20), (2.21), (2.21), and and (2.22) (2.22) now now yields, yields, ifif 00 << ff << 1,1, for Z = 0 , . . . , which implies the desired desired result result in in the the case case 00 << ££ << \.\. Also, it will be needed to give give aa slightly slightly different different estimate estimate for for ||/3||. More precisely, using the first first three three lines lines in in (2.22), (2.22), we we get, get, for for //++ 11<< M, M,

h\\ <

4M ' J (2.23)

32

CHAPTER CHAPTER 2. MAIN RESULTS ON

STABILITY

Now we derive one more estimate for H/^ + 1 ^ \ \ which will be an important ingredient in the proof of (2.17) for \ < £ < | . In fact, putting Xi()), A2(
H(X) = (i + iy\i - x)l+1xt(xi - my1, and integrating by parts, we find

= (-lf(A 2 l

H(X) - H(X)

+ /

(tx^H(A) - jff(A) jff(A) (xi (xi -- my my1) dx

J T>M

With the aid of (2.9) and (2.14), dealing with the case £ > 1, a direct estimation gives, since we then have x^~le~x < C for all x > 0, r«

if £ > i .

(( 22 2 5 )

-

Next, using the argument leading to (2.21) and (2.22), we get I ) - 1 if 0 < £ < l , 1)~^ if 1 < £ < 2,

( 2 2 66 ))

(2.27)

(2.26), and (2.27) that Clearly, in the case 0 < £ < 1 it follows from (2.25), (2.26), 2

^

ifM
Combining this estimate estimate with with (2.19), (2.19), (2.20), (2.20), (2.21), (2.21), (2.23), (2.23), and (2.24) proves (2.17) for \ < £ < 1. It is also seen that if 1 < £ < | , the claim follows by combining (2.19), (2.20), (2.25), (2.26), and (2.27).

2.3. STABILITY

OF DISCRETE DISCRETE EVOLUTION EVOLUTION OPERATORS OPERATORS

33 33

Further, to show the result result for for || << ££ << ||,, we we integrate integrate again again by by parts parts on on the right-hand side of (2.24) (2.24) and and use use (2.25) (2.25) with with the the above above argument. argument. Note Note that one can find in analogy analogy to to (2.23), (2.23), for for ££ >> 1,1, \\H3\\ 0 in aa finite finite number number of of steps. steps.

2.3

Stability of discrete discrete evolution evolution operators operators

In this section we are concerned concerned with with the the general general (non-autonomous) (non-autonomous) case. case. independent This fact means that we we do do not not suppose suppose that that 2l 2ln is necessarily independent of tn. Instead we work with with certain certain Holder-continuity Holder-continuity type type hypotheses hypotheses imimposing restrictions on the the behaviour behaviour of of the the difference difference 2l 2ln — 21;. We assume these hypotheses to hold hold for for all all kk G G (0,K\, tn,ti G Ctk, u G X, and x £ %*i with some i?,i?i G [0,1] and and some some given given seminorms seminorms |||||| |0. |0. n , ||| | / ; n , ||| ||*0;n, and I |||*/;n of which the first first two two are are defined defined on on X X and and the the other other two two on on X*. More precisely, we we will will employ employ the the following following Holder-continuity Holder-continuity type type hypotheses. H

V , t f i ; I ||m;n,m = 0,/,*0) reads

\(X, (On " Kt)u)\ < C and H6L(H(tf,tfi;||

|(X, ( O n - W l

m ; n ,m

= 0,*0,*7) reads

< ( f |

In the sequel a particular particular case case of of both both HOL HOL

($,$1; ||| |m;n>tTi — ®,I, *0)

(1?, -$i; ||| || m;n , m = 0, *0, */) will sometimes and HOL sometimes be be in in use, use, which which isis now stated as a separate separate hypothesis hypothesis as as follows: follows: HOL(tf; 1 || m;n ,m = 0, *0) reads

|(x, (On - W l < Ctf n_,|i«|0;Jixl*0;«. Throughout the present present section section we we assume assume that that TT << 00 00 since since our our considconsiderations will essentially be be based based on on aa discrete discrete version version of of Gronwall's Gronwall's lemma. lemma. It is stated here in the following following form. form.

34

CHAPTER CHAPTER 2. MAIN RESULTS ON

STABILITY

Lemma 2.1. Let z nj and w U}i be nonnegative solutions of the respective inequalities, for tn,ti 6 fife, ti < tn, n-l

Zn,l < Ct~i\+1 + CkJ2 *n q=l

and n-l q=l

with some Ci, C2 G [0,1). Then we have, for all t n,ti € flk, U

and In fact, this proposition proposition follows follows directly directly from from a more general lemma of Gronwall type that is stated in Chapter 8 for the case with a variable stepsize (see Lemma 8.4). On the other hand, Lemma 2.1 can easily be deduced from Lemma 4.1 below. 11 Our subsequent techniques techniques will will be based on using the following telescopic type identities which are trivially verified for all t n,ti € fijt with tj < t n, noting that il g - il ( = fe(2lj - 21,), n-l q=l+l n-l

and n-l 9=' n-l 1 i

xl

l

.

(2.29)

Strictly speaking, one can make this assertion about about the inequality for wn,i only. Note however that both discrete discrete inequalities inequalities are solved in a similar manner.

2.3. STABILITY

OF DISCRETE EVOLUTION OPERATORS

35

Sometimes a slight modification of (2.28) will be used, as follows, for all U < t n, n-l l

l

iln-l,Z+l = K~ ~ +

k

Yl Un-lfi+l&l-^Ur1'1-

( 2 - 3 °)

q=l+l

Denoting p = [ Z ^J and combining (2.28) and (2.29) leads to an additional helpful identity, namely, for all ti
i.^i(^ - ^ W +K-pUl\

(2.31)

which is in subsequent use as well. Apart from everything else, we supplement our below results by showing stability estimates in the situation when the operator in question is under perturbation. More precisely, to problem (1.1) we associate the following perturbed problem, with some *Bn € B(X), (2.32) For subsequent use we introduce the notation

Similarly to (1.4), the solution Yn of (2.32) is then represented by n-l 0

Yn = Un-iflKy + kJ^^n-hj+iQjFj.

(2.33)

3=0

One can easily verify the following identities, for tn,ti G Clk with t;
fin-Li = Un-1,1 - k J^ Un-1,,+ 1® A - l , / q=l

and n-l q=l

(2-34)

36

CHAPTER CHAPTER 2. 2. MAIN MAIN RESULTS RESULTS ON ON STABILITY STABILITY

which, in fact, connect connect the the discreet discreet evolution evolution operators operators ii^/ ii^/ and and UU^^ ofof the problems (2.32) and and (1.1), (1.1), respectively. respectively. Also, Also, combining combining both both these these equalities yields one more more helpful helpful formula, formula, for for titi
n-l

q=l n-l

q=l s=q+l

(2.36) The relations (2.34) (2.36) are are taken taken as as aa basis basis for for applying applying our our perturperturbation techniques when when examining examining the the stability stability of of problem problem (2.32). (2.32). Now we give a series of of stability stability assertions assertions which which differ differ from from each each other other by the assumed Holder-continuity Holder-continuity type type hypotheses. hypotheses. Theorem 2.10. Let seminorms seminorms ]|-|[ ]|-|[n, II \n> and ||| ||i ; n form respectively h\(£\\)-, h.\(l\m)-, and h.\{^\\m)-concordant h.\{^\\m)-concordant pairs pairs with with the the operator operator 2t2tn for some £, 71 G [0,1) and and m m G G N. N. Suppose Suppose further further that that 2l 2ln satisfies the

($,0; ($,0; |||||| |||| m ;ni m — 0,-f, *0), with

^{-condition as well as as hypothesis hypothesis HOL HOL some d G (0,1] and with with

||m;n -

^

I In

*/ tn = 0,

||| ' |||l;n

ifif

Ml*

if tn = *0.

VX VX= = I,I,

(2.37) (2.37)

operator *B *Bn in (2.32), Finally, let the seminorm seminorm ||25 ||25n ||, generated by the operator Then, for for all all form a A* (70)-concordant (70)-concordant pair pair with with 2l 2ln for some 70 G [0,1). Then, k G (0,K], tn,ti G fKfc with ti < t n, and u G X,

PU_i i/ uH n +p n _i i ju][ n < Ct-il+l\\u\\.

(2.38)

Furthermore, suppose that that the the operator operator 2t 2tn satisfies in addition hypothehypothesis HOL(i9; I ||| m;n ,m = 0, *0) and we have ||® n ti|| < Cfc*|til n

/ o r d l l t i e l and ** n G

fifc)

(2.39)

with the same i9 and |||||| | m ; n , m = 0, *0, as above. Then Then (2.38) (2.38) is is valid valid for £ G [0,1 + i9] provided provided that that the the pair pair (][-]|[n;2ln) (][-]|[n;2ln) isis still still Aj(£)-concordant. Aj(£)-concordant. However, for £ = 1 + d, d, the the expression expression t~}_l+1 log(t n_/+2/fc) should be added to the factor t~_l+1 on the right.

2.3. OPERATORS 2.3. STABILITY STABILITY OF OFDISCRETE DISCRETEEVOLUTION EVOLUTION OPERATORS

37

37

Proof. the the casecase where £ G £[0,1), 2l Proof. We We begin begin bybyconsidering considering where G [0,1), 2ln satisfies satisfies H 66 LL H )) ( I ?? ,, 00; ;| |

||m;n,m ||| |||| | m ; nn , m ||m;n,m == 0,7, 0,7,*0)*0)with with m == 0,7,*0, 0,7,*0,given givenby by (2.37), pairpair withwith 2l 2l n . (2.37), and and the the seminorm seminorm||QS||QSn |||| forms forms aaAj(7o)-concordant Aj(7o)-concordant Applying Theorem wewe thenthen obtain, for allfori; all < t i; < tq and Applying Theorem2.1, 2.1, obtain, and uu ££31, 31,

HIHI-

(2-40) (2-40)

It isis not that there exists a seta Tsetof TAi-configuration which iswhich is not hard hardtotoseesee that there exists of Ai-configuration Af(£|l)-properfor A^(l|m)-proper for (|-||n;2tn), and Ai(7i|m)Af(£|l)-properfor(]|[-]|[n;2ln), (]|[-]|[n;2ln), A^(l|m)-proper for (|-||n;2tn), and Ai(7i|m)proper n;2l n ), simultaneously. again the the assumed Holderproper for for (|||(||| |||i|||i ; n;2l simultaneously.Using Using again assumed Holdercontinuity type on on 2l 2ln and thatthat tf tfnn << C, continuity typecondition condition and taking takinginto intoaccount account C, itit follows that 6 >6 0, all tall t n ,tj €€ fife, (T U(T U follows thatwith withsome some > for 0, for fife,uuGGX,X,and andA ^A Int ^ Int

V(6k)), V(6k)),

m

ufi + + Ck\l(XI Ck\l(XI ufi

1 1

< cc (i(i++ fc^iAp fc^iAp - )

IAI !^! IAI1-"1!^!

<

With this applying the the identity, for for t n,ti G With thisfact factininmind mind applying identity, G

+k(XI )-\% +k(XI - kK kKn)-\%

-- 2l2ln) (A7 (A7-- k%Y k%Ym,

(2.41) (2.41)

yields, X Xandand A gAInt (T U(T V(6k)), yields, for forallallu uG G g Int U V(6k)), J(A7 u]|[n << J(A7 -- fe2lj)fe2lj)-mu]|[n

1

1

+fcI(A7 fcSln)-^^ - 2l - 2l n ) (A/ (A/ -+fcI(A7 - - fcSln)-^^

Jfe2li) 1"m«|| Jfe2li) - 2l 2l n ) (AI (AI --

because thepair pair(J[-]|[ (J[-]|[n;2ln) Aj(^|l)-concordant because the ;2ln) isisAj(^|l)-concordant andand the the set Tsetis T is proper for this thispair. pair.The The estimate thatpair the (|[-]|[ pair (|[-]|[n;2ln) is, ;2ln) is, proper for lastlast estimate showsshows that the in fact, fact, strictly strictlyA^(^|m)-concordant. A^(^|m)-concordant. By Theorem 2.1Remark and Remark By Theorem 2.1 and 2.3, we 2.3, we ,ti,t q G then find,for forallall t n,ti,t GO.k O.kand and u u£ £l ,l , then find,

^Ct^hH. ^Ct^hH.

(2.42) (2.42)

38

CHAPTER CHAPTER 2. MAIN RESULTS ON STABILITY

Denoting z n%\ — s u p n ^ - J i b - i ^ n l n , taking taking (2.42) (2.42) and (2.40) into account, and using (2.28), a trivial estimation now gives, for t\ < t n, n-1 |M|=1

< Ct;ll+1 + C* £

Zn,q

)

q=l+l

T h e r e f o r e , w i t h t h e a i d o f L e m m a 2 . 1 w e c o n c l u d e , s i n c e £ < 1 ,, 1 — i 9 < l , and 7 i < 1, that Zn,l < Ctn-l+l

for

tl
which yields that the first term on the left-hand side of (2.38) is estimated as stated. As a consequence of this fact, we note that for all t\
(2.43) (2.43)

and \\
(( 2 - 44 ) = ||Q3n ||

since we can set for ^ = 7o. Applying furthermore (2.34), (2.34), multiplying multiplying from from the left by Q5n, and using (2.44), we get, for all ti < t n, n-l q=l

Applying Lemma 2.1 to the last inequality, it follows that C'Ci+l ||«niin-l,i|| < C'Ci+l

fOT a11

*« ^ *"

With this estimate in mind, multiplying (2.34) (2.34) from from the right by u E X and „ of both sides, an estimation shows the desired taking the seminorm second term on the left-hand side of (2.38), since the same estimation for the first term is already shown. To prove the latter part of the assertion, we assume now that £ G [0,1 + i9], the pair (|[-|[ n;2ln) is Aj(£)-concordant, the operator 2l n satisfies hypothesis HOL(i9; HOL(i9; jjj || m;n ,m = 0,*0), and (2.39) holds with the same i? and (I || m ; n ,m = 0, *0, as above. In this case we have, for allfy< tn and u € X, - St z )«|| < Ct*_J«|| z . (2.45) (2.45)

2.3. STABILITY OPERATORS 2.3. STABILITY OF OF DISCRETE DISCRETEEVOLUTION EVOLUTION OPERATORS

3939

Denoting w wn>l _/+i sup| ^| ??11 and (2.45), Denotingthen then = tt n _/+i sup| M ||il n _ 11)) ^| and using using(2.43), (2.43), (2.45), n>l = M | =1 =1 ||il and Theorem 2.1,2.1, we (2.31), after after aa simple and Theorem we obtain obtainfrom from (2.31), simpleestimation, estimation, for fort;t;<<

. (2.46) (2.46)
Noting next thatthat ^ j ^^ j ^ << 22and ,tj,t g €€ Cl Noting next and i
which further implies which further implies n --ll n

Observing thatthat 1 —191 —19 << 11and 2.1 with Observing and applying applyingLemma Lemma 2.1 with £i £i==0,0,wewe find find that that ti < w n j <
(2.47)(2.47)

Now thatthat are ready ready to to show show(2.38) (2.38) in inthe the case case ££€€ [0,1+$]. [0,1+$]. In In fact, fact,noting noting Nowwewe are (2.47), Theorem 2.1, 2.1, and £— — dd <<11and and using using(2.31), (2.31), (2.47), Theorem andthe the above aboveargument, argument, we have, fort;t;<
q=V q=V

which shows thatthat the _i i{i/][ {i/][ n is the quantity quantity][il][iln_i is estimated estimated as asinin (2.38). (2.38).Further, Further, which shows by Lemma pair pair by Lemma 1.2 1.2the the seminorm seminorm k^^f k^^f |j| j n forms forms aa Aj(1 Aj(1 — — i?/2)-concordant i?/2)-concordant with 2l 2lnn . Therefore, Therefore,noting noting that that the the former formerpart part of ofthe the theorem theorem is is already already with we find, find, for forall allti ti< t< tnn and anduu & & X, X, proved, proved, we

^ iil lMMI .I .

48 ( (2"48 )

40

CHAPTER CHAPTER 2. MAIN RESULTS ON STABILITY

At the same time we have, for all t\ < t n and u G X, that HBniXn-Ljull < Cfc*lU n_1,«u|n < Ck*t-*l+1\\u\\,

(2.49)

since (2.38) for jlitn-i^uln is already shown. Using again again the fact that the quantity filre-i^ufn is under the desired estimate estimate and and with with the the aid of (2.48) and (2.49), a trivial estimation on the basis of (2.36) yields, for all ti < t n and u € X, n-l q=l n-l n - l q=l

s=q+l

< C"£ I+1 ||u||q=l n—1 n—1 q=i s=q+l

which easily leads to the needed estimate for the quantity ]|[it n_iji'u]|[n as well. So the proof is complete.

pair Theorem 2.11. Let a seminorm ][-]|[*;n form a A\(*;^)-concordant pair with 2l n for some £* > 0 and let seminorms ||| || n and ||| | i ; n be under the conditions of Theorem 2.10 for some 71 e [0,1) and m G N. Suppose further that the operator 2t n satisfies the A\-condition as well as hypothesis H O L ~* (i9,0; I || m;n , tn = 0,1, *0) and that the seminorm ||*8 n || is Af (70)concordant with 2t n , wii/i some $ G (0,1], 70 G [0,1) and with ||| ||m;n> m = 0,1, *0, ^wen 6y (2.37). Then, if0<£* < min{i9,1 - 71}, we have, for k G (0,K), tn,ti G fifc such that tt < tn, and x G 3£*, t/iat K-i,j+iXl*;J
(2.50) (2.50)

In the case £* = min{?9,1 — 71}, the last estimate remains true true if one adds log(t n-i+2/k) to the factor t~_l+1 on the right. Furthermore, if 0 < £* < min{i9,1 — 71,1 — 70}, (2.50) holds with iln-i,; iln-i,; substituted substituted for iln-i,/ provided that one adds log(£ n_j+2/&0 *° the factor t~_l+1 on the right if £* = min{i9,1 - 71,1 - 70}.

2.3.

STABILITY

OF DISCRETE EVOLUTION EVOLUTION OPERATORS OPERATORS

41

Proof. By duality it follows from (2.43) that that for all tn,ti G fife with ti < tn, \\K-i,i\\*
(2.51) (2.51)

and

\\K-i,i\U+ tZ l+l\\K-i,iK\U


(2.52)

By (2.51) the desired estimate for ]|[iX^_1 ;+1x|[*;i now easily follows when into their their adjoints, adjoints, using (2.30), with the operators on both sides converted into and applying further Theorem Theorem 2.2 and the above argument. Next, Next, based based on the fact just shown, the quantity ]|[iX^_j j+1xI*;Z i s estimated with the aid of (2.52), using (2.35) with the operators on both sides converted beforehand beforehand into their adjoints. situations which which are also important for appliNow we investigate other situations cations. Theorem 2.12. Let seminorms |||-| n , fl|-||i;n, ||HI*;n, iilo;n, and ||-|*o;n form Aj(7o)-, and Akj(*;^o)-concordant respectively A1} (j)-, A^(7i)-, Aj(*;7*)-, Aj(7o)-, pairs with the operator 2t n for some 7,71, 7*, 70,7*o G [0,1), J G N. Suppose further that 2l n is under hypothesis H O L ($, d\; ||| |m;n> tn = 0, / , *0) with some

i?i G [0,1] such that 7* < ^0 = 1 + mini1!? — 7, d\ — 71} and with

{ II i n */ tn = 0, 1 ||m;n =
(2.53)

I IMII*;n if m=*0, while the operator QSn in (2.32) satisfies the condition, with some i9o G [0,1] subject to 70 + 7*o < 1 + $o> for all k G (0, K], tn G fifcj u E X, and x G X*, | (X, KnU) | < Ck^ |x|*0;nffl«|0;n.

(2-54) (2-54)

Then, if a seminorm ]|[-][*;n forms a Aj(*;^*)-concordant pair pair with with 2l n for we some £* G [O,^o]> have, with e — 0, for all k G (0,K), tn,ti G fife with ti < tn, and x S X*, that

f

i]nxii

if e*<&, (2.55)

42

CHAPTER 2. MAIN RESULTS ON STABILITY

If in addition 70 — $0 < £1 = 1 — 7* + min{i?,i?i}, 7*0 < £,0, and £* < 1 + ^0 ~ 7o, then (2.55) holds for e = 1, provided that the factor t~_t is replaced by t~_ t + t~Z*° log(t n _/ + i/fe) in the case £* — 1 + $0 — 7oAlso, if a seminorm ]|-]|[n forms a Aj(£) -concordant pair with 21^ /or some £ G [0,£i] and i/max{7,7i} < £1, then we have, with e — 0, for all k G (0,K], tn,ti G fifc with t\ < tn, and u G X, that

(2.56)

w/iere 7 71

72 =

(2-57)

if if

/ / in addition 70 < £1, 7*0 < £0 + ^0; «^^ £ < 1 + $0 — 7*0; i^en (2.56) holds for e = 1, provided that the factor t~_l+1 is replaced by t~_ l+1 + Cm

case case £ = 1 +190 - 7*o-

Proof. Assume first that 7 + 7* — i9 < 1 and 71 + 7* — d\ < 1. Using Theorems 2.1, 2.2, and the fact that the operator 2l n satisfies hypothesis

H O L H ) ( 0 , t f i ; | llmjn.tn = 0,/,*0) with I lU, m = 0,1, *0, given by (2.53), we get, for all ti,tq,tn G O, k and x G £*> that (2-58) ^

(2-59) (2-59)

and

||(u?)n(a?-si;)xl|* <

ig-n

SU

P \mu\ii

\\u\\=l

j t, ^

sup |ilpu|i ; / ||u||=l

w + t i C 1 ) l|lxll'*;9- ^2-60) Combining (2.58), (2.59), (2.59), and (2.60) therefore yields, withp withp = for all tq,ti G &k with t; < t g and x € 3£*,

[(q-l-l)/2\,

< ( .»,

(2-61)

2.3. STABILITY OF DISCRETE EVOLUTION 2.3. STABILITY OF DISCRETE EVOLUTION OPERATORS

OPERATORS 43 43

and likewise, and likewise, 71 71 U< 21*) (C t j 7cc--"++ tJiT "^)|»x||U; |»x||U; (2.62) -- 21*) C (tj7" tJiT "^) (2.62) X XU< gg..

Taking now seminorm both ofsidessides of identity with Taking now the the seminorm ||j||jI*./I*./ of both of thethe identity (2.30), (2.30), with and multiplied from the operators beforehand into their the operators converted converted beforehand into their adjoints and multiplied from adjoints an n by > d using using further Theorem 2.2 (2.61), we with the right by x> ax d further Theorem 2.2 andand (2.61), we get,get, with the right

ra-1 ra-1

q=l+l q=l+l

Since — 7* i9< < 1 71 and — $1 + 7* applying 1, 2.1, with — 1 Since 77+ + 7* — i9 1 and + 7* 71 — $1 << 1, applying LemmaLemma 2.1, with nn— 1 substituted for n,this this to inequality yields substituted for n, to inequality immediatelyimmediately yields Zn-1,1 Ct'l] Zn-1,1 << Ct'l] for

tl for < tn.

tl < tn. (2.63)

(2.63)

the seminorm of ]|[-]|[* both of(2.30), (2.30), with the operaTaking Taking next next the seminorm ]|[-]|[*;;;; of both sidessides of with the operaand multiplied the from right by tors converted adjoints tors converted into their into adjoints their and multiplied from the right by applying then(2.63), (2.62), and Theorem (2.63), 2.2, we obtain, for applying then (2.62), and Theorem 2.2, we obtain, for t\
W xU
t7 t7+1 IilUln IilUln ++ *7+iHK«li;n *7+iHK«li;n <
(2.64)

(2.64)

and and

liiUln^Ct^llnll, liiUln^Ct^llnll,

(2.65)

(2.65)

aswell well as as as

U\\ + QqqlKT{n lKT{n U\\ Q nn+

(C (C

%) \\ %) \\hn hn

t )

2 . 6 6 ) t (2.6 )6)(

CHAPTER 2. MAIN RESULTS RESULTS ON ON STABILITY STABILITY

44

and

n - %) Uf n< C

Mn + t

||M| 1;n ) .

(2.67)

Applying (2.64) and (2.66), (2.66), we we take take in in turn turn the the seminorms seminorms | | ||||||n and ||| | i ; n of both sides of (2.29), with with the the expressions expressions multiplied multiplied from from the the right right by by u,u, and we use the above argument argument to to obtain obtain (2-68)

wn,i q=l

and n-l

wl for all t n ,t/ € fifc with t t < tn, with w n;; = sup|| u || =1 |||ii n_1)/'u||ri and w^J — supMu||=1 Ifliln-i^ulijn. As an easy easy consequence consequence of of (2.68) (2.68) and and (2.69), (2.69), we wehave, have, with 73 = max{7,71} and and 74 74 == max{7 max{7 3 - d + 7,, 73 - i?i + 7*}, 7*}, n-l

Thus, by Lemma 2.1 we get, since since 73,74 73,74 << 1, 1,

Combining this inequality inequality with with (2.68) (2.68) (if (if 77 << 71) 71) or or with with (2.69) (2.69) (if (if 77 >> 71) 71) yields (2-70) wn,i < C C J + I (2-70) and Using therefore (2.65), (2.67), (2.67), (2.70), (2.70), and and (2.71) (2.71) leads leads to to (2.56) (2.56) with with ee== 0,0, after a simple estimation estimation based based on on applying applying (2.29) (2.29) with with both both sides sides multiplied multiplied from the right by u. It remains to be proved proved that that under under the the corresponding corresponding conditions, conditions, (2.55) (2.55) and (2.56) hold for e == 11 as as well. well. Since Since both both estimates estimates can can be be established established in a similar manner, we we will will focus focus on on how how to to show show (2.56) (2.56) for for ee== 1.1.

2.3.

STABILITY

OF DISCRETE DISCRETE EVOLUTION EVOLUTION OPERATORS OPERATORS

45 45

To that end, assuming that that 70 70 << £1, £1,7*0 7*0 << £o+"$o £o+"$o and and ££ << l+i9o—7*0) l+i9o—7*0) we we use (2.34) and take (2.54) (2.54) into into account account to to obtain, obtain, after after aa simple simple estimation, estimation, SUp |;O n _i ) /ll|o ; n < SUp IliU-i^loin H|=i IMI=i H|=i n-l q=l

ll«ll=l

X SUp |ii*, ) g + 1 x|*0; 9 SUp |||Ug_i ;i^|||o;g,

with q' — [(n — 1 + q)/2\, for tn,ti G Q,k such that ti < t n. Further, seeing from Lemma 1.2 that the the seminorm seminorm fc^ fc^0 |||*o;n forms a Aj(*;7*o — — $0)$0)(2.56), both both with with ee == 0, 0, we we concordant pair with 2l 2ln and using (2.55) and (2.56), have instead n-l

sup ""'

'"

' " ' "" ^^

""''

By Lemma 2.1, since 70 70 << 11 and and 70 70++ 7*0 7*0 << ll ++ i?0i i?0i this this immediately immediately implies, implies, for ti < t n, SUp lOn-i^loin < Ct-™ +1.

With this fact in mind, mind, estimating estimating similarly similarly sup|| sup||tl||=1][iln_i!/u]|[ri, again on the basis of (2.34), it is is readily readily seen seen that that (2.56) (2.56) holds holds with with ee— —1.1. Using the same method method one one can can prove prove T h e o r e m 2.13. Suppose Suppose that that for for some some 7,7*, 7,7*, 7*1,70,7*0 7*1,70,7*0 SS [0,1), [0,1), and and JJ GGN, N, seminorms ||| || n , ||| |* ; n , ||| |* 1;n , ||| ||o ;n;n, and ||| ||+0 +0;n form respectively Aj(7)-, A5(*-,7*)-; A5(*-,7*)-; A ^ ( * ; 7 * I ) - , Ay(70)-, and Ay(*;7* Ay(*;7*0)-concordant hypothepairs with the operator operator 2l 2l n . Furthermore, let 2l n itself be under hypothe(#,i9i; If || m;Ti,m = 0, *0, */) with some sis H O L some i?,i9i i?,i9i G G [0,1] [0,1] such such that that 7 < £0 — 1 + min{i9 — 7*,i?i — 7*1} and with

f III-In II lm;n = < II |||*;n I I 1*1 ;TI

*/ tn = 0, if m = * 0 , */ m = */,

and let the operator QS QSn in (2.32) satisfy condition condition (2.54) (2.54) with with some some do do GG [0,1] subject to 70 + 7*0 „„ forms 7*0 << 11++ $o$o- Then, Then, ifif aa seminorm seminorm forms aa Aj(£)Aj(£)concordant pair with 2l 2ln /or some £ G [0, £o]> l u e ^ a?;e ; ™ ^ e = 0, for all

46

CHAPTER 2. MAIN RESULTS RESULTS ON ON

STABILITY STABILITY

k G (0,K], tn,ti G f2fc with t\ < t n, and u G 3c, that

(2.72)

/ / i n addition7*0 —#0 < £1 £1 == 1—7+min{t9,#1}, 1—7+min{t9,#1}, 70 70 << £o> £o> and^ and^ << l+i9o-7*o> l+i9o-7*o> then (2.72) holds for ee == 11 provided provided that that the the factor factor t~_l+1 is replaced by case (( = = 11 +19 +190 - 7*0n-?+i log(*n-/+2/fc) m the case , i/ a seminorm ]|[-]|[*;n forms a Aj(*; ^)-concordant ^)-concordant pair pair with with 2l 2ln /or some ^* G [0,£i] and i/max{7*,7*i} i/max{7*,7*i} << ^1, ^1, itit then then holds, holds, with with ee== 0,0, /or /oraZ aZZ fe G (0, if], i n , t/ G ft fe wt/i t; < t n, and x G X*,

tn-1+1

(2.73)

where 7*2=

7* 7*1

if

T?>I?I,

(2-74)

(2.73) holds If in addition 7*0 < £i> 7o 7o << , and£* < l + tf o-7o, for e = 1, provided that that t~5*; t~5*; *s *s replaced replaced by by t~^_* l+1 + ^~2j+i log(£ n_j+2 A ) * n i/ie case £* = 1 + $o — 7oSome further developments developments are are contained contained in in the the next next two two assertions. assertions. seminorms |||||| ||||||n T h e o r e m 2.14. Let there there be be an an operator operator QJ QJn G B(3c) and seminorms and I | i ; n such that the triplets (||| || n ;2l n ;QJ n ) and (||| ||i; n ;2tn;5Jn) ar^ Aj(l)- and Aj(71)-concordant, Aj(71)-concordant, respectively, respectively, for for some some JJ GGNN and and 71 71 GG[0,1). [0,1).

Zef also 2l n satisfy hypothesis HOL HOL

($, ($, 0; 0; |||||| |m;n)^ |m;n)^ == 00,,//,, *0) *0) and and let, let,

for all t n G fifc, « G X, and xx G G 3£*j 3£*j

some d G (0,1] and with with

1;n

*/ m = 0, if m = /, */ m = *0.

2.3. STABILITY STABILITY OF OFDISCRETE DISCRETE EVOLUTION OPERATORS 2.3. EVOLUTION OPERATORS Finally, suppose supposethat thatfor for some some (5 (5n G B(X), B(X), with with the the seminorm seminorm Finally, \\enu\\ M n = \\e

for uu eX, eX, for

47

given by „ „given by (2.75) (2.75)

is strictly strictly A*(£)-concordant A*(£)-concordantfor for some some ££ G G [0,1). [0,1). the triplet triplet ; 2l ;2Jn) is the ; 2l n ;2Jn) Then, putting QJ_i QJ_i== Q?o, Q?o,ww Then, putting iWi-iufn< Ct Ctnil+I \\u\\, itiWi-iuf l+I\\u\\,

(2.76) (2.76)

for all all kk G G(0, (0,K], K],tnt,ti G G£)& £)&with with i; << ttn, and anduu GGX. X. for sake of of convenience conveniencewe we put put lt_i lt_i==Ho Hoand and 2l_j Proof. Forthe thesake 2l_j = = 2lo-2loIt It Proof. For follows from from Theorem Theorem2.4 2.4 and and Remark Remark 2.3 2.3 that that for for all all tnt,ti G G flk flk with with follows and «« 66l l, , ti < ttn and Using further the the accepted accepted Holder-continuity Holder-continuity type condition on 2l 2ln , taking taking Using further type condition on into account account the the properties properties of of the the seminorms seminorms ||j ||||n , |||||| ||ii; nn , and and applying applying into 2.1,we wefind, find, for for ttq, t; t; G Gf^ f^ with with t\
the seminorm seminorm |[-]|[ |[-]|[n is of of aa specific specific construction, construction,the the last last estimate estimatefurfurSince the Since thermore implies, implies,for for all allttn,tq,ti €€ JJ\\ with with U U< < ttq < < ttn, thermore sup """ """ sup !MI=i !MI=i

[t°li+1 +i p + 11 J sup suplUn.^+xQVllUn.^+xQVl- (2-78) (2-78) < C [t°li +1 + Applying next next(2.30), (2.30),with with substituted for for II and andwith with both both sides sidesmulmulApplying I — 1 substituted tiplied from from the the right right by by 5J/_iu, 5J/_iu, and and using using (2.77) (2.77)and and (2.78), (2.78), a simple simple estiestitiplied mation gives gives mation n - ll

for all all ttn,ti G Gfife fife such such that that ti ti < < ttn, with with zn)/ = = Since 1 — — -d -d< 1, 1, 71 71<<11, ,and and^ ^< <1, 1, needed estimate estimatefor for the thefirst first term term Since thethe needed the left-hand left-hand side sideof of (2.76) (2.76) now now follows follows by by Lemma Lemma 2.1. 2.1. With With this this fact fact on the in mind, mind, the the second second term termis is then then estimated estimatedby by using using (2.35) (2.35)and and the theabove above argument. argument.

47

48

CHAPTER CHAPTER 2. 2. MAIN MAIN RESULTS RESULTS ON ON STABILITY STABILITY

Remark 2.4. In the conditions conditions of of Theorem Theorem 2.14 2.14 the the requirement requirement that that the the seminorm ]|[-]|[n must be represented in in the the form form (2.75) (2.75) can can be be avoided avoided if, if,

instead of hypothesis HOL HOL ($, ($, 0; 0; |||||| ||||m;n> m = 0,7, *0), it is accepted accepted that that 2tn — 21; = %fifin,i with some some Sj Sjnii G B(3L), where, for all all tn,ti E ttk and u G X,

theorem. with the same d, j|| || n , ||| | i ; n , and 5Jn as in the statement of the theorem. This result can easily be be shown shown by by using using aa slight slight modification modification ofof the the above proof. Theorem 2.15. Suppose Suppose that that the the conditions conditions of of Theorem Theorem 2.14 2.14 are are satisfied satisfied for some J G N, 1? G (0,1], and 71, £ G [0,1), with with the the difference difference that that we we now accept the Aj(£|l)-concordance Aj(£|l)-concordance of of the the triplet triplet (][-]|[ (][-]|[n;2ln;93n) instead of its strict Aj(£)-concordance. Aj(£)-concordance. In In addition addition we we suppose suppose that that RanQJ RanQJn is independent of t n and the operator 2J n has an inverse 2J" 1 such that for all k G (0,K] andt n,ti G ft fc,

IIQJ"1^!! < C -

( 2 - 79 )

Then the stability estimate estimate (2.76) (2.76) holds. holds. Proof. Using (2.41) with m = 1, 1, with with both both sides sides multiplied multiplied beforehand beforehand from from the right by 93/it, and applying applying furthermore furthermore (2.79) (2.79) and and the the above above argument argument (cf. the proof of Theorem Theorem 2.10), 2.10), we we conclude conclude that that the the triplet triplet (|[-| (|[-| n ;2l n ;9J n ) is, in fact, strictly Aj(£)-concordant. Aj(£)-concordant. ItIt thus thus remains remains to to apply apply Theorem Theorem DD 2.14.

2.4

Stability of discrete discrete Cauchy Cauchy problems problems

Now we apply the above above results results for for showing showing stability stability estimates estimates for for problem problem (1.1). T h e o r e m 2.16. Let TT < < oo, oo, let let seminorms seminorms ]|[-]|[ ]|[-]|[n, ||| ||n> and ||| | i ; n and the operators 2t n, *Bn satisfy the conditions posed posed in in the the former former part part of of the the assertion of Theorem 2.10, 2.10, with with some some dd G G (0,1] (0,1] and and £,71,70 £,71,70 €€ [0,1), [0,1), and and let for ||£n|| + ||JH|| < C for all all kk G G (0, (0,K] K] and and ttn G flk. (2.80)

2.4. STABILITY OF DISCRETE DISCRETE CAUCHY CAUCHY PROBLEMS PROBLEMS

49 49

Then the solution Y n of both (1.1) and (2.32) (2.32) satisfies satisfies the the estimate, estimate, for for all all k G (0,K] and t n G Qk,

]Mn< c Uum V

1=0

JJ

If in addition the operators operators 2l 2l n , *Bn and the seminorm ][-][ n satisfy the conditions imposed in the latter latter part part of of the the assertion assertion of of Theorem Theorem 2.10, 2.10, with with the same -d as above and and with with some some ££ G G [0,1 [0,1 ++##]],, then then the the estimate estimate (2.81) (2.81) remains valid provided that that the the expression expression tltl1 \og(tq+\/k) is added to the factor tq on the right, for for qq — —nn + + 11 and and qq == nn — —I,I, whenever whenever ££== 11++ #.#. This assertion is a direct direct consequence consequence ifif Theorem Theorem 2.10 2.10 isis applied applied to to (1.4) (1.4) and (2.33). Theorem 2.17. Let the the operator operator 2l 2ln and seminorms ]|[-]|[n, ||| \\n, 1 ||*;n> and I | i ; n be under the conditions of Theorem Theorem 2.12 2.12 for for some some JJ GGN, N, ££ >> 0,0, 7,7*,7i G [0,1), and "d,d\ "d,d\ G G [0,1] [0,1] such such that that max{7,71} max{7,71} << £1 £1 == 11 — 7* + min{i?,i9i}, £ < £1, and let (2.80) hold. Then, ifT < 00, the solution solution YYn of (1.1) satisfies the stability stability estimate estimate (2.81) (2.81) in in which, which, J/£ J/£ == £1, £1,tqtq should shouldbe be replaced by tq + tq 12 log(tq+i/h) for q = n + 1 and and qq = = nn — —I,I, where where 72 72 isis defined by (2.57). Furthermore, suppose in in addition addition that that the the operator operator 93 93n and seminorms are II |||o;ri) II |||*0;n subject to the conditions of Theorem Theorem 2.12 2.12 for for some some the above above 7o < £1 and 7*0 < 1 + $0 $0 ++ niin{?9 niin{?9 — 7, i^i — 71}. Then, keeping all the restrictions and assuming assuming in in addition addition that that ££ << ll ++ ^o ^o—7*o> —7*o> (2.81) (2.81) also alsoholds holds for the solution of the perturbed perturbed problem problem (2.32). (2.32). The The last last assertion assertion remains remains valid for £ = 1 + $0 — 7*0 as well provided that that the the factor factor tq tq isis replaced replacedby by tq + tq~10 log(t g+ i/fc) for q = n+ 1 and qq = = nn — —I.I. In the particular case where where 2l 2ln = 21 is independent oft n, (2.81) is valid ;;21) for the solution of (1.1) (1.1) for for any any fixed fixed ££ >> 0. 0. //// in in addition addition the thepair pair 21) is Aj(£)-concordant, then then (2.81) (2.81) for for the the solution solution of of (1.1) (1.1) holds holds inin the the case case T = 00, that is, for n == 0 , 1 , . . . .

The claim follows by Theorems Theorems 2.12 2.12 and and 2.1 2.1 and and with with the the aid aid ofof (1.4) (1.4) and (2.33). Theorem 2.18. Let the , ||| | n > III \\*;n, the operator operator 2l 2ln and seminorms and I |*i;n be under the conditions conditions of of Theorem Theorem 2.13 2.13 for for some some JJ GGN, N, ££ >> 0,0, 7,7*,7*1 G [0,1), and i9,i9i i9,i9i G G [0,1] [0,1] such such that that 77 << £0 £0 == 11++ min{$ min{$ — 7*,$i — 7*1}, £ < (,0, and let (2.80) (2.80) hold. Then, if T < 00, the the solution solution YYn of

50

CHAPTER CHAPTER 2. MAIN RESULTS ON STABILITY

(1.1) satisfies the stability estimate (2.81) (2.81) in which, i/£ — £o, tq should be replaced by tq + tq 1 \og{tq+\/k) for q = n + 1 and q = = nn — I. Furthermore, suppose in addition that the operator 5Bn and seminorms I " llo;n> 1 " |||*0;n &re subject to the conditions of Theorem 2.13 for some 7*0 < 1 + $o — 7 + min{$,$i} and 70 < £o- Then, keeping all the above restrictions and assuming in addition that £ < 1 + $0 — 7*0 > (2.81) also holds for the solution of the perturbed problem (2.32). (2.32). The last assertion remains valid for £ = 1 + do — 7*0 provided that the factor tq is replaced by tq + tq10 \og(tq+i/k) for q = n + 1 and q — n — l.

This result is a consequence if Theorem 2.13 is applied to (1.4) and (2.33). Theorem 2.19. Let an operator QJn G B(3L), seminorms , ||| ||n> If | i ; n , and the operators 2l n , 93n satisfy the conditions of Theorem 2.14 or 2.15 with some J € N, £ > 0, 71 G [0,1), and G (0,1], and let, for all tn E nk, and x € X*, and \\WX\\. < C\\% X\U-

(2-82)

Then, if T < 00 and £ G [0,1), the solution of both (1.1) and (2.32) satisfies (2.81). If in addition the operator 2l n = 21 is independent of tn and the triplet (]|[-]|[n;2l;27n) is Aj(£)-concordant, then then (2.81) (2.81) for the solution of (1.1) is valid for any fixed £ > 0. Moreover, in the case where the triplet — 00, that n) is Aj(£)-concordant, the last assertion holds withT — is, for all n = 0,1,.... The result immediately immediately follows follows when when applying applying Theorems Theorems 2.14 or 2.15 and Theorem 2.4 to (1.1) and to (2.32) and taking (2.82) into account. account. In conclusion we consider an application of the results of Section 2.2, concerning the behaviour of stability constants. With With the same restrictions as in Section 2.2 we are able to deal with the case T = 00. Theorem 2.20. Let the operator 2tn = 21 be independent oft n, let it satisfy the Aj(M)-condition with with some some J G N and M > 1, and and let let (2.80) hold. Suppose further that £ is a fixed number such that that £ > 0 if J — 1 and £ = 0 if J > 2. Then the solution Yn of (1.1) satisfies the estimate, for all k G (0,K\ and n = 0 , 1 , . . . ,

CM

2.5. COMMENTS AND AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL

REMARKS REMARKS

// instead the operator A;2l A;2l satisfies satisfies the the STj(M)-condition, STj(M)-condition, timate holds with K(M) given given by by (2.18) (2.18) in in place place of of M. M.

51 51

then then the the last last eses-

This result follows directly directly as as an an application application of of Theorems Theorems 2.6, 2.6, 2.7, 2.7, 2.8, 2.8, and 2.9.

2.5

Comments and bibliographical bibliographical remarks remarks

Section 2.1: The problem problem of of stability stability of of discrete discrete semigroups semigroups inin Banach Banach space settings has been been extensively extensively examined examined in in the the literature literature by by using using different resolvent conditions conditions on on the the operator operator in in question. question. For For such such reresults in finitely many dimensions, dimensions, we we mention mention the the work work of of Kreiss Kreiss [97], [97], Le Le Veque & Trefethen [103], Miller & Strang [112], Morton [113], Spijker [153], and Tadmor [157, 158]. 158]. As As concerns concerns the the infinite-dimensional infinite-dimensional case, case, we we rerefer to our work [10, 11, 11, 12, 12, 17, 17, 33, 33, 34] 34] as as well well as as to to the the work work ofof El-Fallah El-Fallah & Ransford [67], Kalton Kalton et et al. al. [89], [89], Lubich Lubich & & Nevanlinna Nevanlinna [104], Nagy & Zemanek [114], Nevanlinna [116, 117, 117, 118], 118], Spijker Spijker & & Straetemans Straetemans [154], and Vitse [165] (see also also references references therein). therein). These These results results are are often often inintended for use in rather rather general general situations situations while while in in the the present present book book we we restrict ourselves to the parabolic parabolic case case for for which which itit isis natural, natural, as as pointed pointed out out above, to work with the the notions notions defined defined in in Chapter Chapter 1.1. In In particular, particular, TheoTheorem 2.3, as stated under under the the Aj-condition, Aj-condition, actually actually arises arises from from our our earlier earlier work [10, 11] (see also [17]). [17]). However, However, for for the the aims aims of of this this book book we weprefer prefer toto present our main results results as as stated stated in in Theorems Theorems 2.1, 2.1, 2.2, 2.2, 2.4, 2.4, and and 2.5 2.5 which which further generalize Theorem Theorem 2.3 2.3 and and are are useful useful for for various various applications applications ofof the the theory. Also, we remark that the above above estimate estimate (2.7) (2.7) has has aa counterpart counterpart inin the the theory of holomorphic semigroups. semigroups. More More precisely, precisely, for for aa closed, closed, densily densily defined operator A such such that that AA G GS(ip, S(ip, 0) 0) with with some some ip ip€€ (0, (0,vr/2), vr/2), itit follows follows tA that A generates a holomorphic holomorphic bounded bounded semigroup semigroup e~ e~ and, with any fixed

£ > 0,12

*

M

|| < Cr*

fori>0.

(Cf. Komatsu [93, Theorem Theorem 12.2].) 12.2].) Section 2.2: In our opinion, opinion, the the Aj(M)-condition Aj(M)-condition for for an an operator operator ££ looks looks more natural than the STj STj (M)-condition (M)-condition for for the the operator operator k£, k£, so so that that the the STj(M)-condition is of of more more theoretical theoretical interest. interest. As As itit follows follows from from the the discussion in Section 2.2, 2.2, in in fact, fact, the the STj(M)-condition STj(M)-condition as as stated stated for for fc£ fc£ implies the Aj(M)-condition Aj(M)-condition for for £, £, with with aa larger larger constant constant M\ M\ inin place place ofof 12

The fractional powers A*, A*, ££ >> 0, 0, are are defined defined under under the the stated stated conditions. conditions.

52

CHAPTER CHAPTER 2. 2. MAIN MAIN RESULTS RESULTS ON ON STABILITY STABILITY

M. This fact means that that problem problem (1.1) (1.1) isis naturally naturally parabolic parabolic ififthe the operator operator fc2t is under the 5T/(M)-condition. 5T/(M)-condition. On On the the other other hand, hand, as as itit will will be be seen seen subsequently in the book book (see (see Remark Remark 6.4), 6.4), the the Aj(M)-condition Aj(M)-condition appears appears naturally and should therefore therefore be be thought thought of of as as original, original, not not as as aa produce produce of the 5Tj(Mo)-condition 5Tj(Mo)-condition with with some some MQ MQ<< M M in in place place of of M. M. Nevertheless, Nevertheless, the STj(M)-condition is is examined examined here here for for the the reason reason that that itit isis actively actively discussed in the literature literature (see, (see, e.g., e.g., [48] [48] and and [118]), [118]), where wherethe the question question arises arises in which way the size of of the the constant constant M M appears appears in in stability stability estimates. estimates. ItIt isis worth noting that it has has been been shown shown in in Borovykh, Borovykh, Drissi, Drissi, and and Spijker Spijker [48] [48] (cf. also El-Fallah & Ransford Ransford [67] [67] and and Vitse Vitse [165]), [165]), under under aa condition condition which is equivalent to the the 5Ti(M)-condition, 5Ti(M)-condition, that that ||it ||it n || < CM 2, while the above Theorem 2.8 2.8 gives gives ||iP|| < CMlog(l + M). Also, Also, under under the the same restriction, Yuan [167] [167] has has found found an an estimate estimate which which isis equivalent equivalent toto from Theorem Theorem 2.9 2.9 that that the fact that ||2lit n|| < CMH~\ V while it follows from Apart from everything everything else, else, Theorem Theorem 2.9 2.9 ||2lil n || < CM 2 log(l + M)t~l v n gives an estimation of the the quantity quantity ||2l^il ||2l^il || for fractional values of of ££ >> 0. 0. Section 2.3: Discrete versions versions of of Gronwall's Gronwall's lemma lemma isis aa widespread widespread tool toolofof analysis in discrete problems. problems. As As aa rule, rule, inequalities inequalities like like those those appearing appearing inin the statement of Lemma Lemma 2.1 2.1 have have been been dealt dealt with with in in the the case case Ci Ci — —C2 C2== 00 (see, e.g., Sloan & Thomee Thomee [147] [147] or or Zhang Zhang [168]). [168]). For For our our needs needs we we are are forced to treat the general general case case where where 00 << Ci> Ci>C2 C2<< 11As concerns the stability stability assertions assertions collected collected in in the the present present section, section, we we are not informed of any any related related results, results, except except our our own own work work [10, [10, 11, 11,34] 34] based on assumptions in in aa more more concrete concrete form. form.

Chapter 3

Operator Splitting Problems Problems In this chapter we examine discrete problems problems that that are constructed on the basis of some ideas of operator splitting. For this class of problems the operator in question is decomposed in a special way and our starting starting assumptions assumptions below are stated in terms of restrictions on each separate component component of the original operator. Our investigation here is mainly devoted to the analysis of one of the simplest constructions based based on operator splitting. 1 More precisely, in this chapter we mainly deal with problem problem (1.1) (1.1) in which 2l n is represented as follows

On = a i * + a 2 ; n -

fca 2inai;n,

t 3-1)

with some given operators operators 2lj ;n G B(X), j = 1,2. To settle even this case one one has to employ essentially new techniques in comparison with the analysis carried out in Chapter 2. In order to avoid extra difficulties and to make our presentation readable readable enough, enough, we restrict ourselves to the case where 2ln satisfies the Aj-condition, which does does happen, happen, as it will be seen later, if both 2li ;n and 2l2;n satisfy the A^-condition. 2 For the same reason we assume that the condition 3 £}„ = 9\ — I holds and we use no seminorms which form concordant pairs pairs with with the operators 2ln or 2tj ;n , j — 1,2, and which would differ from from ||(2l ||(2ln + /x 0/)* || or ||(2l j;n + /i 0 /)* ||, £ > 0. At the same time our argument is further extended in Section 3.3 in order to analyse a more involved construction construction which which admits admits second second order order solvers. solvers. However the analysis becomes more more involved involved in the case of general higher :

In applications this kind of splitting leads, at most, to first order solvers. The case with the Aj-condition, J > 2, in principle can be considered but it turns out to be rather technical and it is not discussed here. 3 We could suppose (2.80) instead. instead. 2

53

54

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTINGPROBLEMS PROBLEMS

order solvers, so we leave the corresponding details outside outside of our present scope. In the considered case O.n — 9\ — I, problem (1.1), (3.1) can be equivalently represented as a time stepping method Yn+1

= (I-MQYn + kFn

Y0 = y°, for which the transition operator iXn iXn = I — A;2ln is therefore split into two factors. As a matter of fact, the analysis of problems with a splitting operator becomes rather difficult difficult if the operators 2li ;n and 2l2;n in the splitting construction do not commute, as we will see in the case below. In this situation the concept of commutator will play an important role in what follows. More precisely, given linear operators operators £ i and £2, their commutator Com (£i|£2) is defined as Com (£i|£ 2 ) = £ i £ 2 whenever the expression on the right makes sense. (It is certainly the case if both £1 and £2 are bounded.) In the following considerations considerations we base involve the commutator Com (2li;n|2l2;n)> ourselves on two hypotheses which involve as follows, with some K G [0,1) and no > 0, 0, for for all all kk G G (0,K], tn € fi^, X G X*, and u G X: C O M 1 [ K ] which stands for

u\r\\u\\i-K,

j = 1,2, (3.2)

and COM2[«] which stands stands for 2

|(X,Com (2li ;n|2t2;n)n)| < Cj^ \\(%,n+^I) \\(%,n+^I)Kx\U\\(^-r,n

+ ^I) u\\. (3.3)

Accepting these conditions conditions means means that that even even though though the operators 2li ;n and 2l2;n may not commute, they "almost "almost commute" commute" in the sense that their product, as stated in (3.2) and (3.3). commutator is "weaker" than their product, Also, we remark that making use of functions of ordered non-commutative non-commutative operators is an essential ingredient of this chapter's analysis. The necessary

3.1. ESTIMATES

FOR DISCRETE DISCRETE SEMIGROUPS SEMIGROUPS

55 55

knowledge of such functions functions and and the the corresponding corresponding notation notation are are contained contained in Appendix A.4. Note that due to the non-commutativity non-commutativity of of the the operators operators 2lj 2lj;n) j = 1,2, even if both satisfy the Ai-condition, Ai-condition, the the spectrum spectrum of of the the original original operator operator %n may contain points with with negative negative real real part. part. This This fact fact leads leads to to an an expoexponential growth in stability stability estimates estimates at at large large time time values. values. For For this this reason reasonwe we restrict ourselves to the the case case of of finite finite time time intervals, intervals, that that is, is, where where TT <
3.1

Estimates for discrete discrete semigroups semigroups

In the present section we we examine examine the the stability stability of of the the discrete discrete semigroup semigroup i4j = (/ — /c2ln)', uniformly with respect respect to to tn,ti £ Clk. These results lead lead directly to stability estimates estimates for for the the solution solution of of problem problem (1.1), (1.1), (3.1) (3.1) inin the autonomous case, that that is, is, where where 2l 2ln is independent of t n. In the nonautonomous case the below below assertions assertions are are important important as as well well when when combined combined with suitable frozen-operator frozen-operator techniques techniques which which are are similar similar to to those those already already used in Section 2.3. This first assertion is an an analogue analogue of of Theorem Theorem 2.3. 2.3. T h e o r e m 3.1. Suppose the AjAjSuppose that that the the operators operators 2lj ;n , j — 1, 2, both satisfy the condition and that hypothesis hypothesis C O M 1 [ K ] holds for some K G [0,1). Then, with any fixed /JO > 0 sufficiently sufficiently large large and and with with any any fixed fixed ££ >> 0,0, for for all all k E (0,K] andti,t n € f^.

= Ctrfv where the operator 2l n is specified by 2lj ;n , j — 1,2, through (3.1). Proof. It is convenient to denote denote 4 £ = 4

I A/ - I A; 2tl;n +k 2l2;n ~fc 2l2;n2ll 2l2;n2ll

The newly used notation is is explained explained in in Appendix Appendix A.4. A.4.

(3.4)

56

CHAPTER OPERATOR SPLITTING PROBLEMS CHAPTER 3. 3. OPERATOR SPLITTING PROBLEMS

and and U= = (I(I-fc9C fc9C2;n) Com Com(fc2li (fc2li;n |£). ; n|£).

(3.5) (3.5)

5 Our further argument argumentwill will be based based on on the theuse useofofthe the operator equation equation Our further be operator

n^C n^C

lZU, ++lZU,

(3.6) (3.6)

which is solved solved for for the theoperator operator TZ TZe B(X). B(X). which is Let now now F\ F\ be bea acontour contour6 which which encircles encirclesthe the spectrum spectrum of of the the operaoperaLet Next, using using(A.19) (A.19)with with F\ in in place place of of Si, S i , the theoperator operator ££ is is tor fc2l fc2ln; F\ tor n;i. Next, represented as follows, follows, with withGQ(X;Z) GQ(X;Z) = = ((A ((A-- z)I z)I— —(1 (1 -- z)k$l2;n)~ z)k$l2;n)~1, represented as

C=~ C=~

l G0(X;z)(zI-m (X;z)(zI-m1;n dz. ff G 1;n)~ dz.

(3.7) (3.7)

2vrz7 r AA 2vrz Clearly, there thereexist exista set set TTofofAi-configuration, Ai-configuration,which which is A^-proper A^-properfor for both both Clearly, is 2li;n and and 2l2;m 2l2;m and and aa set setT'T'ofofAi-configuration Ai-configurationsuch suchthat that 2li —v\ v\ ++ v
forall allfjfj£ T £ ,Tj , =j 1,2. = 1,2. for

Further, select selectaa set set T" T"ofofAi-configuration Ai-configurationsuch suchthat that T' C C I nnttTT"" U U {0}. {0}. Further, T' Our aim now nowisistotobound bound the the quantity quantity ||£|| ||£|| for forall allAA^ ^Int Int (T" T>(6ok)) Our aim (T" UU T>(6ok)) with some someSo So> 00 which which will willbe be chosen chosen later. later.In In order order to to perform perform this thisaction action with two cases: cases: |A| |A| << << and|A||A|> >S\,S\, with 5\ 5\ = =SQK, SQK, we distinguish distinguishbetween between we two 5i5iand with where K K appears appears in in the thestatement statement of of the thetheorem theorem and and isisthe theupper upper bound bound where for k. k. Further, Further, we welet letthe thecontour contour T\ T\ in in (3.7) (3.7)coincide coincidewith withthe the boundary boundaryof of for the set TA TA==TT\JV(mm{52\\\,5\52})-, \JV(mm{52\\\,5\52})-, with with some 82> > 00chosen chosen sufficiently sufficiently the set some 82 small in in order order to to satisfy satisfy the the inclusion inclusion small T" U U2>(|A|/2) 2>(|A|/2) D D {z {z ££CC: :z z==vxv+ vv2 - Viv Viv2, vvx £ T TA , vv2 G GT T}}.. T" Since, withsome some5 > >0, 0, Since, with 2

\\(zl - kVlj-^W kVlj-^W <
forzz££IntT, IntT, \z\ \z\> >5k, 5k, for

we find, with with5Q 5Q = 6/62, 6/62,for forall allAA IntT", £1 £1>> |A| |A|>>Sok Sokand and z G we find, ^ ^IntT", zG F\,F\, )-1, 5

The idea ideato to consider considersuch suchequations equations arises arises fromfrom Sobolevskii Sobolevskii [150]. [150]. The will be be chosen chosen later, later,depending depending on on A. A. It will

6

(3.8) (3.8)

3.2. ESTIMATES

FOR FOR DISCRETE DISCRETE SEMIGROUPS SEMIGROUPS

57

Uzi-m^y^KCizi-1.

(3.9)

and

It then follows from (3.7), (3.7), (3.8), (3.8), and and (3.9) (3.9) that that if A 0 I n t T " and ft > |A| > Sok, then

||£|| < C j (|A| + \z\Yl\z\-1 \dz\ < C\\\~\

(3.10)

Also, using similar reasonings reasonings we we see see that that (3.8) (3.8) and and (3.9) (3.9) are are valid valid for for AA ^ Int (T" U V{5{)) as well, well, so so that that we we find find ||£|| < C(X -)- [A|)—x

if A ^ I n t T " and |A| >6V

(3.11)

Combining (3.10) and (3.11) (3.11) thus thus yields yields 1

for A £ IntT", |A| > S Qk.

(3.12)

A^-condition, by by Lemma Lemma 1.1 Next, since both 2li ;n and 2l 2;n satisfy the A^-condition, it follows that j = 1,2, (3.13) (3.13) ||fc2li;n|| < C, which further yields, combined combined with with (3.5), (3.5), \\U\\ < C\\Com (fc2li ;n|£)||.

(3.14) (3.14)

representation To bound suitably the right-hand right-hand side side of (3.14), we use the representation Com (Jfe2l1.n|£) = — / G(X;z)dz,

(3.15) (3.15)

with

G(A; z) = fc2G0(A; z) (1 - z)Com (2t 1;n|2l2;n) G0(A; z) (zl - feS^n)" 1, and with GQ(X;Z) just the same as above. above. In fact, (3.15) is a consequence of (3.7) and of the following following identity, identity, for for ££ii,,££ 2 G B(X), Com (£i|(AI - £ 2 r X ) = (A/ - £ 2 ) " J Com (£i|£ 2 ) (A/ - £ 2 )~ 1 -

(3.16)

enough.7 For our below needs, the the parameter parameter SQ > 0 should be chosen large enough. Observe however that if <5Q <5Q is enlarged, the estimates estimates (3.8) (3.8) and and (3.9) (3.9) remain remain 7

Clearly, So can be taken as as large large as as needed needed choosing choosing the the above above 55 > 0 sufficiently large. Also, we keep the parameter K in conformity with 5o via via the the relation relation 5i 5i = 5QK, where 5i is always unchanged.

58

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTING PROBLEMS PROBLEMS

valid 8 with the same C's as for the former size of So. Moreover, we have, for k G (0, K], X g Int (T" (T" U U V(6ok)), and z G F\, since the operators 2lj ;n , j = 1,2, both satisfy the Af-condition and \z\ > 6260k, that + 60I)\\
(3.17) (3.17)

and ||fe(a1;n + <J0/) (zl - k^n)-11|

< C,

(3.18)

where no constant C depends on o. Also, note that by Lemma 1.5 we can apply hypothesis C O M 1 [ K ] with no — SQ while no constant C in (3.2) depends on SQ, whichever size it has. 9 Using the above observations, we obtain by hypothesis C O M 1 [ K ] and (3.8), (3.9), (3.17), (3.18), (3.18), for k G and z G T A, (Q,K), \£lnt(V'uV(8 0k)),

+ Cfc 1-K||G0(A; z) (zl - m^rY^WGiiX;

z)\\ K,

(3.19)

where d(A; z) = /c(2l1;n + Sol) G0(X; z) (zl -

mnil)-\

In order to bound the norm of G\(\, z) in a suitable way, we proceed from the identity

- z) G0(A; z)Com (a 2 ; n|ai ; n ) G0(A; z) {zl

,-1

which is a consequence of (3.16). Taking norms of both sides of this equation and using next C O M 1 [ K ] , (3.8), (3.9), (3.17), (3.18), (3.18), and the trivial estimate, for all u G X,

we obtain, for k G (0, K], A ^ Int (T" (T"U U V(50k)), and z G T A,

||Gi(A;z)|| < GdAl + l z D ^ + < C([A| 4- kl)- 1 -f8

As pointed out above, the restriction |A| < Si need not, in fact, be imposed for these estimates to hold. 9 The size of <5o should be sufficiently large.

3.1. ESTIMATES

FOR DISCRETE SEMIGROUPS SEMIGROUPS

59

where no constant C depends on <5o<5o- Taking So > 0 sufficiently large we now have (3.20) )-1. (3.20) Combining this equation equation with with (3.19) (3.19) leads leads to the following estimate, for Int (T" U V(6ok)), and z € T A, k€(0,K],\
which further implies, after after a simple estimation on the basis of (3.15), for Jfc € (0, K] and A £ Int (T" U V(S ok)), that 11Com (&2li ;n |£)|| < Ck1~K\\\K~l

< CSQ" ,

where SQ > 0 may be taken as large as needed while no constant C depends gives, with with So So sufficiently sufficiently large, large, since since on So. Inserted in (3.15), this result gives, K < 1, 1

K-\

—

0

— o'

thus, for A g Int (T" U V(S ok)),

(3.6) is uniquely solvable and, It follows from (3.12) and (3.21) that equation (3.6)

for

\£lnt(r"UV(S ok)), < CIA!" 1.

(3.22) (3.22)

shows that that Furthermore, a simple calculation shows (XI - fc2l2;n) Go(A; z) (zl -

fc^)- 1

- (/ - m 2]n) G0(A; z) (zl - k^n)-1 fc2li;n = (zi - mx,n)-1 + (i - m2yn) GO(A; z). With this fact in mind and again using (3.7) (3.7) we we get, get, for A ^ Int (T"{JV(Sok)), (A/ - A:2l2;n) C-(I- fc2l2;n) C fc2li;n

= ^-. I {zl-m1.n)dz + ^-. I (I-m2,n)Go(\;z)dz

60

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTING

PROBLEMS PROBLEMS

since the second integral integral vanishes. vanishes. ItIt isis then then not not hard hard to to see see that that by by the the definition of C and U we we have, have, for for A A££ Int Int (T" (T" U U V(S V(Sok)), {\I-k%n)1l = I.

(3.23) (3.23)

Similarly, it can be shown shown that that there there exists exists an an operator operator itit GGB(X) B(X) (depending (depending for A A ^^ Int Int (T" (T" U U T>(Sok)), T>(Sok)), on A) such that with 5Q > 0 sufficiently large, for n(\i-ksin) = i.

(3.24) (3.24)

Now we conclude_by (3.23) (3.23) and and (3.24) (3.24) (cf. (cf. Hille Hille & & Phillips Phillips [84, [84, Paragraph Paragraph 1.13]) that n = TZ= (Al-fcOn)(Al-fcOn)- 1 for A g Int (T"UX>(<J ofc)), which together with (3.22) shows that the the operator operator 2l 2ln satisfies the A^-condition A^-condition and and the the claim thus follows by Theorem Theorem 2.3. 2.3. Next we show a slightly slightly different different stability stability estimate estimate in in comparison comparison with with (3.4). This new estimate is of of independent independent interest interest and and isisused used in inthe the stability stability analysis of problem (1.1), (1.1), (3.1) (3.1) in in the the non-autonomous non-autonomous case. case. Theorem 3.2. Let the the conditions conditions of of Theorem Theorem 3.1 3.1 be be satisfied satisfied and and let let inin addition hypothesis C O M 1 [ K ] hold for some K G [0,1). Then, with any any fixed no > 0 sufficiently large large and and with with any any fixed fixed ££ GG [0,1], for all k G (0, K] and ti,tn G £}&, we have

3=1

Proof. Let /izo > 0 be chosen sufficiently sufficiently large large in in order order to to make make our our below below argument correct. In particular, particular, this this fact fact yields yields that that the the fractional fractional powers powers (2lj.n + iM)I)*, £ > 0, of the operators (2lj ;n + Mo-0, J = 1,2, are defined defined and and bounded on 36. Also, we we fix fix arbitrarily arbitrarily an an r\r\G G [0,1). [0,1). Let further C, T", F\, and and (?o(A; (?o(A;z) z) be be defined defined in in the the same same way way as as inin the the proof of Theorem 3.1. Using Using then then (3.8), (3.8), (3.9), (3.9), (3.17), (3.17), (3.18), (3.18), the the convexity convexity inequality (A. 18), and the the above above argument, argument, we weget, get, for for AA^^ Int Int (Y"u£>(/zo&)) (Y"u£>(/zo&)) > and z G T\,

||^(2li;n + toiyW - k^n)- 1 II < Clz^1

(3.26)

and

< C(|A| + H)"- 1 .

(3.27)

3.1. ESTIMATES

FOR DISCRETE SEMIGROUPS SEMIGROUPS

61

Also, by (3.27) and (3.7), acting as in the preceding proof and noting noting that that 7] < 1, we find, for A £ Int (T" U V(p, Qk)), ||^(2l2;n + fi0I)vC\\ < CIAI"- 1.

(3.28)

Similarly, changing the order of the resolvents on the right-hand side of (2.4) and further using the following identity, for £i, £2 €

Com (( £i)- 1 (AJ - £ 2 )- 1 Com (-C2I-C1) (A/ - £ 2 ) ~ 1 ( ^ - £ i ) ~ \

it is not hard to see, with the aid of (3.26), COMl[/t], (3.8), (3.8), (3.17), (3.17), and (3.20), that for A g Int (T" U V(fi ok)),

WW^n + ^iyCW^ClXr 1.

(3.29)

Therefore, by virtue of (3.28), (3.29), and (3.21) and using (3.6) we have, for A£lnt(T"uP(MoAO), A£lnt(T"uP(MoAO), P"(2l i ; n + Mo/) T '(A/-/c2l n )- 1 ||
j = l,2.

(3.30)

We need to derive some different resolvent resolvent estimates estimates as well. With C just the same as above and with V given by V = Com solved for 1Z e B(3£), using the operator equation, solved

and acting as above (see the proof of Theorem 3.1 and the preceding part of the present proof), we now obtain, for A ^ Int (Y" 1

,

j = 1,2.

(3.31)

Next we denote Hj = fc(aj!n + fiol) (XI -

and Hj = {xi - m n)-lk{%,n

+ /xoi) {xi -

As a trivial consequence of (3.30) and (3.31), both taken with with 7777 = 1/2, we have, for A £ Int (T" U 2?(/x0fc)), 1

,

j = l,2.

(3.32)

62

CHAPTER3.3. OPERATOR OPERATOR SPLITTING PROBLEMS CHAPTER SPLITTING PROBLEMS

Since 2li 2li;n and and 2l2;n both both satisfy satisfy the A^-condition, A^-condition, it follows follows by Lemma Lemma 1.1 Since ||fc%;n ||
53

u||. (3.33) (3.33) u||.

j=i

We now now apply apply (3.16) (3.16) to obtain obtain H1=H1

(XI -- mnylk2Com + (XI

(2l1;n fc2li;n) (XI -- fc2l fc2ln)-2, (2l 1 ;n|2l 2;n ) (/ - fc2li

aid of of (3.32), (3.32), C O OM M22[[K K]],,1 0 (3.31), (3.31), and (3.33), (3.33), for which implies, implies, with with the aid which \\Hi\\ \\Hi\\

< 2

x 2

1

1

5311^-11, + C/zg- 5311^-11,

(3.34) (3.34)

where no constant constant C depends depends on /^o/^o- Similarly Similarly we get, get, for for A A^^ Int Int(T" (T"UU where

WiH.

(3.35) (3.35)

Therefore, summing summing the inequalities inequalities (3.34) (3.34) and and (3.35) (3.35) and and taking taking /Uo > > 00 Therefore, sufficiently large, large,we find, for A A^^ Int Int(T" (T"UUV(/j,ok)), V(/j,ok)), sufficiently

ill
By Lemma Lemma 1.5 we wecan cantake for po po in in (3.3) (3.3)the thesame same value value as in in the the present present proof. proof.

3.2. STABILITY OF OPERATOR OPERATOR SPLITTING SPLITTING PROBLEMS PROBLEMS

3.2

63 63

Stability of operator operator splitting splitting problems problems

Now we are in a position position to to state state our our results results on on stability stability for for problem problem (1.1), (1.1), 11 (3.1)In the autonomous case, case, using using straightforwardly straightforwardly Theorems Theorems 3.1 3.1 and and 3.2, 3.2, we have the following. Theorem 3.3. Suppose Suppose that that the the operators operators 2lj ;n , j = 1,2, are both indeindesatisfies the the h\-condition. h\-condition. Let Let also also the the pendent of t n and each of them satisfies hypotheses C O M 1 [ K ] and C O M 2 [ K ] hold for some K £ [0,1). Then, with any fixed {1Q > 0 sufficiently large and and with with any any fixed fixed ££ €€ [0,1], the solution estimate, for for all all kk 66 (0, (0,K] K] and and ttn G fife, Yn of (1-1), (3.1) satisfies the estimate,

.7=1 /

n-1

V

1=0

//

At the same time the first first term term on on the the left left can can be beestimated estimated as as stated stated for for any any fixed £ > 0, even though though we we have have not not assumed assumed COM2[/t]. COM2[/t]. Further, we turn to the the general general case case where where the the operators operators 21? ;n , j — 1,2, may depend on t n. Together with problem problem (1.1), (1.1), (3.1) (3.1) we we will will examine examine the the following following perperturbed problem, with some some 23j ;n £ &(%), j — 1> 2, dYn + 2t n y n = Fn

for tn G Clk,

Yo = y°,

(3.37)

where 2ln = 2tl ;n + 2t 2;n - fca2in2il;n and 2lj; n = 2l j;n +
(3.38) (3.38)

Clearly, denoting (^

I

it n < I,

admits the the representation representation the solution Y n of (3.37), (3.38) admits n-l ~\r

it

\j\j

.

7

X

^^ jj

771 771

i

1=0

This formula will be used used in in our our subsequent subsequent analysis. analysis. x

We recall that it is assumed assumed that that £2 £2n = 9t = /.

—

^^ \\

// OO

QQ QQ

\\

64

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTING

PROBLEMS PROBLEMS

Theorem 3.4. Let the operators 2lj ;n , j — 1,2, both obey the Aj-condition Aj-condition and C O M 2 [ K ] be satisfied for some K G and let the hypotheses COMl[/t] and [0,1). Let further the following following Holder-continuity Holder-continuity type type condition condition hold, with some iii > 0,i? G (0,1], and 71 G [0,1), for all j = 1,2, k G G (0,K], i n ,ii € fi/c, and ix € X,

H(aj;n - ai;Z) w|| < Cifn_Z|||(2ij;n +Ml /)u|| + c||(a i;n + tui) u p H 1 - * . (3.40) Apart from everything else, else, suppose suppose that that with with some some 70 70 6 [0,1), for all k 6 (0, K], tn £ fife, and u £ X,

||® i ; n «|| < C||(2l j;n + /xi/) 70 ^||,

j = 1,2.

(3.41)

Then, for any fixed no > 0 sufficiently large and for any fixed £ G [0,1), the solution Yn of both problem (1.1), (3.1) and problem (3.37), (3.37), (3.38) (3.38) satisfies satisfies the estimate (3.36). Proof. Let there be fixed ^0 > 0 sufficiently large and £ G [0,1). By (A.18), (3.40), Lemma 1.5,12 and Theorem 3.2, it follows that for j — 1,2 and tn,ti G fifc such that t\
r ; ||^
(3.42)

Similarly, applying the trivial trivial inequality, inequality, for all tn,ti G f 4 and u G 3t,

u||,

(3.43) (3.43)

with the aid of (3.13) we get, for ti < tn,

Next note that by virtue of (3.13) and (3.40), the above estimate (3.43) implies as well, for all tn,ti G £ \ and u G X, 2

(3.45) 12

If the lemma is applied with 2l j;Tl in place of 2l n .

3.2. STABILITY

OF OPERATOR SPLITTING SPLITTING PROBLEMS PROBLEMS

65

Using now (3.42), (3.44), (3.45), and the identity (2.28), with both both sides sides multiplied from the left consecutively by (2l n + /io-0^ and by (2ln;j + Mo-O^> and applying finally Lemma Lemma 2.1 yields that (3.40) holds holds for the solution of (1.1), (3.1), in view of (1.4). In particular, this fact implies implies that that for ti < t n,

Ct~il+V

(3.46)

which is applied in the sequel. operators iXnj. and il^; are Furthermore, note that the discrete evolution operators connected by the identity, for 0 < t\
iU-l.i = iin-l,J + J2^n-hi+S
(3-47)

q=l

Since with the aid of (3.41) it follows that for ti < t q, 2

? - 1 ,i||.

(3.48)

Recalling now that both both 2li 2li ;n and 2l2,n satisfy the A^-condition, by Lemmas 1.1 and 1.5 we have, for all u G X,

Using this estimate as well as (3.46) with 70 substituted for £, instead of (3.48) we get, for t\ < t q,

Applying this inequality inequality together together with with (3.46) (3.46) for an estimation on the basis of (3.47), with both sides sides multiplied multiplied from from the left by (2ln + /zo-O^, we obtain,

66

CHAPTER CHAPTER 3. OPERATOR SPLITTING SPLITTING

PROBLEMS PROBLEMS

for 0 < U < t n, n-1

||(2tn + M 0 /) € itn-l,/|| < Ct~il+1 + CkJ^ || (2U + M 0 /) S Hn-l, g + l|| t-™+1. q=l

Since £ < 1 and 70 < 1, by by Lemma 2.1 it follows from the last inequality that for ti
Finally, with the aid of these estimates, a trivial estimation on the basis of (3.39), with both sides multiplied multiplied from from the left consequtively by (2ln + /io^)^ and by (2lj ;n + Mo-0^ J = 1>2, shows that (3.40) holds holds for the solution of problem (3.37), (3.38) as well.

3.3

A symmetric operator operator splitting splitting method method

In this section we consider a problem of the form (1.1) for which

On =

22t 1;n + 22l 2;n - k{K\.n + 22t 1;n2l2;n + 22l 2;n 2l 1;n + 2 l | j +fc2(2l1;n2l2.n + 22l 1;n 2l 2;n 2li ;n + Ol ; n a 1 ; n ) -fc32l1;n2l2;n2l1;n.

(3.49) (3.49)

Clearly, using this representation representation leads leads to the following operator splitting splitting method y n + 1 = (/-fc2i 1 ; n )(/-fc2i 2 ; n ) 2 (i-A;2i 1 ; T l )y n + ^ F n ,

tn G fifc) y 0 = y°(3.50) It is seen that the transition operator possesses possesses now a certain symmetry and this circumstance allows allows us to construct second order approximation approximation schemes schemes based on the principle of operator splitting. In the non-autonomous case it is expedient to examine problem (1.1), (1.1), (3.49) (3.49) (or, equivalently, problem (3.50)) with 2lj ;n substituted for %,«, where 2t j;n = 2lj ;n + 5Sj ;n with some Our results here will be that under the same restrictions on the operators 2lj ;n , j — 1, 2, as above, it is possible to obtain stability estimates estimates for problem

3.3. A SYMMETRIC

OPERATOR OPERATOR SPLITTING SPLITTING METHOD METHOD

67 67

(1.1), (3.49), which are analogous analogous to to those those already already found found for for (1.1), (1.1), (3.1). (3.1). The first result concerns concerns the the case case where where the the operators operators 2lj ;n , j = 1,2, are independent of t n and the second one is stated stated in in the the general general case case supposing supposing that the dependence on on ttn is allowed. Theorem 3.5. Let the operators operators ^ij^ij-n, j = 1,2, be subject to the the conditions conditions of Theorem 3.3. 13 Then, with any fixed [IQ > 0 sufficiently large and and with with any any fixed £ G [0,1], the solution Y n of (1.1), (3.49) satisfies the stability stability estimate estimate (3.36). Moreover, the first term term on on the the left-hand left-hand side side of of (3.36) (3.36) isis estimated estimated as stated for any fixed ££ >> 00 even even though though COM2[/c] COM2[/c] isis not not assumed. assumed. Theorem 3.6. Let the operators operators 2l 2l J;n , j = 1,2, be subject to the the conditions conditions of Theorem 3.4- Then, for for any any fixed fixed /io /io >> 00 sufficiently sufficiently large large and and for for any any estimate (3.36). (3.36). fixed £ G [0,1), the solution solution YYn of (1.1), (3-49) satisfies the estimate Furthermore, if*Bj- n, j = 1,2, are small perturbations perturbations to to 2tj ;n , respectively, in the sense that condition condition (3-41) (3-41) is is fulfilled, fulfilled, then then (3.36) (3.36) still still holds holds ifif by by YYn we mean the solution of of the the perturbed perturbed problem problem (1.1), (3-49), that is, with 2t j;n = %j-n +
and

iin = (/-fc2l 2 where 2l j;n = 22l j;n - m2j]n,

i = 1,2.

It is not hard to see that that

The fact that 2li n and 2l2 ;n both satisfy the A^-condition A^-condition implies implies that that the the same is true for 2l J;n , j = 1,2, which can be be shown shown similarly similarly to to (3.12), (3.12), and and that for all A; G (0, K) and and ttn G O,k,

fc2tj;n)-1||
j = 1,2.

68

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTING PROBLEMS PROBLEMS

The last estimate furthermore furthermore shows shows that that with with any fixed /Lto > 0 sufficiently large, for j = 1, 2, u G X, and and \\ £££*> £*> ||

(3.51)

and (3-52) Furthermore, with the aid of the following identity, where where we use the notation £ = Com (2li ;n |2l 2;n ), Com (3Li;n|3l2;n) = 4C - 2A;(2l1;n + 2l 2;n) £ - 2k€ (2l 1;n + 2l 2;n )

using (3.13), (3.51), (3.52), (3.52), and the above permutation techniques techniques we can prove that the hypotheses COMl[/t] and COM2[«;] are still satisfied with 2tj ;n substituted for 2lj ;n - We will show, for example, a possible estimation of the quantity |(x, A;2li;n<£u)| when deriving (3.2) with 2lj ;n in place of 2l,;nBy C O M 1 [ K ] and (3.13) we have, for j = 1, 2,

where, as above, 2lj. n can be replaced by (2li-n + /xol), 14 and further 2lj ;n can be replaced by 2l J;n for j — 1,2, by (3.51) and (3.52). Therefore, with the aid of the above reasonings we conclude that under the conditions of Theorems 3.1 and 3.2, respectively, the estimates (3.4) and (3.25) hold even with with 2l n , 2tj;n) and ii^ substituted for 2l n , 2lj;T1, and iin, respectively. Now using the identity (/ - /c2li;n)2tn — %n(I — ^2li ;n) for any £ € N, it is not hard to see that (3.4) will hold also with iin — (7-fc2li ;ri ) (I-A:2l 2;n ) 2 (I-£;2li; n ) substituted for i ^ for any fixed £ € NU{0} and by the convexity inequality (A. (A.18) 18) for any fixed £ > 0. Also, since ||21 - k%. n\\ < C, we have for j = 1,2 and u e X, X,

It thus follows that the estimate (3.25) holds even even with with iin substituted for On, both for £ = 0 and £ — 1 and, and, applying (A.18), for any fixed £ € [0,1]. 14

Without loss of generality fj.o can be assumed to be positive and sufficiently large.

3.4. COMMENTS AND AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL

REMARKS REMARKS

69 69

Therefore using the above above observations, observations, Theorem Theorem 3.5 3.5 now now straightforwardly straightforwardly follows. Also, employing employing analogues analogues of of the the identities identities (2.28) (2.28) and and (3.47) (3.47) inin the the case where iln plays the the role role of of On On and and following following the the proof proof of of Theorem Theorem 3.4 3.4 shows the assertion of Theorem Theorem 3.6 3.6 as as well. well.

3.4

Comments and bibliographical bibliographical remarks remarks

Sections 3.1 and 3.2: 3.2: The The present present considerations considerations follow follow in in essence essence our our earlier work [13, 15, 34]. 34]. Section 3.3: Similar results results for for the the unperturbed unperturbed problem problem (1.1), (1.1), (3.49) (3.49) have have been announced in [21]. [21].

This Page is Intentionally Left Blank

Chapter 4

Equations with Memory Memory In this chapter we extend the field of our study by including equations with a memory term in our scope. Such equations arise, arise, for example, from discretization of integro-differential equations equations (see Part IV). More precisely, instead of problem (1.1) we now deal with the following discrete Cauchy problem problem iYi + Fn

iovtnenk,

Y Q = y°,

(4.1)

1=0

with ty nj € B(3L) given for ti < t n (depending generally on k) and all other objects retaining their former former meaning meaning (see (see Chapter Chapter 1). In view of the many important applications (particularly, (particularly, in the case where 2l n depends on tn, but not exclusively) it is expedient to consider together with problem problem (4.1) its perturbed version, with with some some *B *Bn € B(X), lYl + Fn

iovtneClk,

Y 0 = y°. (4.2)

1=0

The problems (4.1) and (4.2) differ from (1.1) and (2.32), respectively, by the presence of the memory term SILo^Pn.*-^ SILo^Pn.*-^ m * n e corresponding equation. equation. 1 Obviously, both (4.1) and (4.2) are uniquely solvable. To the problems (4.1) and (4.2) we associate the problems for the corresponding homogeneous equations equations with with floating floating initial initial condition, condition, for tn,tj G fifc with tj < t n, ^n,iYi,

Misgiven,

(4.3)

1=3

the sake of simplicity throughout this this chapter chapter it is accepted that £!„ = 9\ = I.

71

72 72

CHAPTER 4. EQUATIONS EQUATIONS WITH MEMORY CHAPTER 4. WITH MEMORY

and and iYu Misgiven. Misgiven. iYu

(4.4) (4.4)

1=3 1=3

ofthese theseproblems problems notgive giverise riseto todoubts doubtsas as The unique unique solvability of doesdoes not The solvability well. Let 2Bnn,j,j and and2B 2Bnnj be bethe thesolution solutionoperators operators forthe theproblems problems (4.3) well. Let 2B for (4.3) and (4.4), respectively, is, that forYY thesolution solutionof of(4.3), (4.3), and (4.4), respectively, that is, for n the n Yn^VOn-ijYj, Yn^VOn-ijYj,

(4.5)(4.5)

and forYY thesolution solutionof of(4.4), (4.4), and for n the n Yn =Wn-uYj. Wn-uYj. Y n=

(4.6)(4.6)

Then, bylinearity linearityand andcausality causalityit itfollows followsthat that thesolution solutionY Ynn of ofproblem problem Then, by the (4.1)admits admits therepresentation representation the (4.1) n--ll n

Y = =2B 2B_i _iny° ny°+ +kkN NW W_i_i-+iF-+iFY

(4-7) (4-7)

3=0 3=0

thesolution solutiony ynn of of(4.2) (4.2)can can berepresented represented by whilethe be by while n-\ n-\

Y =W =W i ooy° y° i + +kk\ \ 22J 22J_i _i-+iF-+iFY

8) (4 (4 8)

It is iseasily easilyseen seen (4.7)and and (4.8) takethe thesame sameform form as(1.4) (1.4)and and (2.33), It thatthat (4.7) (4.8) take as (2.33), respectively. Therefore allwe we needfor forshowing showing stability isto tofind find aasuitable suitable all need stability is respectively. Therefore estimatefor forthe theoperators operators _ij and and2H 2Hnn_ij, _ij, which whichmeans means infact, fact, estimate 2U2U that,that, in nn_ij ouraims aimsit itsuffices sufficesto toanalyse analysethe theproblems problems (4.3)and and (4.4) only. for our (4.3) (4.4) only. for For furthersuccess success wenow nowderive derivesuitable suitable equations for2B 2Bnn_ij _ijand and For further we equations for 2U Since,by byanalogy analogywith with (1.4), forYY thesolution solutionof of(4.3) (4.3)we we have 2U nn_ij. _ij. Since, (1.4), for have n the n n n --l l

Y

I

I

—i tt nn_ _i i YY-++k ^S~^ ^S~^ k i li nl -ni - zi + iz y+ i^ y^ P^/ Y ^Y P / f fo or r t
1=3 1=3

whereitnj where itnjis isthe thediscrete discreteevolution evolution operator operator for forproblem problem(1.1), (1.1), usingusing (4.5) (4.5) yields, fortjtj<< , t yields, for tn n n-\ n-\

l

l

^n—1,1+1 ^n—1,1+1 // j j

1=3 1=3

1=3 1=3

AUJ AUJ

q-l,ji q-l,ji

73 73 the order ofsummation, summation, or,equivalently, equivalently, or, changing changing the order of n-ln-l n-ln-l

W j --iln-ij iln-ij +k J2 k J2 XXX ^ -^ - U(4.9) - (4.9) Wnn--X + X - W- ^W ^-U Xj 1=3 q=l 1=3 q=l

Similarly, with the (4.6) ofit itfollows follows Similarly, with the aidaid of (4.6) that that n-ln-l n-ln-l

W -\j = iln-lj = iln-lj + k^ J^iL-l,q+lVq^l-l,l, ** * jnn < > Wnn-\j + k^ J^iL-l,q+lVq^l-l,l, *j < >

(4-10) (4-10)

1=3 q=l q=l 1=3

where the discrete evolution for operator problem (2.32). where iinj iinj isis the discrete evolution operator for problem (2.32). Inthe the next two sections we willexamine examine the stability of the problems next two sections we will the stability of the problems In and (4.2) using the equations (4.9) and (4.10) and distinguishing be(4.1) (4.1) and (4.2) using the equations (4.9) and (4.10) and distinguishing between the cases where the operator 2l depends and does not depend on . tween the cases where the operator 2l and does not depend on ttn n. ndepends n The main idea employed isthat that the memory Y^i=j ^Pn,/^ isconsidered considered The main idea employed is the memory term term Y^i=j ^Pn,/^ is (nonlocal) perturbation to the principal l ^ or or 2(2l (2l t nn+ +^& ^& )Y asaa(nonlocal) perturbation to the principal term term 2t nnl^ as n)Y n.. n n Acting in such way leads to estimate discrete analogue Acting in such aaway leads to anan estimate whichwhich presents presents aadiscrete analogue of an integral inequality and solved be by by using suitable discrete verof an integral inequality and cancan be solved using aasuitable discrete version of Gronwall's Note that these are applicable techniques inthe the sion of Gronwall's lemma.lemma. Note that these techniques are applicable in case of finite intervals It is only. therefore assumed T
< Ct~i Y <]>n, <]>n, Wqd Wnj< Ct~i Y ,, Wnj j+1 ++ qW qd j+1 q

(4.11) (4.11)

1=3 1=3

where (f) ,q^^ 00;; 00 <<^ <^ n<— — n 1, 1,are are givenand and satisfy the condition, given satisfy the condition, with with where (f) nn,q some TX,'K G (0,T] [0,1), for rj G kG G(0,K) (0,K) and tj
n,qt~i < whenever ^m < n,qt~i VV whenever ttss^ T\.T\. j+l m < j+l < q=m q=m

Then, for Then, for allall k kG G(0, (0, K) K)and and tj,t tj,t G fife fife such such that tj that tj< t> ,q generally generally depend depend on on k. k. nn,q

(4.12)(4.12)

74

CHAPTER EQUATIONS WITH MEMORY CHAPTER 4. 4. EQUATIONS WITH MEMORY

Proof. Denoting Denotingwnj — — t^_j t^_j+1 Proof. +1wnj, with tjtj << tnt, with

it follows follows from from(4.11) (4.11)that thatfor for tj,t tj,tn E E £lk £lk it n - ll

wn


'S^d> d>n t~^ t~^ 'S^

w w

(4-13) (4-13)

4=3 4=3

Clk,we we put, put,for forI I>>1,1, Further, giventj ££ Clk, Further, given T;

(tj ++(l(l- l)Ti,min{T, l)Ti,min{T,tj + + = (tj

Clearly, by (4.12) (4.12) and and (4.13) (4.13) we we have have Clearly, by l

max wn j <
4.1 The Thegeneral general case case 4.1 the general general case casewhere where2l 2ln may mayvary vary depending dependingon on ttn, our our analysis analysiswill willbe be In the based on on making making use useof ofsome some results resultsof of Section Section 2.3 2.3where where the the stability stability of of the the based discrete evolution evolutionoperators operators andilnj ilnj for for the theproblems problems (1.1) (1.1)and and(2.32), (2.32), discrete iinjiinjand respectively,is examined. examined. In In principle, principle,each eachstability stability assertion assertion of of Section Section2.3 2.3 respectively, can be be employed employed to to get get an ananalogue analogue for for either either problem problem(4.1) (4.1)or or (4.2). (4.2). Here Here can we restrict restrict ourselves ourselvesto to considering consideringsome someselected selected situations only. we situations only. Letseminorms seminorms ]|[-]|[ ]|[-]|[ \\n,11 |i;n |i;nand andthethe operators2l 2lnn, T hheeoorreemm 44..11.. Let \\n, operators n, Q3n satisfy satisfy the the conditions conditions posed posedin in the theformer former part partof of the thestatement statement of of TheTheorem 2.10, with with some some -d-d€ (0,1] (0,1] and and£,71,70 £,71,70 G G [0,1). [0,1). Suppose Supposefurther furtherthat that orem 2.10, an seminorms and |||||| |*3 |*3; nnform form respectively respectivelyK\{^z)K\{^z)- an Ai(*;7*3)seminorms |||||| ||33; nn and d Ai(*;7*3)the operator operator2l 2ln for for some some 73,7*3 73,7*3> 00 such such that that7*3 7*3 << concordantpairs pairswith withthe concordant min{i9,1 — 71,1 71,1 — — 70,1 70,1 — — 73}, 73}, and and that that for for all allxx SS%*> %*> u ££ X, X, and andti,t ti,tn G G Clk Clk min{i9,1 — such that that t\
(4-14) (4-14)

where irirnj > where > 00 satisfy satisfy the the conditions, conditions, for for all allkkSS(0, (0, K] K]and and tj tj< t< < tts < < m t< tn> tn>

for for any any r\ r\>>00there there exists existsT\ T\ G G(0,T] (0,T]such such that that whenever wheneverts-m < < T\, T\, ll vv^^nn- l-.l-.(-7( 37+37+»73»)3 ) a-73 a-73

4.1. THE GENERAL CASE

75 75

and n-ln-l

TTien the solution Yn of both (4-1) and (4-2) satisfies the stability estimate (2.81). Proof. In view of the representations (4.7) and (4.8) it suffices to show the estimate, for all u G X and tj < tn,

Since both terms on the left-hand side of (4.17) are similarly estimated, we will demonstrate our argument for showing the desired result for the first term only. To that end, we first note by Theorem 2.10 that with P — L(n + ?)/2j i forall u € X and tj
and using as well (4.14) and Theorem 2.11

^

u

U

j

.

(4.19) (4.19)

With the aid of (4.18) and (4.19), since ||| | 3 ; n is a seminorm, a simple estimation on the basis of formula (4.9) 3 gives, taking (4.14) into into account, account, for all tj < tn, n-ln-l

SUp lWn-ljuh;n < Ct^j+1 + Ck ^J^ ^ - t ^ ^ =1

ll«ll

l=j l=j q=l

SU

P

M =1

11^1-1,^^.

(4.20) account, Applying now Lemma 4.1 to this inequality and taking (4.15) into account, it follows that since 73 + 7*3 < 1,

sup |2B n _ l j i «| 3 . n < Ct~J j+v tj < tn. (4.21) ll«ll=i this With this fact in mind and using once again the above method, only this „ in place of ||| §3 ;n , instead of (4.20) we obtain the inequality, time with for tj < tn, sup Nl = l 3

' ''

l=j q=l

When estimating |2Un-i,ju||n, |2Un-i,ju||n, (4.10) (4.10) is used instead.

76

CHAPTER CHAPTER 4. 4. EQUATIONS EQUATIONS WITH WITH MEMORY MEMORY

thus the desired estimate estimate for for ][2B ][2Bn_iju][n follows after a trivial calculation calculation using (4.21) and (4.16) as as well. well. T h e o r e m 4.2. Let seminorms seminorms |[-]|[n, ||| || n , ||| | 1 ; n , ||| ||*;n, ||| \\00.nn, and II ||*0;n and the operators 2l Theorem 2.12, 2.12, 2l n , *Bn satisfy the conditions of Theorem with some J G N, £ >> 0, 0, 7,71,7*,70,7*0 7,71,7*,70,7*0 G G [0,1), [0,1), and and T?,#I,I?O G [0,1] SMC/I ^/ia^ 7* < £0, 7o + 7*o 7*o << 11 ++ i?o> i?o> m a x {£>7,7i} < £1, w^ere w^ere £0 £0 == l + min{i9 —7,i?i — 71} and£i = 1 — 7* + min{'#,i9i}. Let further further seminorms seminorms 1 ||3;n anc ^ 1 l||*3;n form respectively Aj(73)- and and Aj(*; Aj(*; 7*3)-concordant 7*3)-concordant pairs pairs with the operator 2l n , for some 73,7*3 > 0 such such that that 73 73 << £1, £1, 7*3 7*3 << £0, £0, and and Zei ^Pn,z be under the conditions conditions (4-14), (4-14), (4-15), (4-15), and and (4-16). (4-16). Then Then the the solution solution Yn of problem (4-1) satisfies satisfies (2.81). (2.81). IfIf in in addition addition 7*0 7*0 << £0, £0, 7o 7o << £i> £i> and and max{£,73} < 1 + i?o — 7*0, then (2.81) remains remains valid valid for for the the solution solution of of problem (4-2) as well. The proof is similar to that that of of Theorem Theorem 4.1, 4.1, with with reference reference to to Theorem Theorem 2.12 for this once.

4.2

A particular case

Here we consider, as aa particular particular case, case, the the situation situation when when the the operator operator 2ln = 21 does not depend depend on on ttn. It will be seen below below that that under under this this requirement some further further developments developments are are possible. possible. Using Using the the notation notation formerly introduced in what what follows, follows, ilil n denotes the discrete semigroup semigroup gengenerated by the operator 21 21and and ilnj ilnj the the discrete discrete evolution evolution operator operator generated generated by2l + > 00 1 G fl k and with ty n>l = (21 + / / o / ) " ^ , / , for all k E (0,K] and tj,t tj,tm,ts,tn

subject to tj < t m < ts < tn, for any 77 > 0 there exists exists T\ T\ G G (0,T] (0,T] such such that that whenever whenever tts-m

Zt=L (Wn-l,l\\ + k EU t-l t-lgWq,l - ^n-l,Hl) < V-

< T\,

(4-22)

We start by examining problem problem (4.1) (4.1) in in the the considered considered case. case. T h e o r e m 4.3. Let the operator^21 satisfy the K kj-condition with some J G GN N and let (4-22) hold. Then the solution Y n of problem (4-1) satisfies satisfies the the estimate, for all k G (0, (0, K] K] and and ttn G fifc; (4-23)

4.2. A PARTICULAR

CASE

77 77

Proof. By (4.7) it suffices to show that ||2Bn-ij|| < C for all tj < tn.

(4.24)

Applying now (4.9) with with 2l n = 21 gives, after a simple calculation, with with?($ ?($nj defined above, for tj < t n, n—ln—l

Wn-u

= an~j + k J2 ^ it"-"-1 (21 +mi) (yq>l -

^

1=3 1=1 n—ln—l

1=3 q=l n—ln—l

J]5]r«- 1Vi,|!!II|.lj.

(4.25)

1=3 1=1

By Theorem 2.3 we have, whenever t\ < t n __i, i,

n-l

n-1 1

9 =Z

q=l q=l

II ^ a " - 9 - 1 ^ ! ! = || ^ i i " - 9 - 1 ^ - u ) | | = ||/ - un-l\\ < c, and n-l

n - ll

IIH"-9-1! < ct n _ t < c.

Using the last three estimates for an estimation on the basis of (4.25), it follows that for all tj < tn, n-l

/

||2fln-lj|| < C + C ^ 1=3 \

_

n-l

\\^ n_U\\+kJ2tn-qWi,l-Vn-l, 1=1

Finally, applying Lemma Lemma 4.1 4.1 to this inequality and taking taking (4.22) (4.22) into into account account gives us (4.24). For the considered class of equations it is worth studying the perturbed problem (4.2) even if 2l n = 21 is independent of tn. The following result is applicable just in this case.

78

CHAPTER CHAPTER 4. 4. EQUATIONS EQUATIONS WITH WITH MEMORY MEMORY

T h e o r e m 4.4. Let the the conditions conditions of of Theorem Theorem 4-3 4-3 be be satisfied satisfied and and let, let, with with a seminorm ||| |||*o;n that forms forms aa A^(*; A^(*;7*0)-concordant 7*0)-concordant pair pair with with 21 21for for some some 7*0 € [0,1) and $0 G (0,1], the operator 2$ n be under the condition, condition, for for all all k e (Q,K], t n e Qk, and x G X*, ;n.

(4.26) (4.26)

Then the solution Y n of problem (4-2) satisfies satisfies the the stability stability estimate estimate (4-23). (4-23). Proof. The claim can be shown shown by by using, using, in in fact, fact, the the same same argument argument as as inin the proof of the preceding preceding theorem. theorem. First of all, applying (4.26), (4.26), letting letting formally formally 77 == 71 71 == 7* 7* — —70 70 == 00 and and tf — °d\ — 1, and leaving leaving 7*0 7*0 and and i?o i?o just just the the same same as as in in the the statement statement ofof the the present theorem, with the the aid aid of of Theorem Theorem 2.12 2.12 itit follows follows that that for for all all tjtj << t tn and tq < t n _ i ,

,

l

< C,

(4.27) (4.27)

since the seminorm ||2l* ||2l* ||* ||* obviously obviously forms forms aa Af Af(*; (*; l)-concordant l)-concordant pair pair with with 21. Also, in view of the identity, identity, for for t\t\
n—\ n—\

q=l

q=l q=l

it is not hard to see that that equation equation (4.10) (4.10) can can be be taken taken into into the the following following one, for tj < t n, n—\n—1

n-l,j + k

^ l=j

^ q=l

n-1

1=3 n—\n—\

+k J2 E2n-l.«+l W ~ ®«) ^n-lM-hJ-

(4-28)

Now the desired result follows, follows, using using (4.27) (4.27) for for an an estimation estimation on on the the basis basis of (4.28) and acting as in in the the proof proof of of Theorem Theorem 4.3. 4.3.

4.3. COMMENTS AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL REMARKS REMARKS

4.3

79

Comments and and bibliographical bibliographical remarks remarks

Actually, Lemma 4.1 is a simple generalization of Lemma 2.2 in Bakaev, Larsson and Thomee [40], and the proof on page 74 follows essentially the ideas from [40]. As concerns the assertions presented in Sections 4.1 and 4.2, they have never before appeared.

This Page is Intentionally Left Blank

Part II II Part

RUNGE-KUTTA RUNGE-KUTTA METHODSFOR FOR METHODS ABSTRACT PARABOLIC ABSTRACT PARABOLIC EQUATIONS EQUATIONS

This Page is Intentionally Left Blank

Chapter 5 Chapter

Discretization by Discretization by Runge-Kutta Methods Methods Runge-Kutta In the preceding preceding part part of the the book we deal with with general general stability stability assertions assertions stated evolution equations equations in discrete discrete time. time. The The main purpose purpose of the the stated for evolution present part is to to use use those results results in the the study study of Runge-Kutta Runge-Kutta methods methods present part applied for discretization discretization in time of abstract abstract linear linear parabolic parabolicproblems. problems. applied

Introductorylight light 5.1 Introductory Runge-Kutta methods are aa widespread widespread tool tool in the the numerical numerical treatment treatmentof Runge-Kutta methods ordinary differential equations. equations.Their Theiruse in in the theabstract abstract framework framework meets, meets, ordinary differential however, some some special special features featureswhich whichmainly mainlyresult resultfrom fromthe fact that that the however, operator question is generally generally unbounded unbounded or is is an an approximation approximation to an an operator in question unbounded one. The Thelast circumstance circumstance should shouldtherefore thereforebe taken taken into intoconunbounded sideration. sideration. given a complex complex Banach Banach space space X and and an an interval interval [0,T], [0,T], Let there there be given < oo. oo.We Weconsider consider the following following abstract abstract linear linearCauchy Cauchyproblem problem 0
y(O) y(O)= y°,

(5.1) (5.1)

A(t), tt ££ [0,T], [0,T], is a closed closed linear linear (generally (generally unbounded) unbounded)operator operatorin where A(t), X, with with domain domain Domi(i) Domi(i) = Vom Vom C CX X independent independent of t,1 f(t) :: [0, T] —> X forcing term, term, and y° y° ££ X X the theinitial initial condition. condition. the forcing 1

This condition condition on DomA(t) DomA(t) is accepted accepted throughout throughout Part PartII. Note that that our theory theory alfully discrete discreteapproximations approximationsto problem problem (5.1) in the case where where Dom A(t) lows us to study fully may depend depend on t but butthe theapproximation approximation to A(t) A(t)has hasdomain domain independent independent of t.

83

84 CHAPTER 5. DISCRETIZATION BY RUNGE-KUTTA METHODS METHODS Our results can also be used in the case where the operator A(t) depends on an additional parameter, h, say. In fact, under the assumption that all the starting restrictions are satisfied uniformly with respect respect to h, our below estimates are then uniformly valid with with respect respect to h as well. This kind of extension is most helpful for applications of our theory. In our considerations the main tool of analysis is reducing the starting discrete problem, which arises arises from from discretization discretization of (5.1), to the Cauchy problem for an evolution equation in discrete time. In order to use the general stability assertions of Part I, we therefore have to verify the restrictions accepted therein. We show that they can readily be verified if the operator A(t) in (5.1) satisfies some reasonable conditions conditions which which are stated below. Our investigation in this part of the book involves certain seminorms seminorms necessarily is furnished which, as a rule, depend on t; any such seminorm necessarily with t as a subscript whenever the dependence on t is generally available. 2 At the same time we keep the style of notation accepted in Part I for objects arising from discretization discretization in time and depending in general on k and t n. In what follows one of the main assumptions will be that A(t) G S(tp, a) for some ip G (0,TT/2) and a G R, both independent of t. This assumption problem. Moreover, Moreover, the allows us to think of (5.1) as a parabolic type problem. discretizations of problem (5.1) that we deal with may be considered as discrete parabolic type problems problems in some sense. At the same time we will employ as well various Holder-continuity Holder-continuity type type conditions conditions on the operator A(t). Now we pose certain necessary hypotheses, hypotheses, all of which are assumed to hold uniformly with respect respect to t, s G [0,T]. First we introduce the following idea, with some some /J,\ > 0, i? G (0,1] and with some seminorm | |t defined on Vom. hol®[#;| |t]: For all u G Vom, \\(A(t) - A(s)) u\\ < C\t - sf\\(A(t) + IHI)U\\ + C\u\t.

(5.2)

If the second term on the right is removed, we say that A(t) satisfies hypothesis hol[i9]. elliptic partial partial differential differential operator operator (or a If A(t) is a multi-dimensional elliptic suitable approximation to such an operator), hypothesis hol®[$; hol®[$; | |t] cannot be verified in some function spaces. 3 This problem is settled by using a weaker hypothesis which which is thought of as a local version of hol®[$; 2

This convention means that that no dependence on t is allowed if t does not appear by the inorm. seminorm. non-verification holds holds in L\, Lao, and in spaces with related norms. 3 For instance, this non-verification norms.

5.1. INTRODUCTORY

LIGHT LIGHT

85 85

More precisely, we say that that A(t) A(t) satisfies satisfies hypothesis hypothesis hol^Ji?; hol^Ji?;| | \t] \t]ifif (5.2) (5.2) A holds for all k G (0, K] and and t,s t,s ££ [0,T] [0,T] such such that that \t\t -- s\s\ << k.k. Further, before stating stating other other useful useful Holder-continuity Holder-continuity type type conditions, conditions, we need to define a new new notion. notion.

Definition 5.1. Let Ao Ao G GC C belong belong to to the the resolvent resolvent set set of of the the operator operator A(t) A(t) for any t G [0,T] 5 and let V* and V be linear linear manifolds manifolds (independent (independent of of t) such that Ran ((A o / - A{t))- 1)* C V* C X* and Vom Vom C C VV C C XX for for all all t G [0,T]. A bilinear functional A(t;x,u), A(t;x,u), xx G G ^*» ^*» uu ££ VV isis then then called called aa testing functional for A{t) A{t) if, if, for for any any tt G G [0, [0,T], T], (x, A(t)u) = A(t; x, u)

whenever whenever xx G GP* P* and and uu G GT>om. T>om.

Incidentally, the setV*xVCX*xX isis thought setV*xVCX*xX thought of of as as the the domain domain ofofA(t; A(t; -,-, )) and we write V* x V = Dom.4(£; Dom.4(£; ,, .. Clearly, given linear manifolds manifolds V* V* C C X* X* and and VV Q Q X, X, A(t) A(t) has has inin general general more than one testing functional == "P* functional A(t; A(t; ,, )) for for which which Dom DomA(t; A(t; "P*xx The question of uniqueness uniqueness of of such such aa functional functional isis outside outside of of the the scope scope ofof this book. Now we introduce more more Holder-continuity Holder-continuity type type hypotheses hypotheses with with some some , i?i G [0,1], A(t; , ) some testing functional for for A(t), A(t), with with Dom.A(t; Dom.A(t; ,, )) == V* x V, and | \^ t, \ |i ;4 , | j^0.t, and | |*i ;t;t some seminorms of which the first two are defined on on VV and and the the other other two two on on V*. V*. hol^)('d,'di;A(t;-,-);\ | m ; i ,m = 0,1, *0): For all x G V* V* and and uu G G V, V,

\A(t; x,u) - A(s; X,u)\< hol{^)('d,-d1;A{t;

C\xU, s (\t - sf\u\$,t

+ \t - s\d>\u\1]t) ,

| m ; t , m = 0 , * 0 , * 1 ) : F o r aall l l x £ V* a n d u G V,

\A(t;X,u) - A(s; X,u)\ < C\u\9.t (\t - s\°\XU,s + \t - s|*|xl«. 4

This hypothesis may look look strange strange because because A(t) A(t) does does not not depend depend on on k.k. However, However,ifif A(i) depends on an additional additional parameter, parameter, h, h, say, say, and and A(t) A(t) isis bounded bounded but but not not uniformly uniformly bounded with respect to h, h, there there are are situations situations when when ('...' substitutes for the second second term term on the right-hand side of of (5.2).) (5.2).)

\\(A(t) - A(s))u\\ < C1>{h) \t - s\*\\(A(t) with ij){h) an unbounded unbounded factor. factor. Then, Then, ifif kk and and hh are are in in conformity conformity by by the the condition condition k^1:4>(h) < C, with some $i $i 66 (0, (0, which is sometimes the case case in in applications, applications, we we come come to hypothesis h61^ } [i? - tfi; | \t]5 We assume that Ao is is independent independent of of t.t. We We might might not not accept accept this this assumption assumption but but such a generalization would would be be worthless, worthless, taking taking into into account account our our subsequent subsequent aims. aims.

86 CHAPTER

5. DISCRETIZATION

BY RUNGE-KUTTA

METHODS METHODS

and hol(#;.4(t;-,-);| |TO;t)m = 0, *0): For all x € V* and u<ET, \A(t;x,u) -A{s;x,u)\

< < C\t C\t -- s|s|tf|u|0;t|xl*0;s-

However, when deriving deriving error error estimates, estimates, if we are interested in getting orders of convergence > 1, it is not natural to be based on Holder-continuity type hypotheses and we employ instead some Lipschitz-continuity Lipschitz-continuity type type conditions. As such we introduce the following. lip: The operator A(t) satisfies hypothesis hol[#] hol[#] for i? = 1, and lip(,4(i;-,-);| |m;t,m = 0,*0): Hypothesis hol(i9;.A(£; hol(i9;.A(£; , ; I \m-t,m = 0, *0) is satisfied with d — 1. In what follows we often use seminorms with special special properties. properties. For the sake of convenience we therefore introduce certain certain new terminology.

Definition 5.2. Let A(t) G S((p,cr) for some ip G (0,TT/2), a G R and let t, £ € [0)^1 be a seminorm with Dom| \t C X. The pair (| is called (£\m\ip, a)-concordant a)-concordant /or some £ > 0 and m G N suc/i £/ia£ m>£if, for all A ^ IntE^ U {0}, t G [0,T], and u£X, Ran((A +
C Dom | | t

and |((A + a ) / - A(t))- mu\t < C\\\*-m\\u\\.

(5.3)

Similarly, if \ |* ;t, £ G [0,T], is a seminorm with Dom | |*;t C X*, £/ie pair a)-concordant for some £ > 0 and m G N (| \*-t; A(t)) is called (*;£|m|<^, a)-concordant i/iai m>£ if, for all A £ Int Int E^, U {0}, t G [0, T), and x G 3£*, Ran (((A + a)7 - A(i))- m )* C Dom and

| (((A + a)I - A(t))-m)*xl.t

< C|Ar m || X ||*.

(5.4)

/ / the size of m is not important in the context, the pairs (| |t;A(£)) and (| \*;t;A(t)) are then called (£\, cr)-concordant, respectively. Apart from from everything everything else, else, if the resolvent estimates (5.3) and (5.4) hold for all t,s G [0, T] with \ \s in place of | \t and with | |*;s in place °f\ ' \*;t, respectively, then the pairs (| \t;A(t)) and (| \^t;A(t)) are called strictly (.. .)-concordant.

5.1. INTRODUCTORY LIGHT LIGHT

87 87

With regard to possible applications applications of of the the abstract abstract theory theory developed developed below, we add a few words words about about classical classical parabolic parabolic PDE PDE problems problemsthat that can can fit into our framework. For example, let Q be a convex, convex, bounded bounded domain domain in in RRd, d > 1, with smooth boundary, and let let A(t) A(t) be be an an elliptic elliptic partial partial differential differential operator operatorofof second order defined by

with Dirichlet boundary conditions, conditions, where where the the coefficients coefficients a,ji(t, x) = aij(t, x) are smooth and real-valued real-valued on on [0, [0,T] T] xx C1J?. C1J?. The The operator operator .A(t) .A(t) isis assumed assumed to be uniformly elliptic in in the the following following standard standard sense, sense, with with some someao ao >> 0,0,

> ao ]T) Mj

for

all /ij G M, j = 1,..., d.

Then, thinking of .A(i) as as aa closed closed linear linear operator operator in in Loo(^) Loo(^) with with suitably suitably chosen domain, it can be be shown shown that that A(t) A(t) G G <S(
88 CHAPTER 5. DISCRETIZATION DISCRETIZATION

BY BY RUNGE-KUTTA RUNGE-KUTTA

METHODS METHODS

case of Loo norm and Neumann Neumann boundary boundary conditions conditions isis studied, studied, and and our our own work [39] where in in particular particular sectorial sectorial estimates estimates in in Holder Holder norms norms are are presented. Furthermore, as concerns concerns the the above above hol/lip hol/lip hypotheses, hypotheses, they they easily easily are are verified for elliptic differential differential and and finite finite element element operators, operators, assuming assuming aa suitsuitable Holder/Lipschitz-continuity Holder/Lipschitz-continuity property property for for the the coefficients. coefficients. Finally, in the matter of of (.. (.. .)-concordant .)-concordant pairs pairs involving involving elliptic elliptic differdifferential operators, such pairs pairs often often occur occur in in interpolation interpolation theory theory (see, (see, e.g., e.g., Komatsu [94, § 1,2] or Triebel Triebel [164, [164, Chapter Chapter 1]). 1]). At At the the same same time time (...)(...)concordant pairs involving involving Holder Holder or or Lebesgue Lebesgue norms norms and and finite finite element element operators are studied in in our our paper paper [39] [39] which which contains contains related related results results for for differential operators as as well. well. In the remainder of the the section section we we show show some some auxiliary auxiliary results results involving involving (.. .)-concordant pairs, for for subsequent subsequent use. use. Lemma 5.1. Suppose that that A{t) A{t) G G S((p, S((p, a) a) and and seminorms seminorms | | \t\t and and \\ |*|*;t form respectively (£\m\ip, (£\m\ip, a)a)- and and (*;£|m|<£, (*;£|m|<£, a)-concordant a)-concordant pairs pairs with with A(t) A(t) for some tp G (0, TT/2), a G M, m G N, and £ > > 00 such such that that m m >> max{£, max{£,1}. 1}. Then, for any fixed integer m\ m\ > > £, £, the the pairs pairs (|(| \\t; A(t)) and (| |* ;t; A(t)) are (^\m\\ip,a)- and {*;£ {*;£t\m\\, (*;£|mi|<£>, <j)-concordance <j)-concordance of of the the latter latter pair pair (|(| |*|*;t;.A(t)) is proved in a similar manner. manner. Observe now that if m\ > > m, m, by by the the (£\m\
=

It thus remains to consider consider the the case case ££ << m\ m\ << m. m. Clearly, Clearly, itit suffices suffices to to show show the claim when £ < mi mi == m m— —1. 1. ItIt isis not not hard hard to to see see that that for for AA^^ Int Int EE v and u G l , o

((A + a)I - A{t))- {m~l)u

- (m - 1) /

where the integral is taken taken along along the the ray ray \x \x== xxee i a r g \ |A| < x < oo. Using this equality, since m — —11 >> ££ and and || \t\t isis aa seminorm, seminorm, aa simple simple estimation estimation

5.1. INTRODUCTORYLIGHT LIGHT 5.1. INTRODUCTORY

89

89

nowyields, yields, forA A^ ^ Int IntE^, E^, now for oo

m l) m ((A +a)I a)I- -A{t))-^ A{t))-^ -l)ul ul ((A +

iargA m iargA m |((ze +a)I a)I- - A(t))A(t))u\.dx <
J\x\ J\x\

roo roo m {m l) m {m dx \\u\\ \\u\\= = C\X\^C\X\^\\u\\, xt~xt~ dx 'l)\\u\\,


whichshows shows again the(£|mi|>C for Definition ipG G(0, (0,TT/2) TT/2) if ifititisisholomorphic holomorphic inthe the opensector sector —S^, aswell wellas asat at0.0. ip in open —S^, as Notethat thatif ifA(t) A(t)G GS(k 0,>using formula (A.10)with with kA(t)inin placeof of£, £ ,one one can definethe theoperator operatoru(—kA(t)) u(—kA(t)) G (A.10) kA(t) place can define G B(X) forany any k (0,K] G (0,K] [0,T], t G with withsome some K >>00 sufficientlysmall. small. B(X) for kG and tand G [0,T], K sufficiently The same sameconclusion conclusion isin inforce forceif ifu(z) u(z)is is (oo oouniformly uniformlywith with respect tozzG GE EV forany any fixed respect to fixed Vll, for we showsome some estimates involving such operators. we show estimates involving such operators. For furtherneeds, needs, forany any rationalfunction function u(z),we we defineits itsintegerintegerfor rational u(z), define For further valued characteristic deg[u;] by valued characteristic deg[u;] by deg[ui] hm deg[ui] ——hm

log\u(z)\ log\u(z)\ log \z\ \z\ log

L 5.2.Suppose Supposethat that A(t)GG S(ip,a) and and seminorms|| \t\tand and L eemmmm a a5.2. A(t) S(ip,a) seminorms \ \|*;;tt |* form respectively respectively (£\m\(p,a)and(*\£\m\(p,a)-concordant (*\£\m\(p,a)-concordant A(t) with form (£\m\(p,a)and pairs pairs with A(t) for someip ip GG (0,7r/2),IT IT 6 K, K,( (> > andm mG GN Nsuch suchthat thatm m >>max{^, max{^, 1}. for some (0,7r/2), 6 0,0, and 1}. Let further furtherUJ(Z) UJ(Z)be beaarational rationalfunction function which which is is(tp)-regular (tp)-regular and andsatisfies satisfiesthe the Let condition deg[w] <— —m. m. Then Thenthere there exists fixedK K >>00such such that that forall all condition deg[w] < exists aa fixed for /cG (0,K],te te [0,T], [0,T], ueX, ueX, /cG (0,K], u{-kA{t))u\
\{u(-kA(t))YxU, > 00ininaddition, addition,then then theabove aboveis isin inforce forcefor forKK== oo. If the oo.

(5.5) (5.5)

90 CHAPTER 5. DISCRETIZATION BY RUNGE-KUTTA METHODS METHODS Proof. We will show only (5.5), (5.5), noting noting that that the latter estimate is proved in a similar way. 6 It follows from Lemma A.2 that with some 0, for all t£[0,T},z<£ I n t ^ j UV(R)), UV(R)), and ! i £ l , \(zl - A{t))-mu\t

< C\z\t-m\\u\\.

(5.6)

Moreover, in the case a > 0, (5.6) holds with R = 0. Using (5.6), we find as well for z $ Int ( S ^ U V(kR)), since z/k £ Int (Ev,1 U V(R)), \{zl - kA(t))- mu\t

= k-m\{{z/k)I

A(t))- mu\t

m

\\u\\ = Ck-t\z\t- m\\u\\.

(5.7)

Next note that the function u(z) has no poles at least in some disk T>(d), d > 0. Further, we set K > 0 subject to K < d/R if a < 0, and andwe we take K — oo if a > 0 (in the last case we have R — 0). Let now F be the contour coinciding with the boundary of the set T,Vl l)V(d). Using now the operator calculus formula (A. 10) with with F in place of H# +£ and with kA(t) in place of £, we get, since u>(oo) = 0, = 7T- I uj(-z)(zI-kA(t))uj(-z)(zI-kA(t))-1dz. 2m JT

(5.8)

Integrating, if m > 2, by parts and multiplying further both sides sides from from the right by u £ X, we obtain

Lu{-kA{t)) u = ^-

f tfm(z) (2/ - kA{t))~ mu dz,

(5.9)

with /

\

rr

-1 -1

Z

J

(2 — Zi) m ~ 2 o;(—21) dzi

if m > 2,

where the path of integration from ooe~ l¥>1 to 2 lies on the contour F. For further success we need to estimate suitably the function \J/ m(z). In the case m — 1 it is evident that - \z\)~l 6

for z 6 F.

(5.10)

In fact, we then have to use the below representation (5.8) with both sides converted into the adjoint operators.

5.1. INTRODUCTORY LIGHT LIGHT

91

Turning next to the case m > 2, we find, since deg[w] < — m, \*m(z)\

< (m-1) [*

|z-zi| m-2|u;(-zi)||

i^—2rii" a-2

< c r

which implies, for all z € F with Qz < 0, o

|*m(z)|
(1 + x)- 2dx

(5.11)

J\z\

Observe that by Cauchy's Theorem, re =

-

/

Jz

and thus (5.11) remains valid valid for all z 6 Y. Using now (5.9) together together with with (5.10), (5.11), and (5.7) and since | |< is a seminorm, a simple estimation gives, for k e {0,K},

\u(-kA{t)) u\t < C f(l + \z\)-1\(zI-kA(t))-mu\ JT

\dz\

< Ck-t [(l + \z\)-l\z\t-™\dz\, JT

thus (5.5) follows at least in the case £ < m. Now let £ = m. It follows from Lemma 5.1 that that the pair (| | t ; A(t)) is, in fact, (£\m + 1|
\uj\(—kA{t))u\t ^ Next, let Ao £ C be subject to A(7m = lim ( |2|->oo

and furthermore let io\(z) be given by {l-Xoz)-m (—1 — \oz)~m

if |argA 0| < TT/2, otherwise.

(5.13)

92 CHAPTER CHAPTER 5. DISCRETIZATION DISCRETIZATIONBY RUNGE-KUTTA RUNGE-KUTTAMETHODS METHODS It is easily easily seen seenthat thatdeg[u;i] deg[u;i]< <—(m —(m and u>i(z) is (ip)-regu\ax (ip)-regu\ax so that that UJI + +1)1) and u>i(z) satisfies (5.12). Since SinceA(t) A(t)£ £S(ip,a) S(ip,a)and andthethe pair ;A(i)) | (£|m|
< Ck-Z\\u\\ Ck-Z\\u\\ ifif | arg | argAA TT/2, < o| < TT/2,

(5.14) (5.14)

and

kA(t))-mu\t \(I - \okA(t))-

< CkS\\u\\ CkS\\u\\ <

|arg(-A0 )|
(5.15) (5.15)

kA(t) substituted substituted for forz,z,we weget getananoperator operator equality (5.13) with with — kA(t) equality By (5.13) kA(t))-m jargAol << TT/2, TT/2, (I + X kA(t))ifif jargAol u)(-kA(t)) =wi(-/cA(t))++^ ^ , , , ,, , .o ., ., , , ,, ,, „„,, u)(-kA(t)) =wi(-/cA(t)) ,,,, K v K vv (-1 ++XokA(t)) XokA(t)) m otherwise. " 11 (-1 otherwise. (5.16) (5.16) Therefore, using using (5.16) (5.16)together togetherwith with(5.12), (5.12), (5.14), (5.15), desired Therefore, (5.14), andand (5.15), the the desired estimate (5.5)ininthe thecase case££==mmfollows follows well, after a trivial estimation. estimate (5.5) asas well, after a trivial estimation. we pose pose separately separatelyananimportant importantcase case Lemma Now we of of Lemma 5.2.5.2. Lemma 5.3. 5.3. Let Let A(t) A(t) ££S( > 00 isissufficiently sufficientlysmall, small,we we have, £ (0, £ T), [0, T), Then, have, forforallall k £k (0, K] K] andand t £ t [0, M-kA{t))\\
(5.17) (5.17)

for some some mm ££N, N,then, then,for forallallk k£ £(0,K] (0,K]and and Furthermore, i/deg[cj]<<— m for Furthermore, i/deg[cj] R&nu(-kA(t)) C Domi Domim (i) and t £ [0,T], [0,T], R&nu(-kA(t))

MkA(t)r<s{-kA(t))\\>0,0, the theabove aboveisisstill stillinin force force forfor KK = oo. = oo. This fact fact follows follows directly directlyfrom fromLemma Lemma5.25.2taking taking 0, m = and 1, and £ =£ = 0, m = 1, (0,7r/2), aa ££ R, R, and andw(oo) w(oo)6 6 Lemma Lemma 5.4. 5.4. Suppose Supposethat thatfor forsome some

—> oo regular and obeys obeysthe the condition conditionOJ{—Z) OJ{—Z) — u;(oo) u;(oo) ++ O^zl" O^zl"1 ) as regular and uniformly with respect respecttotoz z £ £E V1 E1, for for any any fixed fixed ipi ipi ££ (0,(p). (0,(p). Then Thenthethe uniformly with V operatoru{—kA{t)) u{—kA{t)) satisfies satisfiesthe theestimate estimate(5.17). (5.17). operator result can canbe beshown shownbybyusing usingthe thesame same techniques as those employed This result techniques as those employed the proof proof of ofLemma Lemma5.2. 5.2. in the

5.1. INTRODUCTORY LIGHT 5.1. INTRODUCTORY LIGHT

93

93

Leemmmm 5.5. A(t)G GS(ip, S(ip,a) a)and andletleta a(not (notnecessarily necessarily rational) L a a5.5. Let Let A(t) rational) functionfunction u!nn(z), U {0}, {0},be beholomorphic holomorphic (inz)z)inin neighbourhood a ofT.^, T.^,for for u! (z), nn GG NN U (in a neighbourhood of some tpGG(0, (0,TT/2) TT/2) and and Suppose further that forsome somedd >>0,0, the some tp a aG G R. R. Suppose further that for the function (z) obeys obeys theconditions, conditions, forall alln nG GNNUU {0}, function ujnnuj (z) the for {0}, 71 71 \ujnn(z)\
forallz£V(d), forallz£V(d), for allzze eE E V(d), n for all ^ n^ V(d),

and and l^l)"11 for forall allzz G GS Svv . l^l)" T/zen i£holds, holds,with with some K >>0 0 sufficiently small, forall all t [0,T], E [0,T],kk££ T/zen i£ some K sufficiently small, for t E (0,K], K], and and nG GNNUU {0} suc/i nk<< (0, n {0} suc/i that that nk C,C, \\unn(-kA)\\
(5.18) (5.18)

Furthermore, ifaa>> 0 in addition,then then the restriction nk<< is not needed. Furthermore, if 0 in addition, the restriction nk CC is not needed. Proof Note (see Appendix theuioperator {—kA{t))can uican bereprerepreProof Note thatthat (see Appendix A.2) the A.2) operator be nn{—kA{t)) byformula formula(5.8) (5.8) (-)cu substituted forw(-), w(-),where where Yisisreplaced replaced sentedby sented withwith cunn(-) substituted for Y by Tn theboundary boundary ofthe theset set12 12

(d/(2n ++Rk), +Rk), 2) for someR R >>00sufsufof + 2) for some by T (d/(2n n the 7 7 ficiently large. Nextwewe split Tnn into intothree three parts byTT = T^ U T^ U ficiently large. Next split T parts by = T^ U T^ U n n where where : ==d/(2n |z| d/(2n+ +2) 2) Rk, <£ z Int IntS^}, S^}, ++ Rk, z <£ ^^ {{ z ezCe :C|z| d/(2n+ +2) 2) << \z\ G d, Ez^ } , {{ zzeeCC: :d/(2n ++ RkRk < \z\ d,< zG E^}, and and (3) {2 GC C::|z| |z| GzEEvv }}.. T(3) -- {2 G >> d, d, zG T

theintegrals integrals over part ofthe the contourT Tnn Now,estimating estimating separately Now, separately the over eacheach part of contour with theaid aid acceptedconditions conditions onuu (z) and and takinginto into account with the of of thethe accepted on taking account n n(z) the reasonings employed inthe the proofof ofTheorem Theorem theclaim claimfollows follows as the reasonings employed in proof 2.1,2.1, the as in theproof proofof ofLemma Lemma5.2 5.2inin the casem m==1.1. in the the case L 5.6. Let A(t) S((p,a) G and rational a function (z), u>n nG G L eemmmm a a5.6. Let A(t) G S((p,a) and letlet a rational function u>nn(z), N UU{0}, {0},be be{ip)-regular {ip)-regular forany any fixedn, n,for for some>1,1, q 0>and and 0 /or / o rany any fixed Suppose that that for q> fixed ip\ G(0, (0,y?), y?), function {z) ui obeys theconditions, conditions, forall alln nG GNNUU {0}, ip\ G thethe function uinn{z) obeys the for {0}, n[z[ \u \unn(z)\ (z)\ <
Onecan can take R R==00 if if a> a 0. > 0. One take

for forall allzz G G V(d), V(d),

94 CHAPTER

5. DISCRETIZATION \un(-z)\

< Ce~cnW

Un{~z)\ < Ce" c n | z | ~ 9

BY RUNGE-KUTTA

METHODS METHODS

for all z G E V1 n 23(1),

for all z G S V 1 such that

\z\>l,

and

K ( z ) | < Ce cn|z|~* /or aZZ z g Int23(di).

Then, with some K > 0 sufficiently small, the operator operatoru(—kA(t)) u(—kA(t)) satisfies satisfies the estimate (5.18) for all t G [0,T], fc G (0,X], and n G N U {0} sucfr i/iat nA; < C. Furthermore, the condition condition nk < C is not needed if a > 0. Proof. Taking
rn = U5=ir^, where E V l }, £> = {z G C : \z\ = d/(2n + 2) + Rk,z<£ Int E z e C : d/(2n + 2) + Rk < \z\ < 1, z G S V l }, {* e C : 1 < |z| < di(n + I) 1 / 9 , z G E V1 }, and 4 4 ) = { z e C : |z| = di(n+ 1) 1/9 , z ^ IntE^J,

we use the Taylor operator operator calculus calculus (see, (see, e.g., e.g., Taylor Taylor [160]), [160]),so sothat that we wehave have the representation LJn(-kA(t)) = Un(00)I +^~

Now, with this equality in mind, the claim easily follows follows in a manner similar to that used in the proof of of Lemma Lemma 5.2 5.2 in in the the case case m — 1, taking the accepted condtions on to n(z) into account and estimating estimating separately separately the the integrals integrals over over r%\j = 1,2,3,4.

5.2

Discretization

In this section we consider consider the the application application of of the the Runge-Kutta Runge-Kutta discretization discretization to the abstract linear Cauchy Cauchy problem problem (5.1). (5.1). For For subsequent subsequent reference, reference, we introduce the reader to a proper classification of the Runge-Kutta methods methods through a series of definitions. Apart from everything everything else, else, we we give give examples examples which go go into into the the theory. theory. of families of Runge-Kutta methods which

5.2. DISCRETIZATION

95 95

Let Qk — {t n : tn = nk for n — 0,..., N} be be aa uniform uniform grid grid in in the the interinterproblem (5.1) (5.1) be be discretized discretized in in val [0,T], with stepsize fc. 8 Furthermore, let problem time by means of a i/-stage i/-stage Runge-Kutta Runge-Kutta method method9 with Butcher array

bT ) ' where10 b = (&i,... ,b u)T G C , c = (a,... ,c u)T 6 T , and a = (aji)uxu G Cyxu are given. The application application of of the the RK RK method method to to problem problem (5.1) (5.1) leads leads 11 B(X) generally generally to the following discrete problem, problem, with some given 9t G B(X) depending on k, 12

Yn+1 = Yn-kJ2bj

(A(tnj)Ynj - f(tnj))

for t n G Clk,

Yo =

3= 1

(5.19) where Y nj are calculated from the the system system of of equations equations V

Ynj = Yn-kJ2

a jt(A{tnl)Ynl

- f(tnl)),

j = l,...,v.

(5.20)

In (5.19), (5.20): t nj = tn + Cjk and Y n is the approximation to y(tn). The stability function of the the method method isis defined defined in in the the standard standard way wayby by r(z) = 1 + zb T(i - za)~ xe, where i stands for the identity identity {y {y xx i/)-matrix i/)-matrix and and ee == ( 1 , . . . , 1) . Next,

let gji(z), j,l = 1 , . . . ,u, ,u, be be the the entries entries of of the the matrix matrix (i(i — za)~l and let and v(z) = {vi{z),... w(z) = {w 1{z),,..,wl/{z))T {vi{z),... ,vu(z))T be given by

w T (z) = b T (i - zay 1 and v(z) = (i - za)' le. 8

(5.21)

As above, N and k are in in conformity conformity through through Nk Nk — —T, T, and and the the case case TT == oo oocan can be be trivially settled (cf. Section Section 1.1). 1.1). Also, Also,fifc fifcpossesses possesses the the same same meaning meaning as asininSection Section 1.1. 1.1. 9 In what follows we often use use RK RK method method instead instead of of Runge-Kutta Runge-Kutta method, method, for for brevity. brevity. 10 For simplicity we assume assume that that 00 << Cj Cj < < 1, 1, jj; = 1,..., v, throughout. This This restriction restriction is by no means crucial. 11 These equations are a formal formal procedure procedure only. only. Sufficient Sufficient conditions conditions for for problem problem (5.19) (5.19) and (5.20) to be uniquely uniquely solvable solvable are are given given later later in in the the chapter. chapter. 12 It is not in general use to to construct construct Runge-Kutta Runge-Kutta approximations approximations with with correction correctionofof the initial condition y°. In In the the sequel sequel itit isis shown, shown, however, however, that that sometimes sometimes involving involvingD\ D\isis essential.

96 CHAPTER 5. DISCRETIZATION

BY RUNGE-KUTTA

METHODS METHODS

Thus for the function a(z) = 1 — r(z) we have the following representation representation a(z) = -zvsT(z)e = -zbT(i - zo)"^.

(5.22)

The function a(z) will play an important role in the theory below. Also, there are times when we employ the function

p(z) = -z^aiz)

= b T(i - za)~ le.

(5.23)

Clearly, all the functions a(z), p(z), Vj(z), Wj(z), and gji(z) are rational. Throughout this part of the book we will consider consider only only such such RK methods 13 that satisfy the additional condition r(0) = r'(0) = 1, which is therefore always assumed assumed in what follows without explicit explicit mention. mention. Definition 5.4. The RK method is said to be A{
Definition 5.5. We say that the RK method is of type C(ip) for some <^e (0,7r/2) if (i) it is A(ip)-stable; (ii) each of the functions Wj(—z), j — 1 , . . . , v is (ip)-regular and deg[wj] < 0,j = l,...,v. Assume in addition to (i) and (ii) that (in) The eigenvalues of the coefficient matrix a — [aji) vy.v are contained in the sector T.^^^; and (iv) Then we say that the method is o/type V((p) or o/type V*(ip) if it satisfies as well (v) deg[vj]<-l, j = l,...,v, 13

In principle, this condition condition is not needed for showing stability in itself. At the same methods. We We therefore therefore prefer prefer to assume time it is satisfied for all practical discretization methods. this condition throughout, throughout, which which slightly slightly facilitates facilitates our subsequent proofs.

5.2. DISCRETIZATION

97 97

or (vi)

deg[m,-] < - 1 , j = l , . . . , i / ,

respectively. 1^

It is worth noting that unlike unlike some some authors authors (cf., (cf., e.g., e.g., Gonzalez Gonzalez &; &;PalenPalencia [73, 74]), we do not suppose suppose that that the the /L(v?)-stability /L(v?)-stability implies implies (iii). (iii). Also, Also, observe that (iii) and (iv) together together imply imply (ii), (ii), in in view viewof of (5.21). (5.21). Furthermore, Furthermore, if the matrix a is non-singular, non-singular, we we have have <-l,

j , l = l,...,u,

(5.24) (5.24)

thus it immediately follows follows that that deg[i>j] < —1 and degfwj] degfwj] << —1 —1 for for jj — —1,... 1,... ,v. ,v.

(5.25) (5.25)

In particular, it is easily seen seen that that (ii) (ii) isis then then aa direct direct consequence consequence ofof (iii). (iii). Obviously, dealing with A(ip)-stab\e A(ip)-stab\e methods, methods, we we have have only only three three possipossibilities: (Si)

|r(oo)| |r(oo)| < 1,

(Sn)

r(oo) r(oo) == 1, 1,

and |r(oo)| = 1 and r(oo) ^ 1.

Definition 5.6. We say say that that the the RK RK method method belongs to the class Sj, J = I, II, III, if it satisfies the the respective respectivecondition condition (Sj).15 The method is called Aj((p)-stab\e (Aj-stab\e), JJ = = I,I, II, II, III, III, ifif itit isis A(
This terminology may be be justified justified by by noting noting that that the the use useof ofmethods methods ofoftype type C(
98 CHAPTER 5. DISCRETIZATION DISCRETIZATION BY BY RUNGE-KUTTA RUNGE-KUTTA METHODS METHODS Therefore, we have placed placed at at our our disposal disposal aa certain certain classification classification ofof the the Runge-Kutta methods. We We take take this this classification classification as as aa basis basis for for application application of our techniques in this this book. book. When When establishing establishing the the below below results, results, we we nonetheless distinguish between between the the classes classes and and types types introduced introduced by by DefiniDefinition 5.6. Next we recall some widely widely practised practised concepts concepts of of method's method's order, order, which which are needed for our investigation investigation in in Chapter Chapter 7. 7. The The first first definition definition isis rather rather general and may be employed employed for for any any single single step step discretization discretization methods methods but but not necessarily for Runge-Kutta Runge-Kutta ones. ones. Definition 5.7. A single single step step discretization discretization method method isis said said toto have have order order pp if the local error of the method, after one step of length length kk of of its its application application to an ordinary differential differential equation equation dy/dt dy/dt == f(t,y) f(t,y) with with sufficiently sufficiently smooth smooth right-hand side, is O(k p+1) as k —> 0. Dealing with Runge-Kutta Runge-Kutta methods, methods, the the notion notion of of stage stage order order isis useful useful in certain situations as well. well. Definition 5.8. An RK RK method method is is said said to to be be of of stage stage order order qq ifif the the error error of the method, when it is applied with with stepsize stepsize kk to to an an ordinary ordinary differential differential equation with sufficiently sufficiently smooth smooth right-hand right-hand side, side, isis O(k O(kQ+1) as k —> 0 at each separate internal stage. stage. In the case with a constant constant operator operator we we employ employ one one more more concept concept ofof order (cf. Brenner, Crouzeix, Crouzeix, and and Thomee Thomee [50] [50] or or Akrivis Akrivis &: &:Crouzeix Crouzeix [4]). [4]). Definition 5.9. A discretization discretization method method isis said said to to be beof of polynomial polynomial order order qq if, when it is applied to aa linear linear ordinary ordinary differential differential equation equation dy/dt dy/dt ++Xy Xy == f(t), A £ C, the truncation truncation error error vanishes vanishes whenever whenever the the solution solution y(t) y(t) isis aa polynomial in t of degree degree at at most most qq — 1. It is easily checked that that an an RK RK method method isis of of polynomial polynomial order order qq++11ifif itit is of order p and of stage stage order order qq < < p. p. In conclusion, we point point out out some some concrete concrete families families of of the the Runge-Kutta Runge-Kutta methods which are classified classified in in due due course course according according to to the the above above definitions. definitions. We first recall, however, however, some some of of the the so-called so-called simplifying simplifying assumptions assumptions inintroduced in the theory of of the the Runge-Kutta Runge-Kutta methods methods by by Butcher Butcher (see, (see, e.g., e.g., [53]). In fact, for subsequent subsequent use use we we need need the the following following only only three. three.

B(m)

Yl ^ci" 1 = j ~ l for 1 < j < m; s=l

5.2. DISCRETIZATION

C(m)

99 99

] P alsd>~1 = j " 1 ^ '

for 1 < j < m and 1 < I < u;

s=l

and V

T>{m)

Y^ bs4~1(lsi

= J^biO-

-4)

iorl
and 1 < j < m.

It is well known that the fact that the method is of order p implies !B(p) and the fact that the method is of stage order q is equivalent to Q(q) (see, e.g., Butcher [53, Chapter 3]). 3]). First of all we discuss the family of Gauss-Legendre methods methods (see, (see, e.g., Butcher [53], Dekker & Verwer [65] or Hairer & Wanner [81]). Given v>l, for the i^-stage Gauss-Legendre Gauss-Legendre method, method, the coefficients Cj, j — l,...,u, coincide with the zeros of Pu(2c—1), where Pv(x) is the Legendre polynomial polynomial of degree u, while bj, a^; j,l = 1 ,...,!>, are specified by the symplifying assumptions B(^) and C(z/). It is known (see, e.g., [53, 65, 81]) that all methods of this family are ^4-stable and, for any fixed v > 1, the i^-stage Gauss-Legendre method method is of order 2v and of stage order v.l& Also, it follows from Saff & Varga [137] that the eigenvalues of the coefficient matrix a are localized in the set IntS^/2- It is pointed out in Dekker & Verwer [65] that the matrix a is non-singular so that (5.24) and (5.25) hold. Apart from everything else, it is not hard to see that r(oo) = 1 if v is even and r(oo) = — 1 if v is odd. It is therefore clear from the above reasonings that Gauss-Legendre method method is of type Cjj(ip) for if v is even, the corresponding Gauss-Legendre any fixed

With the aid of this equality it follows that for odd u, l-a(oo)-1a(z)

=2 0<m
m\(u-m)\ v

'

which implies directly that that deg[a(.) deg[a(.) — a(oo)] = —1. Also, it is easily seen that a(oo) = 2 for odd u and, hence, r(oo) = —1. Therefore, if v is odd, the The last fact is a direct consequence of the simplifying assumption assumption C(v). C(v).

100 CHAPTER 5. DISCRETIZATION DISCRETIZATION BY RUNGE-KUTTA METHODS METHODS corresponding Gauss-Legendre Gauss-Legendre method method is both of type Vjjj((p) and of type V?n(
1 j This manner of discretization is often referred to as the Crank-Nicolson method. Next we consider the so-called Radau IA and IIA methods. Given ! / £ N , for the z^-stage Radau IA method, Cj, j = l , . . . , i / , are the roots of the equation

Pv-i(2c-l) + P v{2c-l) = 0, while bj and a^; j , I = 1 , . . . , v, are specified by the simplifying assumptions assumptions 'B(u) and T>{v). For the i^-stage Radau IIA method, Cj, j = 1 , . . . , u, are the roots of the equation Pv-i(l

- 2c) + Pu(l - 2c) = 0,

while the other coefficients are specified by the simplifying conditions S(z^) S(z^) and G(u). It is known (see, e.g., [53, 65, 81]) 81]) that the methods of both families are A-stable, all of which satisfy the condition r(oo) = 0 or, equivalently, a(oo) = 1, and, for any v > 1, the i/-stage Radau IA or IIA method is of order 2/^ — 1. At the same time the stage order is v — 1 for Radau IA methods and v for Radau IIA ones. Also, by Saff and Varga [137] the eigenvalues of the coefficient matrix a are localized in I n t S ^ - Moreover, the matrix a itself is non-singular (see, e.g., [65]) so that (5.24) and (5.25) hold. Therefore any Radau IA or IIA method is both of type Vi(ip) and of type Vj(tp) for any fixed if G (0,7r/2). In particular, the widely popular backward backward Euler Euler method method is nothing but the 1-stage Radau IIA method with the Butcher array

/ 1 1\ Further, we discuss the behaviour of the Lobatto IIIA, IIIB, and IIIC methods (see, e.g., [53, 65, 81]). For all these methods, c\ = 0, c v = 1 and Cj, j = 1 , . . . , v, are the roots of the equation

5.2. DISCRETIZATION DISCRETIZATION

101

while the other coefficients coefficients are are specified specified by by the the simplifying simplifying assumptions assumptions rB(u) and Q{v) for Lobatto IIIA methods, by ®(i/) and V(v) for Lobatto IIIB methods, and by 'B(^) and and Q(v Q(v — 1) and the additional condition condition

81]) that that all all these these for Lobatto IIIC methods. methods. It is known (see, e.g., [53, 65, 81]) methods are A-stable and and the the order order of a f-stage method is always 2u — 2. For the ^-stage Lobatto IIIA IIIA and and IIIB IIIB methods methods the the stage stage order order is v and v — 2, respectively, and the stability stability function function r(z) r(z) — —11 — a (z) coincides with matrix a is the (v — l,v — 1)-Pade approximation to e2 but the coefficient matrix singular. Nevertheless, it follows from [65] that deg[ det (i — a)] = v - 1,

which shows that deg[<7j/] deg[<7j/] < 0 and, thus, deg[u;j] < 0. Therefore, for any fixed ip G (0,7r/2), the ^-stage Lobatto IIIA or IIIB method is of type Cniif) if v is even and of type Cu(ip) if v is odd. Next, for the i^-stage Lobatto Lobatto IIIC IIIC method, method, the the stage stage order order is v — 1, and for all methods of this family, the matrix a is non-singular (see, e.g., [65]) [65]) so that (5.24) and (5.25) hold. hold. Also, Also, since since a(oo) a(oo) = 1, it is easily seen that - deg[l

- -2.

The above reasonings show show that that any any Lobatto Lobatto IIIC IIIC method method is both of type Vi(ip) and of type Vf(ip) for any fixed (p (p e (0,TT/2). Finally, we look at the family of 3-stage RK methods for which the Butcher array is given by ( Cl

Cl

C2-7 C3-C-V bi

C2 C3

0 7

C

0 0 V

(5.26)

b2 b3

r i , 3riT 2T3 + Ti + T2 + T3 = 0 , Cj = (1 + Tj) where T\ ¥" T2 ¥"T3 ¥"' j = l,2, 3, and 1

1+3T2T3

°^

3(n-T2) (TI-T3) ' f

7

1+3^iTl 3(r 2 -r 3 ) ( T2-n

T3-T1+3T1T3

2(l+3riT 3 ) '

c=

3

(T1-T3)

2

Clearly, this family is actually described by two two parameters. parameters.

1 + 3TTT2 3(r3-ri)(T3-r2)'

(l+2n) 0(T2

~Tl)'

102 CHAPTER 5. DISCRETIZATION DISCRETIZATION BY BY RUNGE-KUTTA RUNGE-KUTTA METHODS METHODS It follows from [64] that that the the methods methods of of this this family family are are of of order order 44 and and ofof stage order 1; and moreover, moreover, under under the the additional additional condition condition ri > 0 and 1 + 3T!T3 < 0,

(5.27) (5.27)

they are A-stable, and itit then then holds holds that that |r(oo)| |r(oo)| << 1.1. Also, Also, itit isis easily easily seen seen that if (5.27) is fulfilled, fulfilled, the the coefficient coefficient matrix matrix isis non-singular. non-singular. Therefore Therefore all all methods of this family that that satisfy satisfy (5.27) (5.27) are are both both of of type type Vj(ip) Vj(ip) and and ofof type type Vj*(tp) for any fixed tp €€ (0,TT/2). Many further examples of of applicable applicable methods methods can can be be found found within within other other families of RK methods methods (see, (see, e.g., e.g., [53, [53, 65, 65, 81]). 81]). The The interested interested reader reader might carry out the corresponding corresponding analysis analysis by by himself/herself himself/herself similarly similarly toto the above.

5.3

Connection with discrete discrete evolution evolution equations equations

Now we discuss the question question in in which which manner manner the the discrete discrete problem problem (5.19) (5.19) and (5.20) reduces to the the discrete discrete Cauchy Cauchy problem problem (1.1) (1.1) examined examined inin Part Part I.I. In fact, we show that the the reducibility reducibility isis truly truly performed performed ifif the the restrictions restrictions on the operator A(t) and and on on the the RK RK method method suitably suitably conform conform to to each each other. other. Below in some intermediate intermediate estimations, estimations, we we deal deal with with matrices matrices whose whose entries are members of XX or or of of B(X).18 If u = (uji)pxr is such a matrix, we use the notation max \\uu\\, Hull = \\uu\\,

where \\UJI\\ is the vector norm of Uji Uji ifif Uji Uji EE XX and and the the operator operator norm norm ofof 1Q In particular, working Uji if Uji £ B(X). working with with the the product product space space X^ X^ == Xx x 3£ which consists of all z/-dimensional z/-dimensional vectors vectors uu == (u\,... (u\,... ,uj) ,uj) with Uj G X, 1 < j < u, u, for for the the sake sake of of being being consistent, consistent, we we assume assume that that 20 X^ is endowed with the the norm norm ||ul| = max ||tt,-||. 18

In the sequel we also meet meet the the situation situation when when the the entries entries of ofthe the matrix matrix are areunbounded unbounded operators, however such matrices matrices are are involved involved with with no no estimations. estimations. 19 It is always clear from the the context context what what uu isis and, and, thus, thus, inin which which sense sense ||u|| ||u|| isis underunderstood. 20 X^"' may may be be endowed endowed with with any any suitable suitable norm norm composed composed ofof the the Note that the space X^"' norms of the components components in in X. X. All All such such norms norms are are equivalent equivalent to to each each other. other.

5.3. CONNECTION WITH WITH DISCRETE DISCRETE EVOLUTION EVOLUTION EQUATIONS EQUATIONS 103 103 We are confident that using using the the same same symbol symbol |||| |||| differently differently gives gives rise rise to to no no confusion. Also, in any subsequent calculations calculations involving involving operators, operators, aa matrix matrix mmGG C p x r will be identified with with the the operator operator 21 m / : X^ -> £ ( p ) . We first consider the case case with with aa constant constant operator, operator, that that is, is, where where A(t) A(t) is independent of t. (We (We therefore therefore write write AA instead instead of of A(t).) A(t).) Let Let the the RK RK method be of type C((p) C((p) for for some some (p (pG G (0, (0, TT/2). Then, for any fixed a G G M, M, ifif K > 0 is sufficiently small, small, the the functions functions a(—k\) a(—k\) and and Wj(-kX), Wj(-kX), jj == 11, ,. . . ., , v,v, are denned for any k € (0, (0, K] K] and and A AG G (Int (Int S^, S^,U U{0}) {0}) ++ ex, and one can take K — oo if a > 0. It is then then not not hard hard to to see see that that in in the the scalar scalar case case A(i) A(i) == XI, XI, problem (5.19), (5.20) reduces reduces to to the the following following one one

In the operator case A A G G S(tp,a), S(tp,a), dealing dealing with with (5.19), (5.19), (5.20), (5.20), one one should should ,^, from (5.19) and (5.20), (5.20), that that is, is, solve solve (5.20) (5.20) for for eliminate Y nj, J' — lj Ynj and insert them into into (5.19). (5.19). This This procedure procedure certainly certainly isis realized realized ifif the the system of equations (5.20) (5.20) is is uniquely uniquely solvable. solvable. ItIt may may however however happen happen that that (5.20) is not uniquely solvable solvable while while the the operators operators r(—kA) r(—kA) and and Wj(-kA), Wj(-kA), j — 1 , . . . , v, are nevertheless nevertheless defined defined and and bounded bounded on on X. X. In In particular, particular, itit follows from Lemma 5.3 5.3 that that the the definedness definedness and and boundedness boundedness ofof r(—kA) r(—kA) and of Wj(-kA), j = 11,,......,, v, v, are are observed observed for for all all kk GG (0, (0,K], K], with with KK >> 00 sufficiently small, if A G G S(ip, S(ip, a) a) and and the the RK RK method method isis of of type type C(
(5.28)

where

-,/yf^M.

(5-29)

may be thought of as a generalized generalized form form of of problem problem (5.19), (5.19), (5.20) (5.20) ininthe the case case with constant operator. operator. On On the the other other hand, hand, problem problem (1.1), (1.1), (5.28), (5.28), (5.29) (5.29) isis 21

The symbol ® denotes Kronecker's Kronecker's product. product. In the homogeneous case /(£) /(£) == 00 itit suffices suffices to to assume assumethat that the the method method isis^4( > 0. 0. 22

104 CHAPTER CHAPTER 5.5. DISCRETIZATION DISCRETIZATION BY BY RUNGE-KUTTA RUNGE-KUTTA METHODS METHODS follows equivalently represented as follows equivalently represented

Yn+1 = (I-a( (I-a( r(-kA)Yn + k^2wj(-kA)f(t k^2wj(-kA)f(tnj = r(-kA)Y nj)

forttn £ Clk, for

9ty°Yo = 9ty°-

(5.30) (5.30)

form (5.30) (5.30) are arecalled calledrational. rational.In principle principle In the the literature literature methods methodsof the form rational methods for autonomous autonomous equations equations may maybe introduced introduced and and examexamrational methods RKmethod method (5.19) (5.19)and independently, without without connection connectionwith withthe RK ined independently, (5.20). However in this book book we we deal dealwith withthe thenon-autonomous non-autonomouscase caseasas well (5.20). However well common starting starting point pointfor forour ourconsideration, consideration,which which we prefer prefer to keep a common and we problem (5.19), (5.19), (5.20), (5.20),whether whetherthe thedependence dependenceof A(t) on tt is is present present is problem not. Moreover, Moreover, we westrive strivefor creating creating a unified unified terminology terminology within withinthe or not. book which which means means that that we wewant wantrelated relatedterms termsfor different different objects objects to be be in conformity conformity with with each each other. other. In view of these reasons reasons we we introduce introducethe following. following. —>>3t specified specified by the rereDefinition 5.10. The The discrete discretefunction functionYn : Q/- — Definition 5.10. currence (5.30) (5.30) is called called the the extended extended solution solution of the Runge-Kutta Runge-Kuttaprocedure procedure currence autonomous case. case. (5.19) and and (5.20) (5.20) in the autonomous (5.19) Further, we weturn turn to tothe thegeneral generalcase casewhere wherethe theoperator operator A(t)may may depend Further, A(t) depend convenience we weuse use on t. For convenience diag (A(tm),.. (A(tm),.. .,A(t .,A(tni/)), = diag 1

V(t)= =diag V(t)

= {9jl(-kA(t)) (-kA(t))vxv,

and a{-kA(t)). &(t) = k~la{-kA(t)).

(5.31) (5.31)

what follows follows we weconsider consideroperator operatorequations equationsof the form, form, given given u = In what

(«i,..,«,felM, («i,..,«,fel (i + kaDn)z = ou,

(5.32) (5.32)

(i + kaT)n)z = e 0 0 u,

(5.33) (5.33)

form, given given u E X, or of the form,

5.3. CONNECTION CONNECTION WITH DISCRETE EVOLUTION EQUATIONS 105 5.3. WITH DISCRETE EVOLUTION EQUATIONS 105 which aresolved solved for for zz == {z\,...,zj) {z\,...,zj) which are

Vom Vorn. x equation GGVom x x x Vorn. IfIf equation V V

(5.32) (equation (5.33)) isuniquely uniquelysolvable solvable forany any GX^ X^ (for (for any u G (5.32) (equation (5.33)) is for u uG any u G X), we wedenote denotethe thecorresponding corresponding solution operator by(i(i++ka karD Dn)~ )~1a (by (by X), solution operator by -1 (i + + ka1) ka1)n)~ e) so sothat that zz==(i(i++ka ka(D Dn)~ )~lau. au. (z (z==(t(t++fcaD fcaDn)-1 u). (i )~le) e << gg >>u). Now we wepresent presentsufficient sufficient conditions forproblem problem(5.19), (5.19), (5.20) tobe be Now conditions for (5.20) to equivalently representable inthe theform form (1.1). (1.1).In Inwhat whatfollows follows different equivalently representable in wewe useuse different formsof ofHolder-continuity Holder-continuity ofA(t) A(t)and and this main mainidea idea separates thebelow below forms of this separates the assertions from each other. We start, however, bygiving givingan anauxiliary auxiliaryresult result by assertions from each other. We start, however, bein incommon commonuse usefor forthe the corresponding proofs. whichwill willbe which corresponding proofs. Lemma5.7. 5.7.Let Let A(t) S(ip, G a) a) and andlet let the methodbe beof oftype type V( >00 or K sufficiently small, forall allkk G G(0, (0,K] K]and and G[0, [0,T], T],the theoperators operators and matrix sufficiently small, for tG and matrix l operators a(-kA(t)),p{-kA{t)), p{-kA{t)), v(-kA(t)), w(-kA(t)), kaV(t))~ (i(i++ kaV(t))~ , operators a(-kA(t)), v(-kA(t)), w(-kA(t)), M 1 M (i++ka'D(t))ka'D(t))-a, kT)(t)(i (i++kaV^y^e kaV^y^e areare defined onXXor oron on kV(t) (i kV(t) a, kT)(t) defined on X X, with with \\a(-kA{t))\\+ +\\p(-kA(t))\\ \\p(-kA(t))\\+ +\H-kA(t))\\ \H-kA(t))\\ + +Hw(-fcA(t))|| Hw(-fcA(t))||
a\\
(5.35) (5.35)

Furthermore, forall all (0,K] K] and and [0,T], T], k kGG (0, t tGG [0, Furthermore, for (5.36) (5.36)

or or \\kT>(t)w(-kA(t))\\(t)w(-kA(t))\\
(5.37) (5.37)

themethod methodis is of oftype type V((p) V((p)or orof oftype typeV*{ip), V*{ip),respectively. respectively. if the Proof. Sincethe theRunge-Kutta Runge-Kutta method isof oftype type V{j] deg[u>j] < —1, —1, which whichclearly clearly leads usto to(5.37). (5.37). of type < leads us

106 CHAPTER 5. DISCRETIZATION DISCRETIZATION

BY BY RUNGE-KUTTA RUNGE-KUTTA

METHODS METHODS

Now we are in a position position to to state state our our results results on on representability. representability. T h e o r e m 5.1. Suppose Suppose that that A(t) A(t) G G S(ip,a) S(ip,a) and and the the RK RK method method isis ofof type type V(ip), for some ip G (0,TT/2) and a G R. Let further further aa seminorm seminorm || \\t form A(t) and and let let A(t) A(t) itself itself satisfy satisfy hypothesis hypothesis a (711V, cr) -concordant pair with A(t) h61$Ji?;| |t], for some 71 71 G G [0,1) [0,1) and and G (0,1]. Then, with K > 0 sufficiently small, for all all kk G G (0, (0, K], K], the the entries entries of of the the operator operator matrices matrices {T>n-lD{tn)){\ + ka rDn)-le and ( D n - D ( t n ) ) (i + ka'D n)-1 a are defined and bounded on X and problem problem (5.19), (5.19), (5.20) (5.20) can can be be equivalently equivalently represented represented inin the form (1.1) with (5-38) Sin = 2l(*n) + ®n, (5-38)

),

(5.39) (5.39)

Qn = /,

(5-40) (5-40)

and Fn = w T (-fcA(i n )) (i - fc(D n - D(t n )) (i + fcoDn)" 1^ $ n ,

(5.41)

where $ n is ^iuen 6y (5.29). The proof of the theorem theorem is is based based on on the the following following Lemma 5.8. Under the the conditions conditions of of Theorem Theorem 5.1, 5.1, with with KK >> 00 suffisufficiently small, for all k G G (0, (0, K] K] and and ttn G Clk, the entries of the the operator operator kaV(t n))-le, matrices k(V n - !>(*„)) (i + A;aD(£ n ))-y k(T>n - £>(*„)) (i +

(i + fcaD n)-y (i + ka1) n)-le, k(D n - D(t n )) (i + kaV n)-la,

and k(l) n -

with 1>(tn)) (i + fcaB n)-1e fteZon^ to B(X), with n

- V(tn)) (i ^ C ^ + fc 1"71) (5.42)

and \\k{Vn ^ + k1-^1).

(5.43)

Furthermore, for all k G G (0, (0, iC] iC] and and ttn G fifc; ^ e entries of the operator operator r 1 matrices kT>n(\ + ka Dn)~ a and kT) n(i + kaT) n)~le are defined and bounded bounded on X, and the following following representations representations hold hold

(5.44)

5.3. CONNECTIONWITH WITH DISCRETE EVOLUTION EQUATIONS 107 5.3. CONNECTION DISCRETE EVOLUTION EQUATIONS 107 and and = kV(t (i = kV(t n) (i (i --fc(!D fc(!Dn - £(t £(tn )) )) (i (i ++ x (i V(tn))v(-kA(t ))v(-kA(t )). xfc(Dn -- V(t xfc(D n)).

(5.45) (5.45)

Proof. First of of all, all,since sincethe theseminorm seminorm| | t forms formsaa (-fi\(p, (-fi\(p,cr)-concordant cr)-concordantpair pair Proof. First with A(t), similarly similarlyto to (5.7) (5.7)ititholds, holds,for for all allkkGG (0,K] and with A(t), (0,K] and u Gul G , l,

\(I+ +kA^^ult kA^^ult < om T>omis is independent independentof of t,t, we weprove prove Next, DomA(i) = uniquesolution solution [z\,..., zzu} such suchthat that that equation equation(5.32) (5.32) possesses aa unique zz == [z\,..., that possesses Zj G GVom, Vom,j j==11,,..... ., ,v, v, for forany anyu uGG X^\andand further we weshow show (5.43). (5.43).More More X^\ further Zj precisely,since since GT>om T>om componentwise, componentwise, (5.32) is rewritten rewrittenin in the theform form precisely, zz G (5.32) is + kaV(t kaV(tn))z ))z + +ka(T) ka(T)n -- T>(t T>(tn))z ))z = = cm, cm, (i + which givesas as well well which gives = --((ii++kaV(t kaV(tn))))-lak(T> ak(T>n z=

D(tn ))z ))z + + (i(i++kaV(t kaV(tn))))-lau. au. -- D(t

(5.46) (5.46)

24 Now weput put24 3n = = fc(D fc(Dn - D(t D(tn )) )) (i (i+ +kaT)(t kaT)(tn))~ ))~la and andmultiply multiplyboth bothsides sides 3 Now we theleft left by by k(T> k(T>n - T)(t T)(tn)) to to obtain obtain of (5.46) (5.46)from fromthe of

k(Vn -- V(t V(tn))z ))z = = -S -SnkCD kCDn -- 2)(t 2)(tn))z ))z + + ggn u. u. k(V Applying further(5.42), (5.42), wetherefore thereforefind, find,if if K K >>00isis sufficientlysmall, small, that Applying further we sufficiently that for all all kk€€(0, (0, K] and G Q, Q,k, for K] and tn tG

IKi+ +S Snnrr' 'l l l^^CC IKi

(5.47)(5.47)

D(*n))z ))z = = (i(i++Sn^SnU. Sn^SnU. kCDn - D(* kCD

(5.48) (5.48)

and and Insertedinto into(5.46), (5.46), yields Inserted thisthis yields = --((ii++ka'D{t ka'D{tn))))-1a(i a(i + + ggn)"" )""1Snu + + (i(i++ z= 24 24

It follows thatthat followsfrom from(5.35) (5.35) the theentries entriesof of $$n are arebounded boundedon onX. X.

1 1

108 CHAPTER 5. DISCRETIZATION DISCRETIZATION BY BY RUNGE-KUTTA RUNGE-KUTTA METHODS METHODS Observe that, as already already shown, shown, the the entries entries of of the the operator operator matrix matrix (i(i ++ into T>om, hence the above argument argument is is ka'D(tn))~1a are mappings of X into correct. Further, by (5.35), (5.35), (5.42), (5.42), and and (5.47) (5.47) itit follows follows that that the the entries entriesofof the operator matrix (i ++ kaT> bounded on on X. X. Now Now kaT>n)~1a are defined and bounded a simple estimation of the the expression expression (i(i ++ Sn)~ Sn)~1SnU leads, with the aid of of (5.42), (5.47), and (5.48), to the the desired desired estimate estimate for for the the first first term term on on the the left-hand side of (5.43), and and the the same same for for the the second second term term follows follows by byusing usingaa similar argument. It is also also seen seen from from the the above abovereasonings reasoningsthat that the theentries entriesofof the operator matrices kT)n{\ + kaT> n)~la and kT) n(i + kaT) n)~1e are defined and bounded on X. It remains to show (5.44) and and (5.45). (5.45). We We denote denote 55n = k(T> n-T>(tn)) (i+ l ka'Dn)- a. It follows from (5.46) that that (i + kaVn)-^

= -(i + ka'D{t ka'D{tn))-1aBn + (i +

fcoD^n))-^.

(5.49) (5.49)

Multiplying now both sides sides of of this this equality equality from from the the left left by by kT>(t kT>(tn) and using the identity kDitn) (i + kaT){t n))-la = i - (i + / c a D ^ ) ) " 1 ^ which can easily be verified verified due due to to the the fact fact that that aa and and (i(i++ ka.T)(t ka.T)(tn))~l are permutable, we find that

Inserting this result into the the right-hand right-hand side side of of the the identity identity

kVn{i + fcoDn)-^ - kV(t n) (i + kaV n)-1a+

S n,

leads immediately to (5.44). (5.44). Further, acting similarly to to the the manner manner leading leading to to (5.50) (5.50) and and (5.44), (5.44),we we get kVnii + kaDn)-^

= kV(t kV(tn) (i + kaV(t n))-1e + (i + fccd)^))" 1 xfc(D n -D(t n ))(i + fca2) n)-1e (5.50)

and (5.51) Thus we see that (5.45) follows follows when when inserting inserting (5.51) (5.51) into into the the right-hand right-hand side of (5.50) and using (5.48) (5.48) and and the the trivial trivial identity identity Now the proof of the lemma lemma isis complete. complete.

5.3. CONNECTION

WITH WITH DISCRETE DISCRETE EVOLUTION EVOLUTION EQUATIONS EQUATIONS 109 109

Y n G X{u) be denned by Proof of Theorem 5.1. Letting Y )

I n—

we rewrite problem (5.19), (5.19), (5.20) (5.20) as as follows follows Yn+l =Yn- kbTT>nYn + kbT$n

for tn G Clk,

Yo = 9fy°

(5.52)

fca$ n.

(5.53)

and Yn = e®y n-fcaD nYn +

Since, by Lemma 5.8, the the entries entries of of the the operator operator matrices matrices (i(i++ ka karDn)~1a and 1 (i + feoDn)~ e are defined on X for /c G (0,-ftT] and t n G fifc, it follows from (5.53) that Y n = (i + kaVn)-^

® y n +fc(i+

fcaD,,)-^^.

(5.54) (5.54)

Inserting further (5.54) (5.54) into into (5.52), (5.52), we we find find that that problem problem (5.19), (5.19), (5.20) (5.20) can can be rewritten in the equivalent form

Yn+l = (l-kbTVn{i

1

)

T

+kb (i - kVn(i °

(5.55) (5.55)

the above above Therefore, to show the claim, it suffices to combine (5.55) with the representations (5.44) and and (5.45), (5.45), noting noting that that bbT'D(t)(i + ka'D(t))' 1e = 2l(t) D D and b T (i + kaV{t))- 1 = w T ( - i t ^ ( t ) ) . 2 5

R e m a r k 5.1. In particular, the assertion of Theorem 5.1 remains valid valid if, instead of holfLfi?; | \t], it is based on either of the more restrictive hypotheses hol®[i9; | | t ] or hol[i?]. So we have shown that under under some some reasonable reasonable conditions conditions problem problem (5.19), (5.19), (5.20) can be equivalently represented represented in the form (1.1), that is, as the Cauchy problem for an evolution equation in discrete time. In particular, this result yields that the then the unique unique solvability solvability of problem (5.19), (5.20) then convergence only. only. We We conconfollows and it remains to examine stability and convergence centrate on these questions questions in subsequent chapters of the the book. book. 25

Note that, as follows from from Lemma Lemma 5.8, 5.8, the the operators operators 2l(£ 2l(£n), 58n, 2ln are bounded on on X for k € (0,K] and tn 6 Clk.

110 CHAPTER CHAPTER 5. DISCRETIZATION BYRUNGE-KUTTA RUNGE-KUTTA METHODS 110 5. DISCRETIZATION BY METHODS

5.4 Comments bibliographical 5.4 Comments and and bibliographical remarks remarks Section The proofof ofLemma Lemma5.2 5.2follows followsthat that ofLemma Lemma2.2 2.2ininour our Section 55..11: :The proof of paper[37]. [37]. paper Section5.2: 5.2:For For generalbooks books inthe the field of ofRK RK methods,we werefer referthe the Section general in field methods, reader toButcher Butcher[53], [53], Dekker &Verwer Verwer[65], [65], Hairer, N0rsett, andWanWanreader to Dekker & Hairer, N0rsett, and ner [80], [80], Hairer& &Wanner Wanner[81], [81], and Stetter[155]. [155]. Although Althoughwritten written in ner Hairer and Stetter in the context ofordinary ordinarydifferential differential equations, these manuals wide present the context of equations, these manuals present aa wide rangeof ofideas, ideas,notions, notions, terminology. Moreover, in[53, [53,65, 65,81, 81, 155], range terminology. Moreover, in 155], theythey pay pay attention tothe the analysisof ofdiscretizations discretizations ofstiff stiffproblems. problems. mention analysis of We We also also mention attention to recentbook book ofHundsdorfer Hundsdorfer &Verwer Verwer[86] [86] which discretizations a recent of & which dealsdeals with with discretizations of both bothODEs ODEs andPDEs. PDEs. The The bookscited cited above aretherefore thereforerelated related to of and books above are to the present present which isintended, intended,in inparticular, particular, forapplications applications inorder orderto to the oneone which is for in examine discrete versions ofpartial partialdifferential differential equations. of Most the modmodexamine discrete versions of equations. Most of the ern approaches approaches tothe the studyof ofdiscretization discretization methods deal abstract with abstract to study methods deal with ern andthe the situationof ofpartial partialdifferential differential equations can differentialequations, equations, differential and situation equations can thenbe besettled settledas asaaparticular particular case. Inthe the abstract abstract case however, then case. In case theythey meet,meet, however, new features features ofwhich whichthe thecentral centralone oneisisthe the problemof ofinfinite infinitedimension. dimension. new of problem As concerns concerns different aspects ofgeneral generalRK RKmethods methodsconsidered considered inHilbert Hilbert As different aspects of in and Banach Banach space settings andapplied appliedto tolinear, linear,quasilinear, quasilinear, andfully fully nonnonand and and space settings linearproblems, problems, wemention mentionour ourwork work[16, [16, 19,20, 20, 22, 27, 30,36] 35, 36] linear we 19, 22, 25,25, 26,26, 27, 30, 35, and the thework workof ofBeyn Beyn& &Garay Garay[47], [47], Gonzalez &Ostermann Ostermann[71], [71], Gonzalez and Gonzalez & Gonzalez et al.[72], [72],Gonzalez Gonzalez& &Palencia Palencia[73, [73, 74,75], 75], Gudovich& &Terteryan Terteryan [78], et al. 74, Gudovich [78], Karakashian [90], Keeling [92], Lubich &Ostermann Ostermann[105, [105, 106,107], 107], NakNakKarakashian [90], Keeling [92], Lubich & 106, aguchi& &Yagi Yagi [115], [115],Ostermann Ostermann &Roche Roche[123], [123],and andOstermann Ostermann &ThaiThaiaguchi & & hammer[125]. [125]. hammer Section5.3: 5.3: Ourpresentation presentation is,in infact, fact, an animproved improvedversion version ofour our Section Our herehere is, of earlier work (These arealso alsoannounced announced in[16, [16,22, 22, 26].) 26].) earlier work [19,[19, 35]. 35]. (These results results are in

Chapter 6

Analysis of Stability In this chapter we further investigate the Runge-Kutta discretizations discretizations discussed previously in Chapter 5. We concentrate here on examining stability for discrete problems of the form (5.19), (5.20) by using seminorms | \t that form concordant pairs pairs with with A(t) in the sense of Definition 5.2 above. It has already been pointed out before that our techniques are based on reducing the starting problem (5.19), (5.20) (5.20) to the discrete Cauchy problem (1.1). In fact, under the assumption that A(t) is (<£>, u)-sectorial with some

6.1

Some resolvent estimates estimates

Our main aim in this section is to show certain weighted resolvent resolvent estimates estimates properties of the operator &2l(£) = a(—kA(t)) by using seminorms with properties regulated by Definition 5.2. In fact, such estimates are related related to the Aj(.. .)concordance of certain new seminorms derived from the original ones. 1

lt is seen from our below investigation that the operator 2l(i n) can be thought of as the principal part of 2ln in some sense.

Ill

112

CHAPTER CHAPTER 6. 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

Perhaps, it will not be a good style if it is often used to state, one after the other, even two assertions which are formally different but the latter of which is obvious as soon as the former is stated and proved. We therefore accept the following.

Convention. In what follows there often often occur occur different different assertions assertions which which involve seminorms and operators in the space X and possess, at the same time, dual versions in X* in which all objects in X are replaced by their doubles in X*. For example, the former estimate in Definition 5.2 possesses a dual version which is just the latter estimate involving involving the corresponding objects in X*. In such situations we will not state very often the dual version explicitly, marking marking by [*] the corresponding assertion. assertion.2 Furthermore, if we need to refer to the dual version subsequently, subsequently, we refer to the explicit result (in X) marked by *. For example, Theorem 6.1 below possesses a dual version which may be referred to in the sequel, although not stated explicitly, as Theorem 6.1*. Now we present a series of helpful resolvent estimates. estimates. Theorem 6 . 1 . [*] Suppose that A(t) G S(ip,a) and a seminorm \ \t forms a (£\m\tp, a)-concordant pair pair with with A(t), for some ip G (0,TT/2), a G K, £ > 0, andrn G N such thatm—l < £ < m. Let also the RK method be A j (if)-stable and let a rational function u(z) be ((p)-regular and subject to deg[cj) < —m. Then there exist a set T of Ai-configuration and numbers 5 > 0 and K > 0 such that for all k G (0,K), t G [0,T], A £ Int (TUV(8k)), andu£X, \u(-kA(t)) (XI - k%{t))k%{t))-mu\t < C|A|*- m|M|.

(6.1)

In the case £ = m — 1, under the extra requirement that the pair (| \t;A(t)) is (£|£|, 1 and \u\t < C\\u\\ for all t € [0,T] and u G GX X ifif ££ == 0, 0,

^^ '' ''

it is enough to suppose that deg[o;] < —£ instead of < —m. If a > 0 in in addition, then the above still holds for 5 = 0 and K = oo. Proof. Assuming first that (6.2) is not generally satisfied, we then act under the condition deg[u;] < —m. Let 0 be the same as in the proof of Lemma 5.2. Observe that since the method is A(<^)-stable, the poles of the functions a(—z) and p(—z) lie outside of S ^ - 3 Let further 2

This symbol appears just just before before the statement. We recall that the function p(z) is defined by (5.23).

3

6.1. SOMERESOLVENT RESOLVENT ESTIMATES 6.1. SOME ESTIMATES

113

>00be bechosen chosensufficiently sufficiently small sothat that p(z), p(z),a(z), a(z),and and ui{z)have haveno nopoles poles d> small so ui{z) V(d). Also, Also,we wechoose chooseK K< 0>and K =Koo=if oo R= in By the theaccepted acceptedconditions conditions there exists setT' T'ofof Ai-configuration such By there exists aa set Ai-configuration such that the thefunction functiona(—z) a(—z) maps intoT'. T'. Let LetTTbebe a set Ai-configuration of that maps T,V1 into a set of Ai-configuration V1T, such thatT' T' \\ {0} {0}CC IntT. We Weselect select aa fixed fixeddo do >> 00sufficiently sufficientlysmall small to such that IntT. to have have do\p(z)\< < ^^ do\p(z)\

for X>(d). e for allall z ez X>(d).

(6.3) (6.3)

Let nowF\ F\bebethe the contourcoinciding coinciding the boundary boundaryof of the theset set Let now contour withwith the V(min{do\X\, d})US^. We Wethen then apply applythe theoperator operatorcalculus calculus formula (A.10) V(min{do\X\, d})US^. formula (A.10) 1 with kA(t) kA(t)ininplace placeof of££ and andwith withT\ T\ in inplace placeof ofH# H#+e toget, get,with with5 5 ==dQ dQ R, with R, +e to for all allkkGG(0, (0, K], 0 Int U (T V(6k)), and anduu€€X,X, for K], A 0AInt (T U V(6k)), m m m m G :=u(-kA(t)) u(-kA(t))((XI ((XI - a(-kA(t)))(A- -a(oo))a(oo))J) u G := - a(-kA(t)))-- (A J) 1 = 7r^ 7r^ [ [ am a(X;z)u:(-z)(zI-kA(t))(X;z)u:(-z)(zI-kA(t))udz, = udz, m

(6.4) (6.4)

where where m m (A; z) z) == (A (A- -aa((--zz) ) — ) —-- (A (A- - a(oo))" a(oo))" . 5m m(A;

Integratingby byparts partson onthe theright-hand right-handside side of (6.4), (6.4),we weget getinstead instead Integrating of G G

m m yym (X;z)(zI-kA(t))udz, udz, m(X;z)(zI-kA(t))-

(6.5) (6.5)

where where

{

am (X;z)uj(-z) m(X;z)uj(-z)

if mif= 1, m = 1,

m 2 m (A;z1 )a;(-zi)^ )a;(-zi)^ (m m--11) ) / / (z-2i) (z-2i) - 5m ifif mm>>2,2, m(A;z 1 tv>1 tv>1 while thepath path of of integration integrationfrom from ooe~ to zz lies lies on on Y\. Y\. while the ooe~ to Assumefor for aamoment momentthat thatwe wehave have shown shownthe theestimate, estimate,for for A A^^Int Int Assume TT and GT\, T\, and zz G

IAMzl)IAMzl)-1

ifif mm= =l , l ,

((

..

( tt66))

Sincethe Since thepair pair (| (| \t;A(t)) ;A(t)) is is (^|m|
m m <
(6.7) (6.7)

113

CHAPTER 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

114

If A £ Int T, 6k < |A| < CMQ \ noting that £ < m, it follows follows from from (6.5), (6.5), (6.6), (6.6), and (6.7) that

\G\t < CAT* f \z\t- m-1\dz\ \\u\\ < Ck-*\\\*

u .

(6.8)

(6.5), In the case where m > > 22 and and A A ££ IntT, IntT, |A| |A| >> dd dd0 1 , we derive, using (6.5), (6.6), (6.7), and the fact that that 11 << (( < < m, m, \G\t

1

""1 / Iz^dAI + lzD^ldzHHI

<

f\z\t-m-1\dz\
(6.9)

Finally, if m = 1, since since ££ G G [0,1), [0,1), by by (6.5), (6.5), (6.6), (6.6), and and (6.7) (6.7) we we have, have, for for all all

A i IntT, |A| >dd^\ 1

\G\t<

/

u .

(6.10)

Observe furthermore that that a(oo) a(oo) // 0, 0, since since the the RK RK method method isis ofof class class Sj. Sj. Also, it necessarily holds a(oo) a(oo) G G T'. T'. Therefore, Therefore, |A - aioo)^ 1

< C ( 1 + |A|)- 1

for all A g Int T.

(6.11) (6.11)

By Lemma 5.2 and (6.11) (6.11) we we then then obtain, obtain, for for A A^^ IInnttTT and and uu GGX, X,

n

\\u\\.

(6.12)

The claim thus follows by by combining combining (6.4), (6.4), (6.8), (6.8), (6.9), (6.9), (6.10), (6.10), and and (6.12). (6.12). It remains to show (6.6). (6.6). Since Since a(—z) a(—z) G GT' T' whenever whenever zz GGSS^^,, we we have, have, for all A ^ Int T and z G GH HV1,

Next, by the i4(y>)-stability i4(y>)-stability of of the the method, method, the the zeros zeros of of a(—z) a(—z) lie lie outside outsideofof the set S^j \ {0}. Also, Also, 00 is is aa simple simple zero zero of of a(—z).4 These observations yield that cx(—z) admits admits the the two-sided two-sided estimate estimate

(1 + N T 1 < \a(-z)\ < ClzHl + lzl)'1 4

for all z G E Vl .

We recall that r(0) = r'(0) r'(0) == 11 for for all all RK RK methods methods under under consideration. consideration.

(6.14)

z

6.1. SOME RESOLVENT ESTIMATES

115

GT Thus, since deg[a(oo) ] < - 1 , with the aid of (6.11), (6.13), a(-z and (6.14) a simple estimation gives, for all A ^ IntT and G £>(min{d 115 6.1. SOME RESOLVENT1, ESTIMATES (6.6) immediately follows by (6.15), (6.16), and

= |a(oo)z - a(-z)\ |A - c^ \~ Thus, since deg[a(oo) ] < - 1 , with the aid of (6.11),- (6.13), and (6.14) l a(-z)]- gives, 2|Ar a simple estimation for all A ^ IntTz and z G E ^ , (6.15) m— for (6.3) that Furthermore, it follows from l - a(-z = |a(oo) - a(-z)\ |A - c^ |A|the +\~|zi|, identity dd^ In view of |A < for all |A|,d}). (6.16) 0 only. Assuming therefore the last J (6.15) It is now seen that 1if o (6.11), since \u(-z)\ , Furthermore, it follows from (6.3) that . Let further m > 2. Assume first that |A| < A |A - a(-z)]-1 < 2|Ar J Qz for (6.16) < all z G £>(min{d o|A|,d}). 1 G It is now seen /that{ zif- m — 1, (6.6) immediately follows by (6.15), (6.16), and z ) - a { \ ;1) z i )and u ( - (1 z + ) d |A|) z (1= +~ (6.11), since \u(-z)\ < C -fora (z- G TA. = 5i(z)((A 1 m 2 . In view of the identity Let further m > 2. 1 Assume m first that |A| 1 < dd^ 1 it suffices to examine the case z restriction and using next the identity /

(z)

2 { z - z 1 ) m --(m a m {-\ ; z i ) u ( - z 1 ) d z 1

< C J

= ~

... + (A - a(

t to examine andsuffices the above estimates (6.13), it the(6.11), case Qz < 0 (6.15), only.

T n S E ^ and sinceusing deg[w] < the identity restriction next

/

,

and (6.16),therefore we find, the for last Assuming dV(d

(A - into a( account the con= 5i(z)((A - a ( - we also have...to+take lies on <9D(do|A|), and the above estimates (6.11), (6.13), (6.15),Dand (6.16), we find, for z G 2 + + (1 TA n S E ^ since deg[w] < -(m (1 + + |A|) (1 + \zi\) > |A|G+ |zi|, x ((|A| + - 1) and = t(z)

A

< C J 2

z

< C J z

z)\ < C^ x ((|A| +\z\)-\

\z\

z

T\ z

+ (1 +

+

(6.17)

Clearly, if dT>{do\\\). It is easily seen tribution coming from the integration over then that |* (A; so that (6.17) is still valid for \X\) since do\\\. These reasonings (6.17) 1 \z\)-\ show (6.6) for |A| <

< C J

m

1

o

Clearly, if z lies on <9D(do|A|), we also have to take into account the contribution coming from the integration over T\ D dT>{do\\\). It is easily seen so that (6.17) is still valid for z G dV(do\X\) then that |*m(A; z)\ < C^1 1 since \z\ = do\\\. These reasonings show (6.6) for |A| <

116 116

CHAPTER 6. ANALYSIS ANALYSISOFOFSTABILITY STABILITY CHAPTER 6.

1 thecase case |A| |A|>>dd^ dd^ similarargument argument leads tothe thefollowing followingestimate. estimate. In In the , aasimilar leads to 5 For zz G GF\, F\, since since \z\\ \z\\>>\z\ \z\>> For d,d,

z

C < C

j

This result resultagain again implies This implies (6.6).(6.6). wediscuss discussseparately separately thecase case£ £ == m m— — 1. 1. Note Notethat thatall allthe theabove above Finally,we the Finally, for the theestimate estimate(6.12), (6.12), arein inforce force even evenif if deg[a>] deg[a>]< < conclusions,except except conclusions, for are —(m (m — —1) 1) == —£. —£. However, However, using thelast last restriction restriction onuu and and assumingin in — using the on assuming additioncondition condition (6.2), byLemma Lemma5.2 5.2ititfollows followsthat that(6.12) (6.12) holds aswell, well, addition (6.2), by holds as whichshows showsthe theauxiliary auxiliaryassertion assertion inthe theconsidered consideredcase. case. in which theproof. proof. This completes This completes the The following followingresult result is aaconsequence consequenceof ofTheorem Theorem 6.1. The is 6.1. Theorem6.2. 6.2.[*] [*]Let Let A(t) S(tp,a) G andlet letthe the method be be Ai(ip)Ai(ip)Theorem A(t) G S(tp,a) and RKRK method tp G G(0, (0,TT/2) TT/2) and andaaGGM.M.Then Then there thereexist exist setTTofof stable, withsome some stable, with tp aa set A\-configurationand andnumbers numbers6 6> >00and and such 0 that thatfor for all allk kEE (0,K], K], A\-configuration KK> > 0 such (0, te [0,T], [0,T],and andA A <£lnt <£lnt(T (T UV(8Jc)), UV(8Jc)), te 1 l ||(A7- fcaft))fcaft))!! < < C\\\~ C\\\~ ||(A7 !! .

(6.18) (6.18)

>00ininaddition, addition, then then(6.18) (6.18) for88==00and and = oo. still still holdsholds for KK = oo. If cc > The resultimmediately immediately follows if one one takes takes£ £ == 0,0,mm— —1, 1, and and| | \t = = |||| |||| The result follows if in theconditions conditionsof of Theorem Theorem 6.1. in the 6.1. Further, Further,we weconsider considermethods methods of ofclass classSJJ. SJJ. Theorem 6.3.[*] [*]Let Let A(t) <S(, G a), a), let letthe the RK method be be Au{ip)-stable, Au{ip)-stable, Theorem 6.3. A(t) G <S(, RK method and let let aaseminorm seminorm || \t\tform form aa (£\m\tp, (£\m\tp,a)-concordant a)-concordant pairpair with with A(t), A(t),for for and some (pG G(0,vr/2), (0,vr/2),aa G GM, M,£ £> >0, 0, and mNGsuch such N that thatm m— —1 1 << ££<<m. m. some (p and m G Suppose also alsothat that rationalfunction functionLO(Z) LO(Z)is is ( > £(ro £(ro++1)1) if£if£ = m-lorm>2 and £ and m -+ 1), l/(w /« + i1), < T = m-lorm>2 = m£ - = l/(w n\ otherwise, ?> > £(ro £(ro++1)1) otherwise, 5

As above, above,without without ofgenerality generalityit it can canbe beassumed assumedthat thatQz Qz<<0.0. of As lossloss

/« i n\

6.1. SOMERESOLVENT RESOLVENT ESTIMATES 6.1. SOME ESTIMATES

117 117

where w == — —deg[a]. deg[a]. Then Then setTTof of Ai-configuration and where w therethere exist exist aa set Ai-configuration and numbers and such 0 that thatfor forall all (0,K], tt GG[0,T], [0,T], A A££ numbers 88 >>00and K K > 0>such k kG G (0,K], Int (TU UV(5k)), V(5k)), and and Int (T uue e X,X, m m \to(-kA(t)) (XI- -k%(t))k%(t))u\t \to(-kA(t)) (XI u\

m
(6.20) (6.20)

In the theparticular particularcase case £==00and andm m==1,1,(6.20) (6.20)still still holds i/deg[cj] <0, 0, In £ holds eveneven i/deg[cj] < instead of<<— —1, 1, provided providedthatthat instead of \u\t <
(6.21) (6.21)

cr> >00in inaddition, addition,then then theabove aboveis isin inforce forcefor for66== andK K ==oo. oo. If cr the 00 and If Proof. Wefirst first prove prove themain mainassertion assertion ofthe the theoremwithout without assuming Proof. We the of theorem assuming (6.21).The Theparticular particular 1, and (6.21)holds holds isdiscussed discussed (6.21). casecase wherewhere ff ==0,0, mm == 1, and (6.21) is separatelyin inthe the final part partof ofthe the proof. separately final proof. Sincethe theRK RK methodis isof ofclass classSJI, SJI,we we havea(oo) a(oo) = 0, 0,which whichis isequivequivSince method have = alent tothe the inequality w >>1.1.This Thiscircumstance circumstance toextra extra difficulties difficulties inequality w leadsleads to alent to whendealing dealing with methods ofclass classSu Su for for the reasonthat that a(z)vanishes vanishes when with methods of the reason a(z) both at at00and and at at oo.oo. both Let T, T,T\, T\,d,d, 8, and >dd$ dd$l. Therefore, Therefore, do we allows us case we do 1 1 no more morethan than toconsider considerthe thesituation situationwhen when |A|< >11and and usingthe theoperator operatorcalculus calculus formula Now,noting noting Now, thatthat <; using formula (A. 10),(A. 10), forall all (0,K], K],A A£ £ Int Int(T (T UV(5k)), V(5k)), and and we have, have,for we kk GG (0, U uue e X,X, m m G:= :=u(-kA(t)) u(-kA(t)) (XI-- a{-kA(t))) a{-kA(t))) u G (XI u

= —. —.I I =

m 1 m 1 (zI-kA(t))udz. u(-z)(X-a(-z))u(-z)(X-a(-z))(zI-kA(t))udz.

(6.22) (6.22) byparts, parts,we weget get well Integrating by asas well Integrating G= = G where where

(

uumm(z) (z) m 2 2 m um (zl)dz )dzl (m-1) (m-1) / / (z-z(z-z m(z l) - u

if m if =m 1, = 1,

m>2, ifif m>2,

CHAPTER 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

118

with um(z) = u)(—z) (A — a(—z)) m, while the path of integration integration from from 1 ^ to z lies on T\. Let m — 1. Then noting that that for for zz ££ T\, T\, (6.23)

and (6.24)

by (6.7), (6.13), and (6.16) (6.16)6 we have, since 0 < £ < 11 and and <<;; >> 1, 1, for for all all

\(£Int(ruT>(6k)), -l\dz

|G|t < C

-1 + cfi

dx

log(2

C

which shows (6.20) for m m= = 1, 1, due due to to the the fact fact that that qq>> £(zu £(zu++ 1)1) and, and, thus, thus, the last two terms of the the expression expression in in parentheses parentheses are, are, in in fact, fact, dominated dominated by the first one. Now we study the case m m >> 22 ,, m m -- ll << ££ << m m — l/(tu + 1). For short we put £{h) = log(2 + 1/h)

for for hh > > 0. 0.

(6.25) (6.25)

We first show a suitable estimate estimate for for |0 |0 m (A;z)|. By (6.13) and (6.23) (6.23) itit follows that for z e T ^ f l dd EE^^,,

-H-z)\rm C(\X\ + \z\)-m C\z\'wm C|A|-m

if if

\<\z\<

if

>

Using this estimate and (6.24), (6.24), since since m{w m{w ++ 1) 1) — 1 > ? > £(ZJ + 1) > m, we 6

Clearly, (6.13) and (6.16) (6.16) hold hold for for methods methods of of class class SII SII as as well. well.

6.1. SOME RESOLVENT ESTIMATES ESTIMATES

119

have, 7 for z € V\ n dT,(pi, \z\ < min {l,

, Qz < 0,

|6m(A;z)| < C J"

< c?u

,m-2

Jz

r,wm+m—2—<;

dx

o

+C|A|- m /

dx

-l + ci(\x\) 6

L—mj/zu—m

(6.26)

where (5^ is the Kronecker Kronecker delta. delta. Clearly, Clearly, the the last last estimate estimate remains remains valid valid as as 1 well for z eV x such that z € &D(d Q\X\) or z € S S ^ and 1 << |z| |z| << llA A))-- ^ . (The first integral in the the second second line line does does not not enter enter ifif \z\ \z\ >> 1.) 1.) Besides, Besides, as as in the proof of Theorem Theorem 6.1, 6.1, the the restriction restriction $sz $sz << 00 isis not not needed. needed. AA similar similar argument also yields since since m m< < ?? ++ 1, 1, for for zz €€ T\ T\ such such that that \z\ \z\ >> |A|~ |A|~1//ro, c

< c\x\~

m

(x-\z\)

m-2

dx

J\z\

o

\

x

dx

l

-\

(6.27)

With the aid of (6.26) and and (6.27) (6.27) we we then then obtain, obtain, for for all all AA££ Int Int (TuV(6k)), (TuV(6k)),

\G\t < C\\u\\

which leads to (6.20), since since the the second second term term of of the the sum sum on on the the right-hand right-hand 8 side is dominated by the first first one, one, in in view view of of (6.19). (6.19). Let now m > 2 and m — l/{w + 1) < £ < m so that that ?? — {w + l)m. It is then not hard to see that for for zz G GT\, T\, \z\ \z\ < < min min {l, {l, , the second term 7 8

The second integral in the the second second line line does does not not enter enter ifif |A| |A| >> 1.1. It is easily seen that \G\t \G\t is is bounded bounded as as stated stated for for |A| |A| >> d^n7^^tD

as well.

120

CHAPTER6.6.ANALYSIS ANALYSISOFOFSTABILITY STABILITY CHAPTER

in the the second second line line and and the theterm termininthe thethird thirdline lineof of(6.26) (6.26) bounded areare bounded by by C and and C|A| C|A|x/ro, respectively, respectively, and, and, thus, thus,both bothadmit admita acommon commonupper upperbound bound C^l"1 . Therefore, Therefore, instead instead ofof (6.26) (6.26)we wehave have|6|6 (A;z)| << Cl^l"" Cl^l""1 for all all m (A;z)| 1//ro T\ with with \z\ \z\ <<min min{l, {l,|A|~ |A|~ } since since the the first firstintegral integralininthe thesecond secondline line z G T\ ofof z)\ is is is under under the the same samebound boundasasabove. above.Clearly, Clearly, quantity (6.26) is thethe quantity |©m|© (A; z)\ 1 estimated in in the the same sameway wayfor forallallz zGG with1 < 1 <\z\\z\ jAl"" too. Noting Noting estimated T\T\with << jAl"" ^ too. m further that that (6.27) (6.27) remains remainsunchanged unchangedasaswell, well,wewe that\G\t \G\t C|A|^~ further seesee that < < C|A|^~ Int (T (T U UV(Sk)), V(Sk)), which whichimplies impliesagain again(6.20). (6.20). for A ££ Int remains to to show show that that ifif££==0,0,mm= =1,1,and and(6.21) (6.21)hold, hold, estimate It remains thethe estimate is still still valid valid under underthe theweaker weakerrestriction restrictiondeg[o;] deg[o;] Since case (6.20) is << 0. 0.Since thethe case actually settled settled ininthe thepreceding precedingpart partofofthe theproof, proof, without without deg[u;] << — 1 is actually deg[u;] of generality generality itit isisassumed assumedthat thatu>(z) u>(z) = 1. Also, Also, from from now nowon onwe weaccept accept loss of (6.21) holds. holds. Let Let further furthera*a*GGCCbebedefined definedbyby that (6.21)

lim a* = lim

(zpa(z)),

given by and let FFA and FFA be given F A = cA U lA+) U [A-) and FFA = cA U lA+1 U lA~] U? A ,

where where r

ii

cA = }{ zz G GC C :: \z\ \z\ == - | A / aa**||

[f = jz ee C : \z

,, /p

] ]


,

(

GC C :: \z\< \z\< ddo |A|, | a r g ( -- zz))|| < ir -?A = {z G and

\zeC: lf] = \zeC:

do\X\<<\z\ \z\<<-|A/a*| -|A/a*|1 / p , arg^ arg^ == do\X\

x

suppose that that lAl and and ((A are oriented oriented downwards, downwards,which whichfurther furtherspecifies specifies We suppose orientation of ofthe thecontours contoursFF FA . Using Using next next the the operator operatorcalculus calculus the orientation A and F formula (6.4) (6.4) with with mm — —11and andnoting notingthat thata(oo) a(oo)= =0 now, 0 now, formula wewe get,get, for for all all (T U UV(8k)) V(8k)) and uu G G X, X, A (f: Int (T G

:= :=

-

\~1u+—

2-KI

[ ((X-a(-z))((X-a(-z))-1 -A"1) (zI-kA(t))(zI-kA(t))-ludz.

JTx

2 the norm norm of of the theintegrand integrandisisO(|z|~ O(|z|~ follows from from Cauchy's Cauchy'sTheoTheoSince the ), it follows that after after aa suitable suitabledeformation deformationofofthe thecontour contourT\,T\,thethe formula rem that lastlast formula is is

6.1. SOME RESOLVENT ESTIMATES 6.1. SOME RESOLVENT ESTIMATES

121

121

in force with +/ /rr2 2 in inplace place of Observe that the resolin force with JJrrii + of JJ ..Observe alsoalso that the resolrr l (zl— —kA{t))~ kA{t))~ is is holomorphic inaa neighbourhood ofthe theset setsurrounded surrounded vent(zl vent holomorphic in neighbourhood of by Y\.Hence, Hence, by Cauchy's Theorem, by Cauchy's Theorem, by Y\. 1 1 1 1 {zI-kA(t))udz = = O. !! \-\{zI-kA(t))udz O.

M M Wetherefore therefore obtain We obtain 1 1 ll G = = (2m)(2m)( ( A - a ( - z11)-A" ) -A" - 11)) (zl (zl- -kA{t))~ kA{t))~ udz G [[ ((A-a(-z))udz

H H

f

1 1 ll (X-a(-z))(zI-kA(t))udz (X-a(-z))(zI-kA(t))udz

u ++ X-xxXu

11

1 1

u. u.

(6.28)

(6.28)

Sincethe the pair(| (|\\tt;;A(i)) A(i)) is(0|l|
(6.29) (6.29)

Also, itisis easilyseen seen for A^ ^IInnt T t Tand and F\, Also, it easily that that for A zz 66F\, m 2 11 11 2 m (A" (A-- aa((- z- )z) )- xx)| -<
theaid aid of(6.29) (6.29)gives, gives, for A£ £Int Int(T (T UV(6k)) V(6k))and and ueeX, X, whichwith with the of for A U u which

HGiH^CIAr^HI. HGiH^CIAr^HI.

(6.30)

(6.30)

Next,to toestimate estimate Gion on theright-hand right-hand of(6.28), (6.28), observe that inside Gi the side side of observe that inside Next, of the theset set(of (of points) z ofwhich whichY\ Y\isis theboundary, boundary, theoperator operator function of z points) of the the function l 11 —kA(t))" kA(t))" is is holomorphic thefunction function — (Aa(—z))~ a(—z))~has has finitely (zl — holomorphic whilewhile the (A — finitely (zl manypoles. poles. One One pole poleis islocalized localized in inan an annulus annulus many eC: di\X\ di\X\ < \z\< < \z\ <^2!Aj}, <^2!Aj}, < di,d di,d =const const>>0,0, {z {z eC: 2 = 2

and the otherones ones lieoutside outside ofa adisk diskV(ds), V(ds), d%= = const const>>0.0. Thenorm norm of and the other lie of d% The of thecontribution contribution ofthe thefirst firstpole pole tothe the totalsum sum ofresidues residues ofthe theintegrand integrand the of to total of of does notexceed exceed CjA| 1|tt||-while while the otherpoles poles contribution whose does not CjA| -111|tt|| the other give give aacontribution whose not exceedC\\u\\. C\\u\\.It Itthus thusfollows follows for A^ ^ Int Int(T (TU U V(6k)) V(6k)) normdoes does not exceed that that for A norm and uG GX, X, ^ ^ and u

HGall^ClArVH. HGall^ClArVH.

(6-31)

Finally, using and and (6.31) (6.31)as aswell wellas as(6.21) (6.21)for for an an estimation estimation of ofthe the Finally, using (6.30)(6.30) right-hand right-hand sideside of of(6.28) (6.28)readily readily leadsleads to to(6.20) (6.20)in inthe thecase case£ £==0 0 and andm—1. m—1. The proof The proof is isnow now complete. complete.

(6-31)

122 122

CHAPTER 6. ANALYSIS OF STABILITY CHAPTER 6. ANALYSIS OF STABILITY

The followingis, is, in infact, fact, aa consequence consequenceof of Theorem Theorem6.3. 6.3. The following Theorem 6.4. 6.4. [*] [*]Let LetA(t) A(t)G G S((f,a) and andlet letthe theRK RKmethod method be be Au(tp)Au(tp)Theorem S((f,a) cpG G (0,w/2) (0,w/2) and andaa GGR.R. Then Then there thereexist existaa set set TTofof stable, with with some somecp stable, Ai-configuration andnumbers numbers 5 5> >00 and andKK>> such that thatfor for all allkkGG(0, (0, K], Ai-configuration and 00 such K], te[0,T}, and\£ and\£ Int Int(T (TUU V(6k)), V(6k)), te[0,T},

o~ > > 00 in inaddition, addition, then thenthe the last last estimate estimatestill stillholds holds for 66==00and and If o~ for KK= = oo.oo. This result is is easily easily seen seenif if one onetakes takes ££ == 0, 0, mm ==1,1,w(^) w(^) 1, and This result = = 1, and the conditionsof of the thepreceding precedingtheorem. theorem. | |t|t == |||| || ||ininthe conditions Remark 6.1. 6.1. [*] [*] The Theresults results of of Theorems Theorems 6.1 6.1 and and6.3 6.3actually actually show showthat that Remark 9 the accepted acceptedconditions conditions theseminorm seminorm |||||| |||n = = \u(—kA(t \u(—kA(tn)) \t \tnn forms forms under the under the A{(£|m)-concordantpair pair As concerns concerns Theorems Theorems6.2 6.2 and and6.4, 6.4, a A{(£|m)-concordant withwith 2l(i n2l(i )- As their assertions actually actually yield for Aj(tp)Aj(tp)- and andAu(tp)-stable Au(tp)-stable methods, methods, their assertions yield thatthat for the operator operator2l(t 2l(t the A\-condition, in particular, the A\-condition satisfies ) satisfies the A\-condition, in particular, the A\-condition the n ifo> 0. 0. ifo- > Remark 6.2. 6.2. [*] [*] Under Under the the conditions conditions of of Theorem Theorem 6.3, 6.3, in in the thecase case £ £ == Remark —l/(w l/(w ++1) 1)and andq q £(m + 1), resolvent estimate estimate(6.20) (6.20) holds m — = =£(m + 1), the the resolvent holds withwith + 11 in inplace place of of m. m. m + This fact fact can canbe beshown shown by by using using the thesame same reasonings reasoningsas as those those leading leadingto to This (6.20). (6.20). In conclusion, conclusion,we wediscuss discuss methods methodsof of class class5 //////.. Theorem 6.5. [*] [*]Let LetA(t) A(t)G G S(tp,o~),let let the theRK RKmethod method be be Auj(ip)-stable, Auj(ip)-stable, Theorem 6.5. S(tp,o~), let aa seminorm seminorm || ||t form, form, aa (£\m\(p,cr)-concordant (£\m\(p,cr)-concordant pair A(t), for for and and let pair withwith A(t), some (p (pG G(0,TT/2), (0,TT/2), a aG GR, R,££>>0,0,and and G such such N that thatm m— —1 < < ££ << m. m. some mm GN Also, suppose that thataa rational rational function functionOJ{Z) OJ{Z) is is (ip)-regular (ip)-regularand and satisfies satisfies the the Also, suppose condition deg[w] deg[w]< < — —<< for some some <; <; > > m. m. Then Then there thereexist existaa set set TT ofof A2condition rrfor A2configuration, with withsingular singular points £1 == 00 and and£2£2= =a(oo), a(oo), and and numbers numbers configuration, points £1 5>0 andK andK >0 >0 such such that thatfor for all allkkGG(0,K], (0,K], t£ t£ [0,T], [0,T], A Agg Int Int(TUX>(<JA:)), (TUX>(<JA:)), 5>0 and uG GX, X, and u

\to(-kA{t)) (XI (XI -\to(-kA{t)) a(oo)|)*|A- a(oo)r a(oo)ri n {{00' ( c -« / r oo- m }}|M|, |M|, (6.32) (6.32) - a(oo)|)*|A The main mainrestriction restriction is that that the the RK RKmethod method is is of ofclass class Si Si or orof ofclass class5//. 5//. The is

6.1. SOME RESOLVENT RESOLVENT ESTIMATES ESTIMATES 6.1. SOME

123 123

wherezu deg[a(-)-a(oo)],K = = (<5 (<5?,(^+i)m ,(^+i)m wherezu = -- deg[a(-)-a(oo)], and £(fo) £(fo) is is defined defined by by (6.25). (6.25). In In the the case case £ = = m m— —l/(w l/(w ++1)1)and and<; — — and m{w 1) — 1, the the last last estimate estimate holds holdswith withK = 00 and andTO£/I TO£/I m + + 11 substituted substituted m{w + 1) m. Also, Also, for for ££ == 00 and andmm==1,1,(6.32) (6.32) is still still valid valid even evenwhen when > 00 inin provided that addition, the the above above holds holdsfor for 66 — —00 and andKK==oo. oo. addition, Proof. For A's A's bounded bounded away away from froma(oo), a(oo),the the proof proof is similar similar to to that that of Proof. For Theorem 6.1. For For A's A'sbounded bounded away awayfrom from0, the the claim claim follows followsin aa manner manner Theorem 6.1. similar that used used in the the proof proof of Theorem Theorem 6.3. 6.3. similar to that Now we state state a result result of particular particular character characterwhich whichis intended intended for for subsubNow we sequent use. use. sequent Theorem 6.6. [*] [*] Let LetA(t) A(t) € €£( > 00 and andKK >>00such such that that with with any any Ci = 00 and fixed andmm 66NNsuch such that that £ < <m m and andany anyfixed fixed no no > > 00 sufficiently sufficiently fixed ££ > 00 and for all allkk €€ (0,K), (0,K), t G G [0,T], [0,T], and and A A££ Int Int (TUV{Sk)), (TUV{Sk)), large, for large,

m

|A-- a ((oooo))| |m i nn ^ / r o - m > ) , (6.33) (6.33) + |A

where —— deg[a(-) deg[a(-) — a(oo)]. a(oo)]. In particular, particular,11 for for all allkk ££ (0, (0,K\, K\,t t6 6[0,[0, T], where zu — T], and Int (TuV{8k)), (TuV{8k)), and A ££ Int

a^))"1 !! < C C (lAl"" (lAl""1 + |A A-- a(oo)r a(oo)rx ) . \\{XI- a^))" \\{XI

(6.34) (6.34)

o~ > 00 in in addition, addition, the the above above holds holdsfor for 55 == 00 and andKK==00. 00. If o~ Proof. First First of all all note note that that the the latter latter stated statedestimate estimate(6.34) (6.34) consequence Proof. is aa consequence Theorem 6.5 6.5 in in the theparticular particular case casewhere where£ = = 0, 0, m m == 1,1,| | \t\t==|| || ||, ||, and of Theorem and L0(z) 1. L0(z) = 1.

Further, Further, for for A's A's bounded bounded away awayfrom froma(oo), a(oo),(6.33) (6.33) follows follows by by (1.2) (1.2)and and the convexity convexity inequality inequality(A. (A. with = ££ and and(3 (3— —mmsince since the the operator operator 18)18) with a = u)(—kA(t)) is uniformly uniformly bounded boundedwith withrespect respect (0,K] K]and andt tGG[0,T], [0,T], u)(—kA(t)) to kk €€ (0, 10

It is therefore therefore allowed allowedthat thatit would would hold holddeg[w] deg[w]= 0. 0. "When u{z) "When u{z) = = 1,1,££==0,0,mm==1 1

124

CHAPTER CHAPTER 6. 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

by Lemma 5.3. For A's bounded away from 0, (6.33) is a direct consequence of the estimates and ||(AJ - k%(t))-mLo{-kA(t))\\

< C\X - a(oo)| m i n { ( W r o - m } ,

(6.35)

of which the former one follows from (1.43) with 2l(£) substituted for £. 12 As concerns the latter inequality (6.35), for q < mw it is obtained by applying directly the representation (6.22) and next using the argument employed in the proof of Theorem 6.3. At the same time, if ? > mm, it is seen that as in the proof of Lemma 1.12,

||(AJ - m{t))-m{a{-kA(t))

- a(oo)I)m\\ < C,

while using Lemma 5.3 gives, with ip(z) = (a(z) — a(oo))~~mui(z), {-kA(t))\\
since deg[^] < 0 and ij){z) is ( mw. Remark 6.3. [*] Let ||| |||n = \u(—kA(tn)) \tn. It is then seen that under the conditions of Theorem 6.5, the pair (||| | n ;2l(^n)) is hi^it^m)-concordant for q > £(t37 + 1) unless £ = m — 1 or m > 2 and £ = m — l/(w + 1) in which case the strict inequality q > £(m + 1) is required. In the particular case £ = 0 and m = 1 it is however allowed that q = 0 provided that (6.21) is satisfied. At the same time the pair (||| !„; 2l(£n)) is Arj^lm + 1)concordant for £ = m — \/{w + 1) and q — £(m + 1). As concerns Theorem 6.6, under its conditions the former stated estimate (6.33) actually actually shows shows that the pair (||(2l(*n) + fjLoI)^u(—kA(t n)) ||;2l(£n)) is A^)-concordant for q > ro£ while the latter estimate (6.34) yields that that the operator 2l(£n) satisfies the A|- condition. Remark 6.4. In some applications (M\(p,o~)-sectorial (M\(p,o~)-sectorial operators operators are of interest. More precisely, letting £ : Dom£ C X —* X be a closed linear operator, we write £ € S(M;ip,a) and call the operator £ (M;tp, a)-sectorial for some tp € (0, TT/2) and a 6 R if the sectorial estimate (1) holds with some M > 1 substituted for C. 13 Assuming, instead of the requirement 12

The estimate (1.43) can be applied due to the fact that 5l(t n) satisfies the A^-condition. As in Section 2.2, we denote by M a constant whose size, unlike that of C, is essential for our analysis and should be under control. 13

6.2. HOLDER-CONTINUITY HOLDER-CONTINUITY TYPE TYPE RESULTS RESULTS

125 125

A(t) G S(ip,a), that for some ipi G (0, 1, the assertions of the above Theorems 6.2, 6.4, and 6.6 remain valid with CM substituted for C in the corresponding stability estimates. estimates. In fact, this can be shown by using the same reasonings as above.

6.2

Holder-continuity Holder-continuity type type results results

Here we show that if the operator A(t) satisfies one of the Holder-continuity then the operator type conditions introduced introduced in the preceding chapter, then 1 2l(£n) = k~ a(—kA(tn)) also satisfies a certain Holder-continuity Holder-continuity condition, condition, with the classes 5/ 5/ among those posed in Part I. The below assertions deal with and Theorem 6.7. Let A(t) G S( tn = 0,1, *0, * 1 , are (j\ip,a)-, (ji\ tively, for some 7,71,7*,7*1 G [0,1). Then there exist m = 0, / , *0, *J, such that the pairs (||| |||m;ri; 2l(£ n)), m = 0, / , *0, */, are respectively Aj('yjl)-, Ai(7i|l)-, Ai(7i|l)-, Af(*;7*|l)-, Af(*;7*|l)-, and h\(*;ry*i\l)-concordant and there hold the following implications involving involving hoi hypotheses for the operator A{t) and HOL hypotheses for the operator 2t(t n ) substituted for 2l n , with any fixed i9, d\ G [0,1], i;

Ait- .,.);|- k t , m = 0,1,*0) 6 M ( t f , i ? i ; 1 | m ; n ,m = 0,/,*0), (6.36)

)

(#,#i;.4(t;-,-);Hm ; t,m = 0,*0,*i) \

'

J - ' II

III n T j / l j

*-')'-')

/1 /1

\\ ""

//

and hol(i?; A(t; , ; I |m;t, m = 0, *0) =

HOL(tf; ||| |||m;n (6.38) m;n, m = 0, *0).

Proof. We will show the implication (6.36) only, only, noting noting that that (6.37) (6.37) and (6.38) can be proved with the aid of the same reasoning.

126

CHAPTER CHAPTER 6. 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

First of all observe that deg[p] = — 1 since the RK method is of class Sj. We therefore have, with with some some b\, bo, by G C, Jo

p(-z)-

1

mi

= bxz + b0 + Y, J2 M * - zi)~j 1=1

b z

i + bo + P(z),

(6-39)

3=1

where zi, I — 1 , . . . , lo, are the poles of p{—z)~ l and mi is the multiplicity of a pole z\. Next, using the fact that A(t) G S(ip, a), it follows from Lemma A.I that ||(A7 - Ait))-1]] < C|A|- X for all A £ Int (E V1 UV(R)), with some R > 0 and if i G (0, 0 sufficiently small, the operators (kA(t) — 2;/)" defined on X and uniformly bounded with with respect respect to k G (0, K] and t G [0, T], and, clearly, the same is true for the operator P(kA(t)). representation for the difference 21 (t) — Below we need to get a suitable representation 2l(s). To that end, we put g(t) = p(-kA(t)) and ?(t) = P(kA(t)) and note that using (6.39) leads to the identity

which gives - 2l(s) =

(bik2i(s) + bQg{s) boQ(t) + 7(t)g(t))

=:

Q1 + Q2.

(6.40) (6.40)

Further, for convenience' sake we put (j>i{t) = (kA(t) — zrf)'1, gij(t) — ((£) C have Q\ = bog(s) (A(t) — A(s)) g(t). Also, it is easily seen that Ran£>(£) Vom C V and Ran^*(i) C V* since deg[p] = —1. Using the fact that A(t) satisfies hypothesis hoV'~ hoV'~>'('d,'&i;A(t\ , ; | |m;t,tri = 0,1, *0), we therefore obtain, for all x G X* and « ee ll ,, \(X,Qiu)\ = \b0\ \AA{t,s;Q*(s) X,Q(t)u)\

s)xUs (\t - sf\g(t)u\ $.t + \t- s\^\g(t)«| 1;t)

. (6.41)

6.2. HOLDER-CONTINUITY HOLDER-CONTINUITY TYPE TYPE RESULTS RESULTS

127 127

Arguing as above and applying applying the the identity, identity, for for all all xx GG3t* 3t* and and uu €€X, X, m

'o

i

j

bl

, Q2U) = E E (0

iE

A 4 t s; a

" ('

th

m;

EE

+ E E kjAA(t, s; Qij(s) x, g(t) u), i=\

j=i

we have as well l0

mi

1=1

j=l

IQ

mi

\(x,Q2u)\
X 2_\ /_. (I* ~" S^lQljit)

\%;t + I* ~ s l

\Ql,j{t) u\l;t)-

(6-42)

1=1 j=0

Now, putting III ' l*0;n

=

/ . / . {\au(tn)

' l*0;tn + \Ql,j(t n)

|*0;tn) + \Q*(tn)

|*0;tn,

1=1 j=0

and ;0

mj

it follows from (6.40), (6.41), (6.41), and and (6.42) (6.42) that that the the implication implication (6.36) (6.36) holds. holds. cr)-concordant, Note that since 7, < 1, the the pair pair (|(| \^t; A(t)) is (*; 7*|l|, cr)-concordant, thanks to Lemma 5.1. Using Using then then Theorem Theorem 6.1 6.1 and and taking taking into into account account that deg[p] = —1, it is readily readily seen seen that that the the pair pair (\g*(tn) |*0;tn;2l(£n)) is A?(*i 7*|l)-concordant. Acting Acting after after the the same same pattern, pattern, itit can can be be shown shown that that the pairs (la^-^n) |*0 ;tn ;2l(t n )) and (|^*j(i n ) I*0;tn;2t(*n)), I = I, j — 1 , . . . ,mi, are A^(*; 7*|1)-concordant. 7*|1)-concordant. Therefore Therefore the the pair pair (|H|*0 (|H|*0;n;2l(^n)) that form form is Aj(*;7»|l)-concordant Aj(*;7»|l)-concordant since since |||||| |||*0;n is the sum of seminorms that Aj(*;7*|l)-concordant pairs pairs with with 2l(i 2l(i n ). Similarly, we find that the the pairs pairs (||| (||| |||m;n;2l(*n))) |||m;n;2l(*n))) tn tn == 00,,//,, are are A^('yjl)A^('yjl)and Aj(7i|l)-concordant, Aj(7i|l)-concordant, respectively. respectively. This completes the proof.

128 128

CHAPTER 6. ANALYSIS OF STABILITY CHAPTER 6. ANALYSIS OF

STABILITY

The result is helpful when dealing methods of class Thefollowing following result is helpful whenwith dealing with methods of class SJJJ SJJJ .. Theorem 6.8.6.8. Suppose that A(t) the a) RK and method Theorem Suppose thatG S(tp, A(t) a)Gand S(tp, theisRKAiu(tp)method is Aiu(tp)stable, forforsome tp G tp G (0,TT/2) M..M.. Let Let further A(t; , A(t; ) be a , testing stable, some (0,TT/2) and anda aG G further ) be a testing functional forfor A(t)A(t) with Dom.A(£; , ) = V* x, V. that there also are that there are functional with Dom.A(£; ) =Suppose V* x V.alsoSuppose *0,*0, *1, firstfirst two are on given seminorms | | | | m;t m==0,1, 0,1, *1, of ofwhich whichthethe twodefined are defined on given seminorms m;t , m V and other two on so V* that so the that pairs the (\ | pairs (\ | m; A(t)), m = 0,1, *0, *1, V andthethe other twoV*on t; A(t)), m = 0,1, *0, *1, m; are (l/2\tp,cr)-, ji\(p,a)-, (*;l/2\ip,a)-, and (*;and 7*i|<£>,(*;a)-concordant, reare (l/2\tp,cr)-,( (/ ji\(p,a)-, (*;l/2\ip,a)-, 7*i|<£>, a)-concordant, respectively, withwith somesome 71,7*171,7*1 G the the condition deg[a(-)deg[a(-) — [0,1/2]. Then, Then,under under condition — spectively, G [0,1/2]. m m a(oo)] = =—1, therethere exist exist seminorms | | || n> = 0,/, *0,*I, that form rea(oo)] —1, seminorms ||| ||m; n> = 0,/, *0,*I, that form rem; spectively and A2(*;7*i)-concordant pairs spectivelyA2(l/2)A2(l/2)- ; AK71)-, AK71)-,A2(*;l/2)-, A2(*;l/2)-, and A2(*;7*i)-concordant pairs with 2l(£ ), and holdhold the following implications involving hoi hypothe- hoi hypothewith 2l(£n), andthere there the following implications involving ses for A(t) A(t) and HOL for the operator ses forthetheoperator operator and hypotheses HOL hypotheses for 2l(i the operator 2l(i n ) subsubstituted forfor %n,%n, with with any fixed $1 G 1?, $1 G [0,1], stituted any1?,fixed [0,1],

H6L HH V,tfi;IHIL;n,m V,tfi;IHIL;n,m = 0,I,*0), H6L = 0,I,*0),

and and hol(tf;.4(t;-,-);| | | mm;t; t ,m ,m ==0,*0) = >= > HOL(tf; ,m ==0,0,*0). HOL(tf; ||||| |||| mm;n; n ,m *0). hol(tf;.4(t;-,-);| 0,*0) The proof to that Theorem 6.7, using 6.7, as well Theorems 6.5 Theorems 6.5 The proofis similar is similar to ofthat of Theorem using as well and 6.3 and the and fact that = —1. deg[p] = —1. and 6.5*, 6.5*,Remarks Remarks 6.36.3*, andand 6.3*, the deg[p] fact that We a more concreteconcrete assertion assertion which will be neededwill in be needed in We state stateseparately separately a more which the the sequel. sequel. Theorem Let A(t) G S(fixed 0 sufficiently fixed HO >> 00sufficiently sufficiently andany with K > 0 small, sufficiently small, we have, all kallG k(0,K], t,s G [0,T], £ X*, and X, that we have,forfor G (0,K], t,s Gx [0,T], x £u GX*, and u G X, that -QL(s))u)\
(6.43) (6.43)

6.2. HOLDER-CONTINUITY HOLDER-CONTINUITY TYPE TYPE RESULTS RESULTS

129 129

If in addition A{t) satisfies hypothesis hol[i9], hol[i9], then then the term \(I+kA(t))~ lu\t does not enter.

Proof. The result can be shown following the proof Theorem 6.7 and noting, with the aid of Lemmas 5.3 and 1.5, that with no > 0 sufficiently large, for all t G [0,T] andu G X, \\A{t)g{t)u\\ + \\g{t)u\\ < C(||2t(i)n|| + ||n||) < C||(2l(t) + ^I)u\\ and \\{I + kA{t))u{-kA{t))\\ < C, for any rational function u(z) which is ((^)-regular and satisfies the condition D deg[w] < - 1 . D In the remainder of the section we give two auxiliary results which show show the Holder-continuity of a bounded rational function function of the operator A{t) under the assumption that A(t) itself satisfies a Holder-continuity type condition among those stated stated before. before. Lemma 6.1. Suppose that A(t) G S(
rational function which is (ip)-regular and satisfies the condition deg[u;] < 0. Then, with K > 0 sufficiently small, for all k G (0, K] and t,s G [0,T], \\u(-kA(t)) - u{-kA(s))\\ < C\t - sf. Proof. We can assume without loss of generality that deg[a;] < — 1, otherwise we might deal with the function w(z) — w(oo). Having therefore accepted accepted the last inequality, we let ip\ G (0, 0 be just the same as in the statement of Lemma A.I with A(t) substituted for £. Let also d > 0 be chosen such that u)(z) has no poles in V(d). Further, we set K > 0 subject to K < min{T,d/R}. Let next F be the contour coinciding with 9(2^! U V{d)). Since u;(oo) = 0 now, we can be based, as in the proof of Lemma 5.2, on formula (5.8) which clearly yields that cj(-kA(t)) -~uj{

= — [

LU(-Z)

((zl - kA{t)Y l - (zl - kA(s))- 1) dz. (6.44)

Using the fact that the domain of A(t) is independent of t, we have as well {zl - kA{t)Y l - {zl - kA{s))~ l = {zl - kA{t))~ lk{A{t) - A{s)) {zl - kA{s))~ 1.

(6.45)

130 130

CHAPTER 6. ANALYSIS ANALYSISOF OF STABILITY STABILITY CHAPTER 6.

A(t)G G S((f,a), S((f,a), similarly similarly to(5.7) (5.7) followsthat that Since to it it follows Since A(t)
ffoor ar al llz lEE TT. .

(6.46) (6.46)

Also, usingthis thisestimate estimate andthe the fact that thatA(t) A(t)satisfies satisfieshol[i?], hol[i?], it isisnot not Also, using and fact it hard tosee see that with withsome some fi\ > >0,0,for for z ET, hard to that fi\ allall z ET, 1 1 \\k(A(t)-A(s))(zI-kA(s))\\ \\k(A(t)-A(s))(zI-kA(s))\\ l < Ck\t Ck\t sf\\(A(s) + ml) - kA(s))~ || < - -sf\\(A(s) + ml) (zl - (zl kA(s))~

1 1 < C\t C\t- -s\° s\° + ||(z + (zl fc/ii) (zl -- kA(s))kA(s))\\) \\) < (1(1 + ||(z + fc/ii)

< C\t C\t-- sf. sf. <

(6.47) (6.47)

Combiningtherefore therefore (6.44), and(6.47) (6.47)gives, gives, after simpleestimaestimaCombining (6.44), (6.46),(6.46), and after aa simple tion, tion, l \\u{-kA{t)) -- u(-kA(s))\\ u(-kA(s))\\ < (-z)\ \u>(-z)\ \z\~\z\~ \dz\, \\u{-kA{t)) \dz\,

which shows theclaim claimin inview viewof ofthe theevident evidentestimate estimate which shows the

w(-^)|<
L mmma a6.2. 6.2.Let Let A(t) S(, (*;7*|,a)-concordant, a)-concordant, respectively, andA(t) A(t)satisfies satisfieshypothhypoth(j\ip,cr)- and respectively, and esis hol($; A(t;, , ;;|| ||m;t m= = 0, 0,*0), *0),for forsome some7,7* 7,7*ee [0,1) [0,1)and and ee (0,1]. (0,1]. esis hol($; A(t; m ; t,,m Finally, letu)(z) u)(z)be bea a rational rationalfunction function which is((p)-regular ((p)-regular andsatisfies satisfiesthe the Finally, let which is and conditiondeg[w] deg[w] < 0. 0.Then, Then,with with K> >00sufficiently sufficientlysmall, small, forall allk kE E(0, (0, K] K] condition < K for andt,s G G[0,T], [0,T], andt,s 111?? 1 \\u{-kA(t)) Lo(-kA(s))\\
Proof. Theresult result is is proved provedin inaamanner mannersimilar similar tothat that used usedin inthe theproof proof Proof. The to 6.1,with with the thedifference differencethat that instead of (6.45) (6.45)we wenow now applythe the of Lemma Lemma6.1, instead of apply of identity, for identity, forall allxxS S X* X*and anduuEX, EX, l kA(s)Y kA(s)Y ) u) u) 1 1 1 1 =kAA(t; =kAA(t; (z*I (z*I--fcA^t))fcA^t))*, *, {zl {zl~~ kA(s))kA(s))u). u).

6.3. 6.3. STABILITY STABILITY ESTIMATES ESTIMATES

131

131

Next, estimating on on the the rightright by analogy with with the proof of Next, estimatingthe theexpression expression by analogy the proof of Theorem 6.7, all all z 6z F,6 F, Theorem 6.7,we wefind findthat thatforfor \\(zl -- kAit))' \\(zl kAit))'1

-- (zl (zl -- ^^( s( s) ))-) - 1 ! ! << C\t C\t --

ll+lf 2 s\°\z\' s\°\z\'ll+lf '- .

Using next for for an an estimation on the of (6.44) leads leads Using next the thelast lastinequality inequality estimation on basis the basis of (6.44) thethe proof of the preceding lemma. to the the desired desiredresult resultasasin in proof of the preceding lemma.

6.3

Stability Stabilityestimates estimates

central partpart of the whole chapter, we examine In this this section, section,which whichis isthethe central of the whole chapter, we examine the stability problem (5.19), (5.20). To prove the problem, the stability ofofthe thediscrete discrete problem (5.19), (5.20). To prove the problem, in Part I as Iwell as the resultsresults we use use here herethe thetheory theorydeveloped developed in Part as well as preparatory the preparatory found sections. found inin the thepreceding precedingtwo two sections. We begin in in thethe casecase withwith a time-independent operator, We beginwith witha adiscussion discussion a time-independent operator, settled trivially, afterafter the above preparations. for which which the thequestion questionis is settled trivially, the above preparations. Theorem 6.10. A(t)A(t) be independent o/t o/t 1 4 and and let let Theorem 6.10. Let Letthetheoperator operator be independent C

(0,K}. (0,K}.

for for alike alike

(6.48) (6.48)

Suppose is ofis type Cj(ip)Cj(ip) or ofortype Suppose further further that thatthetheRKRKmethod method of type of C//( 0>sufficiently largelarge and with (5.19), (5.20), (5.20), we wehave, have,with withany any fixed 0 sufficiently and any with any for all all kk ee (0, (0,K]K]and andt tn G GCl Clk, fixed ££ >> 00 ; for fixed

L

max max ||/(ty)ll) ||/(ty)ll)

(6-49) (6-49)

Moreover, 0 and /^o /^o = 0, valid Moreover, ififininaddition additiona a> > 0 and 0, then then the thelast lastestimate estimateis is valid 15 and KK ==oo ooasaswell. well. for TT == oo oo and Proof. itit follows t h atth a t Proof. Since Since deg[tOj] deg[tOj] < < 0,0,j j= = l,...,v, l,...,v, follows from fromLemma Lemma5.35.3 ||U;J(—fc.A)|| by by (5.28), ||U;J(—fc.A)|| < C, C, which whichfurther furthergives, gives, (5.28),

\\w \\w T(-kA)$ (-kA)$ n\\


for for ttn e

titik-

(6-50) (6-50)

j=l,...,v j=l,...,v

Clearly, valid for for Ai(ip)or AJJ methods Clearly, the thelast lastestimate estimateremains remains valid Ai(ip)or (
We can 21 instead of A(t) and and 2l(£),2l(£), respectively. We can therefore thereforewrite writeA Aandand 21 instead of A(t) respectively. That is, That is, for for all alln n= = 0 ,,11,,..... .

15

132

CHAPTER6.6.ANALYSIS ANALYSISOFOFSTABILITY STABILITY CHAPTER

satisfies the the Aj-condition, Aj-condition, ininparticular, particular,the theAi-condition Ai-condition case satisfies in in thethe case a >a 0.> 0. conclude the the proof, proof, it remains remains to to apply apply Theorem Theorem2.17, 2.17,noting notingthat thatthethe To conclude (||(2l+/zo-O^ 'Ih^l) 'Ih^l)isisAf Af(^)-concordant, (^)-concordant,inin particular, Ai(£)-concordant pair (||(2l+/zo-O^ particular, Ai(£)-concordant Lemma 1.12. 1.12. if a > > 00 and and /J,Q — 0, by Lemma Theorem 6.11. 6.11. Let Letthe theoperator operator A(t)==A Asatisfy satisfythethe conditions of Theorem Theorem A(t) conditions of Theorem 6.10 and and let let the the RK RK method methodbebeofoftype typeCui(
(6.51) (6.51)

where u>(z) u>(z) is a rational rationalfunction function which whichisis((p)-regular ((p)-regularand and under requireunder thethe requirewhere with some some ??eeNN, , ment, with deg[w] -<;, deg[w] < -<;,

(6.52) (6.52) 16

then, assuming assuming that that the RK RK method method itself itself satisfies satisfies the condition condition deg[wj]<-q, deg[wj]<-q,

= l,...,u, l,...,u, jj =

(6.53) (6.53)

holds for any any fixed £ € [0,q/w], [0,q/w], where where w — — — deg[o;(-) deg[o;(-) — —a(oo)]. a(oo)]. (6.49) holds Moreover, if a > > 00 and and /io /io== 0,0, the theabove above is valid for T — —K K == 00 00 as aswell. Moreover, Proof. follows from from Theorem Theorem 6.6 6.6that thatthe theoperator operator2121 satisfies Proof. It follows satisfies thethe A|-A|condition, in particular, particular, the theA2-condition A2-conditionif ifa a> >0. 0.Note Note also that (6.50) condition, in also that (6.50) holds for for methods methods ofoftype typeCjjj(ip) Cjjj(ip)asaswell. well.Therefore, Therefore, even if (6.51) —(6.53) even if (6.51) —(6.53) are not not necessarily necessarilysatisfied, satisfied,(6.49) (6.49)with with£ £—— follows Theorem 2.17 since 00 follows byby Theorem 2.17 since the pair pair (|| (|| ||;2l) ||;2l) isisthen, then,obviously, obviously,A2(O)-concordant A2(O)-concordant(A2(0)-concordant (A2(0)-concordant if if a > 00 and and fio fio== 0). 0). Now we we turn turn to to the the case casewhere where(6.51) (6.51)and and(6.53) (6.53)hold, hold, under above under thethe above OJ(Z), including including (6.52). (6.52). By By (6.53) (6.53)ititthen thenfollows followsfrom fromLemma Lemma conditions on OJ(Z), conditions on 5.3 that that the the operators operators (/(/++kAywj(-kA), kAywj(-kA), j j= =1,... 1,...,u,,u,areare defined X for defined on on X for k
with Q = (I (I + +kA)-* kA)-* and andFFn = (I + +kAyv/T'(-kA)$ '(-kA)$n, (6.54), (6.54), ||Fn||
This condition condition is not not needed needed if f(t) == 0. 0.

where, by (5.28) (5.28) and where, (6-55) (6-55)

6.3. STABILITY ESTIMATES ESTIMATES

133 133

Now, by Theorem 6.6 (see (see also also Remark Remark 6.3) 6.3) we we conclude conclude that that the the pairs pairs (||(2l + / x o J ) M - * ^ ) -II;21) a n d ( | | ( * + / x o / ) € ( / + fcA)-< - I I ; » ) a r e A § ( 0 concordant if £ < > 00 and and jU jUoo== 00 isis obvious. obvious. DD Remark 6.5. //, instead instead of of the the condition conditionAA €€ S(ip,a), S(ip,a), one one requires requiresthat that A G S(M; ipi,a), M > 1, for some fixed
134

CHAPTER CHAPTER 6. 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY

hol®[i?; | |t], for some 71 G G [0,1) [0,1) and and $$ G G (0,1]. Apart from everything else, else, let (6.48) hold. Then, with any fixed ££ €€ [0,1) [0,1) and and with with any any fixed fixed /io /io >> 00 sufficiently large, we have, have, for for all all kk G G (0, (0,K] K] and and tn,ti G fifc with ti < t n, ||(2l(tn) + iJorfUn-ijW < Ct~i Ct~il+V

(6.56)

problem (5.19), (5.19), (5.20), (5.20), and for Y n the solution of problem

||(2l(tn) + Mo I)^n|| ^ C K I J ^ H + fc^Xii fc^Xii max max ||/(ty)ll) ||/(ty)ll) .. (6.57) (6.57) fixed Furthermore, with any fixed fixed 00 << £* £* << min{i9,1 min{i9,1 — 71} and with any fixed pbQ > 0 sufficiently large, for for all all kk G G (0, (0, K] K] and and tn,ti G flk such that t\
< Ct~i].

(6.58) (6.58)

remains valid valid ifif one one adds adds In the case £* = min{i9,1 min{i9,1 — 71} the last estimate remains log(£n_/_|_i/fc) to the factor t~_ t~_t. If in addition the operator operator A(t) A(t) satisfies satisfies hypothesis hol[#], then (6.56) (6.56) and and (6.57) (6.57) are are valid validfor for any any fixed fixed££ GG[0, [0, Moreover, they retain their their shapes shapes even even for for ££ == 11++ $$ provided provided that that t~ t~ isis 1 log(t j+i/k). replaced by tj* + tj Proof. First of all it follows from from (5.41), (5.41), (5.34), (5.34), and and (5.43) (5.43) that that | | F n | | < C max ||/(i n j )||.

(6.59) (6.59)

Also, observe that the operator operator 2l(i 2l(i n ) satisfies the A^-condition, A^-condition, by by Theorem Theorem 6.2. Further, consider the general general case case where where the the operator operator A(t) A(t) satisfies satisfies hyhypothesis h61®[$; I |t]. Observing Observing then then that that for for methods methods of of type type Vj((p), Vj((p), deg[vj]

< - 1 = deg[p],

j =

l,...,v,

and noting that the function function p(z)"1 is ( 0 sufficiently large, for

< C\\(p(-kA(t))r\(-kA(t))\\ C\\(p(-kA(t))r\(-kA(t))\\

\\A(t)p(-kA(t))u\\ \\A(t)p(-kA(t))u\\ ++ C\\u\\ |. (6.60)

6.3. STABILITY

ESTIMATES ESTIMATES

135 135

Using now (5.39) and taking taking into into account account the the estimates estimates (5.34), (5.34), (5.43), (5.43), and the accepted Holder-continuity Holder-continuity type type condition, condition, with with the aid of (6.60) we obtain, for all u £ X, V

||2M! <

cY/\\('Dn-'D(tn))vj(-kA(tn))u\\ (

n) + lilI)Vj(-kA(t n))u\\ + \Vj(-kA{t n)) u\tn

J=l v

^

.

(6.61) (6.61)

Since 2l(t n ) satisfies the Aj-condition, Aj-condition, by by Lemma Lemma 1.1 1.1 we we have have ||fc2l(i ||fc2l(in)|| < C, which further implies

+ MO/) U\\ < C||(2l(tn) + MI/) uW1-*\\uf. Using the last estimate estimate and and applying applying Lemma Lemma 1.7, 1.7, it easily follows that the seminorm fc^||(2l(i n) + Mo/) II forms a Aj(l — $|l)-concordant pair with Also, applying Lemma 5.1 shows shows that that the the seminorm seminorm | \t is, in fact, with , cr)-concordant with A(t) since 71 < 1, which therefore yields, with v \tn forms the aid of Theorem 6.1, that the seminorm X^=i \ j(~kA(tn)) a Aj(7i|l)-concordant pair l,...,u. pair with with 2t(i 2t(i n ) since deg[uj] < —1, j = Further, by Lemma 1.8 we we conclude conclude that that the the seminorms seminorms A; A;1'||(2l(tn) + MO/) II and X]^=i l v j ( ~ ^ ( * n ) ) '\tn both form A^(7o|l)-concordant A^(7o|l)-concordant pairs pairs with with 2l(i 2l(i n ), where 70 — max{l — i9,7i}. These observations clearly clearly yield yield that that the pair (||!Bn-1|; 2l(t n )) is Aj(7o|l)-concordant. Also, Also, using using Theorem Theorem 6.9 6.9 gives gives that that 1 8 2l(t n ) satisfies hypothesis HOL HOL (->) (i9,0; ||| | m ; n , m = 0,/,*0) with ||| ||0.n = ||(2t(t n) + Mo/) || and ||| \\^.n = || ||* while the seminorm ||| | / ;; nn = |(/+fcA(i n ))~ 1 \tn forms a Aj(7i|l)-concordant pair with with 2l(i 2l(i n ), by Theorem 6.1. By the above reasonings, reasonings, applying applying Theorems Theorems 2.10, 2.10, 2.16, 2.16, and and 2.11, 2.11,all allofof them with 2l(i n ) substituted for Qln, and taking (6.59) into account, account, we we see see that the estimates (6.56) (6.56) and and (6.57) (6.57) hold hold for for ££ G G [0,1) [0,1) and and the theestimate estimate (6.58) (6.58) holds for £* e [0, min{i9,1 — 71}] since the seminorms seminorms ||||(2l(t (2l(t n ) + Mo/)^ II and ||(2t*(t n) + Mo/)^* ||* are respectively Aj(£|l)- and Ai(*;£»|l)-concordant with 52l(tn) for any fixed £ € [0,1], by Lemma 1.12. 18

One has to take into account account that that \\(p{-kA(t)))* \\(p{-kA(t)))* ||» ||» < C for all k G (0, K] and t G [0, T], by Lemma 5.3 and duality. duality.

136 136

CHAPTER 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY CHAPTER 6.

the case case where whereA(t) A(t) satisfies satisfies hypothesis hypothesishol®[#], hol®[#], instead of (6.61) (6.61) we we In the instead of have, for for all all uu GGX, X, have,

so tthhaatt (6.56) (6.56) a nndd (6.57) (6.57) hold hold for for ££ ee [1,1 [1,1++tf]tf]asaswell, well, by by T hheeoorreemmss2.10, 2.10, 2.16, a nndd 6.9, 6.9,m oorree precisely, precisely, by by tthhee latter l a t t e r ppaar rt t of of tthhee s t aatteem meennt t of of each each 2.16,

one. one. Theorem 6.13. 6.13. Suppose Suppose that thatA(t) A(t) G GS{ip,a) S{ip,a) and and the the RK RK method method is is both both Theorem type Vi((p) Vi((p) and and of of type type Vf(ip), Vf(ip), for for some some ip ip G G (0,TT/2) (0,TT/2) and and aa GGM. M. Also, Also, of type seminorms \ |® |®;t and and \\ | m | ;;tt, m m == 0,1, 0,1,*0, *0, form form respectively respectively(7®|y, (7®|y, cr)-, let seminorms cr)-, m (j\ip,a)-, (71 (71
some ix-y ix-y> 00 and and some

I(A(t) ++miy^u^.t miy^u^.t < < C\\u\\ k^' I(A(t) C\\u\\

(6.62) (6.62)

and and *

I)-TyXU,t ttt1I)-Ty

< C\\x\\*C\\x\\*<

(6.63) (6.63)

Next, letting letting A(t;-,-) A(t;-,-) be be aa testing testing functional functional for for A(t), A(t), suppose suppose that that A{t) A{t) Next, satisfies the hypotheses hypotheses hoi®. hoi®.Ji9@; Ji9@; |-|© |-|©;t] andh.or'('d,'d\\A(t\ andh.or'('d,'d\\A(t\ ;|-|m;t)'Ti 'Ti = = satisfies the , , ; |-|m;t *0) for for some some i9©,i9 i9©,i9G (0,1] (0,1] and andd\d\ GG[0,1], [0,1], where where > i?. i?.++ 1?*.. 1?*.. Apart Apart 0 , 11,, *0) >

from everything everything else, else,suppose suppose (6.48) holds. the stability stability estimate estimate from thatthat (6.48) holds. ThenThen the (6.58) holds holdsfor for any any fixed fixed fj,o fj,o> > 00 sufficiently sufficiently large largeand and for for any anyfixed fixed £* £*G G (6.58) [0, £o)> £o)> where where£0 £0 == 11++min{i9 min{i9 — — 7, i?i i?i — — 71}, 71}, and and itit remains remains valid validin in the the case case — £0 £0 ifif one oneadds adds t~2.}log(t t~2.}log(t i/k) to the thefactor factor t~_ t~_t. Furthermore, Furthermore, if £* — to n^i+i/k) in addition, addition, the the stability stability estimates estimates (6.56) (6.56)and and (6.57) (6.57) hold holdfor for any any $1 > 00 in fixed no no > 00 sufficiently sufficiently large largeand and for for any anyfixed fixed £ G G [0,£i), [0,£i), where where£1 £1 == fixed 1— — 7* 7* ++ min{$,T9i}. min{$,T9i}. In In the the case case £ = = £1, £1,both both estimates estimates retain retaintheir theirshapes shapes with tq tq replaced replacedby by tq tq ++ tq~ tq~12 log(t log(tq+i/k), +i/k), where where72 72 is is given given by by (2.57). (2.57). with Proof. First Proof. First of of all, all, to to make make our our below below argument argumentcorrect, correct,we we assume assume that that is extrapolated extrapolated by by A(t) A(t) ==A(T) A(T)for fort > t >T.T. A(t) A(t) is Further, notethat thatthe the estimate estimate (6.59) (6.59)is unchanged unchanged in in the the considered considered Further, note case. At the thesame same time timehypothesis hypothesishor~^($, h o r ~ ^ ( $$1; , $1; A(t; ,, ;;| | | m; |m;t, t,tn tn == 0,1, 0,1,*0) *0) case. At A(t; >)) (—> for A(t) A(t) implies, implies, by by T hheeoorreemm 6.7, 6.7,t h aatt hypothesis hypothesisH O O LL (#, ( # $, 1 ;; ||| | m ;; nn , m m == *0)is isfulfilled fulfilled for for 2l(i 2l(in ) s uubbssttiittuut teeddfor for 22lln , where w h e r e t hhee seminorms s e m i n o r m s ||| | m ;; nn, 0, / , *0) 19

Since Since A(t) A(t) €€ S( > 0, 0,are aredefined defined on X X ififfii fii >>00isissufficiently sufficiently large large(cf. (cf. Appendix AppendixA.3). A.3). and and bounded boundedon

6.3. STABILITY ESTIMATES ESTIMATES

137 137

m = 0,7, *0, form respectively respectively A* (7)-, Af(7i)-, and Af(*;7*)-concordant Af(*;7*)-concordant pairs with 2l(£ n), due to the conditions imposed imposed on on || || m;t , m = 0,1, *0. Next, since degf^j] < —1 —1 and and deg[i«j] deg[i«j] << —1 —1for for jj — —1,... 1,... ,u, ,u, applying applying Theorems 6.1 and 6.1* yields yields that that the the seminorms seminorms .0;n

and 3=1

form respectively A* (7)- and A^(7i)-concordant A^(7i)-concordant pairs pairs with with the the operator operator 2l(t n). Moreover, recalling that the the pair pair (|(| l^; l^; A(t)) A(t)) isis strictly strictly (*;7*|y>, (*;7*|y>, cr)concordant, it is readily seen seen that that the the seminorm seminorm n))y

U-tn+Cjk

forms a Aj(*; 7*)-concordant 7*)-concordant pair pair with with 2l(t 2l(tn). Apart from everything everything else, else, assuming that we have shown shown the the estimate, estimate, with with the the operator operator 25 25n given by (5.39), for all t n G ftk, x G X*, and u e X, (6.64)

it holds, with i9o = min{$,$i}, min{$,$i}, 70 70 == i?o i?o++ max{7 max{7—i9,7i —i9,7i — ^1}, and ||| |.. n =

Therefore, noting that by by Lemma Lemma 1.2, 1.2, the the seminorm seminorm |||||| |.|. ; n forms a A^ concordant pair with 2l(£ 2l(£n) and taking the above reasonings reasonings into into account, account, the claim follows by applying applying Theorems Theorems 2.12 2.12 and and 2.17 2.17with with2l(i 2l(in) substituted for 2ln, with 23 n given by (5.39) and with with 7*0 7*0 == 7*7*It thus remains to show (6.64). (6.64). To To that that end, end, note note that that as as itit follows follows from from (5.39), the operator 25 n can be represented as Q3 Q3n = ©i ;n + 232;m where

and
k(V n - ©(*„)) (i + x (D n - !D(t n))v(-fcA(tn)).

(6.65) (6.65)

138 138

CHAPTER6. 6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY CHAPTER

It is obvious, obvious, since sinceA(t) A(t) is isunder under hypothesis hypothesish ho ol ^l (^i(9i,9$1; , $1; A(t; ,, ;;| | | mm;;tt, m m == A(t; 0,1, *0),that that (6.64) (6.64) holds holdswith withQ3i Q3i substituted for *B therefore suffices 0,1, *0), substituted for *B . It therefore suffices ;n n place of 5B 5Bn, which which is clearly clearly achieved, achieved, with with to show show (6.64) (6.64) with withQ52 Q52 ;n in place aid of of (6.65) (6.65) and and hypothesis hypothesis hhool ^l ^( i(?i ,?i,?ii?; i^;(^t (; t ;,, ;;|| | m; t,m = = 0,1, 0,1,*0), *0), the aid m ;t,m if we we prove prove that that with withe, — —{Sji,..., {Sji,..., —11,,......,, v, for for all allt tn G G fifc fifc ) ) , jj — (u\,..., Uu) Uu) G GVom Vom xx xxVorn, Vorn, u = (u\,..., V

V(tn))u\. ))u\.]tn < cY,\ cY,\ui\'-,tn, i\'-,tn, n - V(t j = ll

(6-66) (6-66)

3=13=1

&1 where is denoted denoted | |.|.; t = = k^\ k^\ \^\^t + + k' k'&1 |x;t- For Forshowing showing (6.66), (6.66),we weuse use where it is \ |x;tformula (5.49),with withboth bothsides sides multiplied from the right right by by k(D k(Dn — D(£ D(£n ))u, ))u, formula (5.49), multiplied from the to obtain obtain l n

D(tn ))u = (i (i ++ kaV(t kaV(tn))-1ak{'D ak{'Dn -- D(t

T>{tn))u -- T>{t

mind, it is is not not hard hard to see, see,applying applying the the kicking-back kicking-back With this this equality equalityin mind, With verified if we we show show instead instead that thatfor for all all ttn G &k &kand and trick, that (6.66) (6.66)will willbe verified trick, that u = ( uu jj ,,......, , uu)

G G Vom Vom xx

x xT>om, T>om,

V

n

0)(tn))u|. ))u|.;tn < -- ]]££ |^|. | ^ | .; t n . - 0)(t

(6.67) (6.67)

Since the Since the entries entries of the the matrix matrix (i -- zaj^a zaj^a vanish vanishas \z\ \z\ —>>oo, oo, (6.67) (6.67) is, is, in in fact, consequence of the the following following fact. fact. For For any any fixed fixed rational rational function function fact, a consequence ijj(z) which which is ( 0, 0, for for all allkkGG(0, (0, K],t G t G [0,T], and and uu G G Vom, Vom, with with some some e > K], [0,T],

Cke\u\,, \u\,,u \u(-kA(t))k{A(t ++ck) ck)- -A(t))u\., A(t))u\., \u(-kA(t))k{A(t t < Ck

(6.68) (6.68)

so that that the the factor factor Ck Cke on on the the right right can can be be taken taken arbitrarily arbitrarilysmall smallat the the expense choosingsuitably suitablyK > > 0. 0. Observe Observenow nowthat that if K K> > 00isissufficiently sufficiently expense of choosing the negative negative fractional fractionalpowers powers(/ ++ kA(t))~ kA(t))~T, r >> 0, 0, of ofthe theoperator operator small, small, the I + kA(t) kA(t) are aredefined defined and and bounded bounded on on 3£ 3£for for kk GG(0, (0,K] K](cf. (cf.Appendix AppendixA.3). A.3). Furthermore, since7 7 1 7+*7 *< 1, 1, there there exist exist 7J >> 71 71and and7^ 7^>>7*7*such such that that Furthermore, since 1+ 7i + 7* 7* == 1.1. Using Using the the fact fact that that the the seminorms seminorms | |i|i;t and and || \^. \^.t form form respectively (71 (71|y, a)a)- and and strictly strictly (*;7*|v, (*;7*|v,cr)-concordant cr)-concordantpairs pairs with with A(t), A(t), respectively

6.3. STABILITY STABILITY ESTIMATES ESTIMATES

139 139

similarly to Lemma Lemma 1.9 1.9 itit can can be beshown shown that that for for all all kk GG{0,K], {0,K], t G G [0,T], [0,T], similarly X, and and xx GG£*; £*; u e X, sup | ( // ++ ^ ( tt )) )) ^^nn| |0 ; tt + FF** sup sup fc» sup IMI=i llxll.=i IMI=i llxll.=i

i i

(6.69) (6.69)

Also, follows from from Lemma Lemma5.3 5.3 that that Also, it follows ||wi(-fcA(t))|| < C C for for aUfce aUfce {0,K} {0,K} and andii GG[0,T], [0,T], ||wi(-fcA(t))||

(6.70) (6.70)

where 0)1(2) = (1 (1 — z)a>(,z). z)a>(,z). where 0)1(2) Now, with with the the aid aid of ofthe theidentity identity20 Now, u(-kA(t)) == (I(I u(-kA(t)) applying the estimates estimates (6.62), (6.62),(6.63), (6.63),and and (6.70), (6.70), we therefore therefore obtain, obtain,for all all applying the (0,K], £ G G [0,T], [0,T], and and itit G GPom, Pom, k G (0,K],

cfc) cfc) -- A(t))u\\ A(t))u\\ yl(t + cfc) A(t))u\\ ck) - A(t))u\\ (6.71) | 1 ; t ) . (6.71) similar manner manner since since7^ + + 7^ 7^ — 1, applying applying the the identity identity In a similar

using (6.69) (6.69) and and (6.70) (6.70) and and the the above above argument, argument, we we get, get, for for all allkkGG and using (0,K], G [0,T], [0,T], and and uu G G Vom, Vom, (0,K], t G to(-kA(t)) k(A(t k(A(t ++ck) ck)- - A(t))u\i A(t))u\iitit to(-kA(t)) tf

) (6.72) (6.72)

—min{i9 min{i9 —19. —19.— ?9*., ?9*.,1 — 71 — 7*} 7*} by by combining combining Finally, (6.68) (6.68) follows followswith withe — Finally, (6.71) and and (6.72), (6.72), noting noting that thate > > 00 since since d, -f-1?*. -f-1?*. < i9 i9 and and 71 71++7* 7*<<1.1. (6.71) Theorem 6.14. Theorem 6.14. Suppose Suppose that that A(t) A(t) G GS((p, S((p,a) a) and andthe theRK RKmethod method is both bothof Vj(ip) and and of of type type Vf(, (7©|,cr)-, cr)-, seminorms 20

recall that that 7 + + 7. 7. == 1.1. We recall

140 140

CHAPTER 6. ANALYSIS OF STABILITY CHAPTER 6. ANALYSIS OF STABILITY

(*;7*|y>, &)-,(*;7*i|y, (*;7*i|y, andstrictly strictly (^(p, (^(p,a)-concordant a)-concordant pairs pairswith with A(t) (*;7*|y>, &)-, cr)-cr)A(t) t and for some7®, 7®, 7*, 7*1,7 G[0,1) [0,1)such such that that7 7 ++ 7* 7*==11and and 7 7*1 + 7*1 < < 1, 1, and andlet let for some 7*, 7*1,7 G 7+ the inequalities(6.62) (6.62) and(6.63) (6.63) hold holdfor for some some JJL\ JJL\ > > 00 and and G[0>l)[0>l)the inequalities and . .G Also, letting A(t; A(t;-,-, ) )6e6ea atesting testing functional functionalfor for A(t), A(t), suppose suppose that thatA(t) A(t) Also, letting satisfies the thehypotheses hypotheses hol^Ji?©; hol^Ji?©; |-|mm;t>m ;t>m = = satisfies |-|®|-|® 11^ ^( i(?i,?i ?, ii ?; >i ;l > ( tl; ( t,,; ;;|-| ;t] aannddhh66 0, *0, *0,*1) *1)for for some some tftfffi ,i9 G (0,1] and t? [0,1], w/iere i9 > 1?. 1?*.. ylpart ,i9 G (0,1] and t? € [0,1], w/iere i9 > 1?. + 1?*.. ylpart ffi x from everything everythingelse, else, suppose (6.4.8) Then thestability stability estimates estimates from suppose thatthat (6.4.8) holds.holds. Then the (6.56) and (6.57) (6.57) hold holdfor for any anyfixed fixed /J,Q /J,Q> > 00 sufficiently sufficiently large large andfor forany any (6.56) and and fixed G[0, [0,£o)> £o)>where where £0 £0== 11++min{i9 min{i9— — 7*,i?i 7*,i?i — — 7*1}, 7*1}, and andthey they remain remain valid valid fixed ££ G 7 in the thecase case £ £ == £0 £0ififone one adds £<J~ £<J~log(t log(tg ++i /k) /k) to tothe thefactor factor tq. tq. Furthermore, adds Furthermore, in addition addition $1 $1>> 0,0,the thestability stability estimate estimate (6.58) (6.58) holds for any anyfixed fixed /J,Q /J,Q> > 00 if in holds for sufficiently largeand andfor forany anyfixed£* fixed£* G G[0,£i), [0,£i), where where£1 £1 = = 1— 1—7+min{'#,'#i}. 7+min{'#,'#i}. sufficiently large In the the case case £* £* == £i; £i;£/ws £/ws estimate retains retains its its shape shape with with t~_ t~_t replaced replaced by by estimate t~l*i + t~Zf t~Zf log(t log(tn-i+i/k), -i+i/k), where7*2 7*2«s «s#wen #wen by by (2.74). (2.74). t~l*i + where The proofis is similar similar to to that that of of Theorem Theorem6.13. 6.13. The proof In principle, principle,in in the thestatement statement of of Theorem Theorem 6.13 6.13we wemay maywithdraw withdrawthe the In assumptions (6.62)and and (6.63) (6.63) since sinceusing usingthe the argument argument employed employed in the the assumptions (6.62) in proof of Theorem Theorem6.12 6.12as aswell well as as hypothesis hypothesishol^Ji?©; hol^Ji?©; \t]leads leadsto to the thefact fact proof of | \t] that (2.54) (2.54)holds holdswith with #0 == 7*0 7*0==0 0and and = — — {1i?©,7©}. i?©,7©}. Therefore, Therefore, #0 7070 = {1 that applyingagain againTheorem Theorem 2.12,one onecan can state an an analogue analogueof of Theorem Theorem6.13 6.13 in in applying 2.12, state which theconditions conditions(6.62) (6.62) and(6.63) (6.63) are areremoved removedbut butititisisrequired requiredinstead instead which the and that 70 << 11— — 7* 7*++ min{i9,i?i}, min{i9,i?i},£* £* << 11— — 70, 70,and and££<<1 1 additionto to what what that 70 inin addition is already alreadyaccepted, accepted, where in the thelimiting limitingcases cases£» £» == 11— —70 70 and and££==1,1, the in the where stability estimatesare are accordingly accordinglycorrected. corrected.In In aa similar similar manner mannerone onecan can stability estimates show aa modification modificationof of Theorem Theorem6.14 6.14by byapplying applyingTheorem Theorem 2.13. show 2.13. We do not notintend intend to to examine examinemethods methodsof of class class SJJ SJJ in inthe thecase case with withaa We do variable operatorfor for the thereason reason that thatsuch suchaa consideration considerationwould would contain variable operator contain the results results do do not notseem seem to to be be many additionaltechnical technical difficulties the many additional difficulties whilewhile widely applicable.Now Nowwe weturn turn to to the thestudy study of of methods methodsof of class class SS1///. ///. widely applicable. Theorem 6.15. 6.15.Suppose Suppose that thatA(t) A(t) GGS(tp,a) S(tp,a) and andthat that the the RK RK method method is is Theorem both of of type type Vjn((p) Vjn((p) and andof oftype type Vf VfH{}p) {}p) and andsatisfies satisfies the thecondition conditiondeg[a(-) deg[a(-) — both — a(oo)] = = —1, —1,for for some some ip ip G G(0,TT/2) (0,TT/2) and andaa GGR.R. Suppose Supposefurther furtherthat that a a(oo)] a seminorm forms aa (71 (71|y, |y,a)-concordant a)-concordantpair pair with the operator operatorA(t) A(t)and and seminorm \\ \\t forms with the A{t) itself A{t) itselfsatisfies satisfieshypothesis hypothesis hol®[i9; hol®[i9; \-\t), \-\t), for for some some 71 71G G[0,1) [0,1)andd andd G G (0,1]. (0,1]. Finally, Finally, let letLO(Z) LO(Z) be beaa rational rationalfunction functionwhich which is is (ip)-regular (ip)-regularand andsubject subject to to deg[a;] deg[a;] < < —1. —1. Then Thenit it holds, holds, with with2J 2Jn = = uj{—kA(t uj{—kA(tn)) and andQ3_i Q3_i==2Jo, 2Jo, with with sufficiently large, large,and and with with any any fixed fixed £ £G G [0,1), [0,1), for for all all any any fixed fixed [IQ [IQ > 00 sufficiently

6.3. STABILITY ESTIMATES ESTIMATES

141 141

k G (0, K] and t n,ti G fife such that ti < tn,

|| < Ct~il+V

(6.73)

i/ i/ie operator 91 in (5.19) is given by DK = u>(—kA(0)), then the solution Yn of (5.19), (5.20) satisfies the stability estimate (6.57). (6.57).

Proof. First of all note that by Lemma 5.3, it suffices to show the claim in the particular case u(z) = (1 — z)~l which is equivalent to the fact that QJn — {I + kA(tn))~l for t n G (lk- From now on we therefore accept the last assumption. Further, by Theorem 6.6 the operator 2l(t n ) satisfies the A^-condition, while using Theorem 6.9 yields that 2l(i n ) substituted for Qln satisfies hypothesis HOL ~* ($, 0; 1 | m ; n , m = 0, / , *0), where with some no > 0,

||m;n

\\((I + kAitn))" 1)*

if m = 0, if m = / , ||* if m = *0,

since the seminorm \\(p(—kA(t n)))* ||* appearing in (6.43) is replaced by ||((/ + kA(tn))~1)* ||*, by Lemma 5.3 and duality. Observe that the triplet (||(2l(i n ) + / V ) ||;2l(^n); {I + kAitn))-1) is A|(l)-concordant, by Theorem |t n ;2l(^); (/ + kA(tn))~l) is A^( 7 i)6.6, while the triplet (|(/ + kA^))-1 concordant, by Theorem 6.5. Also, a slightly more delicate calculation calculation than than with the aid of (5.39) and (5.43), for the one leading to (6.61) gives, again with all tn <E Clk, X G £*, and u e X,

3= 1

where the seminorms Y!j=i W(.wj(-kA(tn)))* ||* and Y!j=\ \vj(~kA(tn)) \tn may be replaced, by Lemma 5.3, respectively by ||((I + kAftn))^ 1)* ||* and by !(/ + kA{tn))~l \tn, preserving the needed properties since deg[u;j] deg[u;j] < —1 and deg[«j] < — 1 for j = 1 , . . . , v. Furthermore, it is easily checked, using the assumed Holder-continuity Holder-continuity

142 142

CHAPTER 6. ANALYSIS OF STABILITY CHAPTER 6. ANALYSIS OF STABILITY

hypothesis for A(t) A(t) and andarguing arguing as as above, above, that thatfor for all alltn,ti G G Cl^, Cl^, hypothesis for

=

||I++k(A(ti) k(A(ti) -- A(t A(tn)) (I (I ++ ||I

<

Ck\\{A(ti) + + fill) fill) (I (I + + 11++ Ck\\{A(ti) 11 +Ck sup sup KZ KZ +Ck

<

C. C.

(6.74)(6.74)

Therefore, using usingthe the above above reasonings, reasonings,the the stability stability estimate estimate(6.73) (6.73) follows Therefore, follows 2.15since since the thetriplet triplet (||(2l(i (||(2l(i + /i 0 -0 C ll;2l(*n)". ll;2l(*n)".(I (I + + kA(t kA(tn))))-1) by Theorem Theorem2.15 by n) + A2(^|l)-concordant, in view view of of Theorem Theorem6.6. 6.6. is A2(^|l)-concordant, in To completethe the proof, proof, F Fn and andYQ YQ are are represented representedby by To complete = 2J 2Jn F n and andyy0 = = 9J-ig°, 9J-ig°, Fn = andy° y°==(I(I+ +kA(0))9iy kA(0))9iy0. and

Fn = = (I (I ++kA(t kA(tn))F ))Fn with F with (5.43) gives gives (5.43)

||Fn || <
Usingnow now(5.41) (5.41)and and Using (6.75) (6.75)

3=l,...,v 3=l,...,v

by (6.74), (6.74), we we have have while, by while, I I 55°°IIII<
(6.76) (6.76)

With the the aid aidofof(6.75) (6.75) and and (6.76), (6.76), and andthe theabove above observations, observations, it is is readily readily With it nowthat that (6.57) (6.57)follows followsby by Theorem Theorem2.19. 2.19. seen seen now 6.16.Suppose Suppose that thatA(t) A(t) 66 S(ip,a) S(ip,a) and andthat that the the RK RK method method is is T hheeoorreemm 6.16. both of of type type Vju(ip) Vju(ip) and andof oftype type Vjjj((p) Vjjj((p) and andsatisfies satisfies the thecondition condition deg[a(-) deg[a(-) both — a(oo)] = —1, —1,for for some some ip ip G G(0,TT/2) (0,TT/2) and andaa 66M. M.Let Let further A(t; A(t; , , ) )be be a(oo)] = further aa testing functional functionalfor for A(t), A(t), let letseminorms seminorms || |© |©;t and and \\ | m; | t, t, m m == 0,1, 0,1,*0, *0, testing m; form respectively respectively(/y®\(p, y®\(p,a)-, a)-, (l/2\ip, (l/2\ip, a)-, a)-, (yi\(p,cr)-, (yi\(p,cr)-,and andstrictly strictly (*;l/2\ip,a)(*;l/2\ip,a)form concordant pairs pairswith with A(t), and andlet letthe the operatorA(t) A(t) itself itself satisfy satisfy the thehyhyconcordant A(t), operator potheses h61®fc) ; |©;t] and ho\^\-d,^;A(t; , ; \m-t,m = 0,1, *0), for potheses h61® [tf I |©;t] and ho\^\-d,^;A(t; , ; I \m-t,m = 0,1, *0), for fc) e some some ^^,,ii99e G G (0,1], (0,1], i?i i?i €€ [0,1], [0,1], 7 7 e €€ [0,1), [0,1), and and71 71ee [0,1/2). [0,1/2). Also, Also,supsuppose that that the the conditions conditions (6.62) (6.62)and and (6.63) (6.63) hold holdwith with7 7 == 7* 7*==1/2 1/2and and with pose with G [0,1) [0,1)such such that that"d, "d,+ .. < < -d. -d. Apart Apart from from everything everythingelse, else, some i?,,!?*, i?,,!?*,G some let u{z) u{z) be beaa(tp)-regular (tp)-regularrational rational function.Then, Then, 2Jn = = uj(—kA(t uj(—kA(t let function. with with 2J n)), any fixed fixed /xo /xo >> 00sufficiently sufficiently large largeand andfor for any anyfixed fixed £* £* G G[0,£o), [0,£o), where where for any for = 11++min{'i9 min{'i9- 1/2, 1/2,i?i i?i- -71}, 71}, if if deg[u] deg[u]< < -£*, -£*, we wehave, have, for for all allkkGG (0,K] (0,K] ^0 = and Gflk flk such such that thatti ti < < ttn, and tn,ti G

6.3. STABILITY ESTIMATES

143

6.3. STABILITY ESTIMATES

In the case £* = £o, the last estimate is still in force with t~_t replaced £

143

-| Icy

t -| Icy In the£ ; case by £~_* + t n _j log(i n _j+i/fc). Furthermore, if in addition i9i > 0 replaced and the (5.19) satisfies condition (6.4.8), then we have, for any fixed operator ; 9Kn in n £* log(i = £o,_j+i/fc). by + t _j Ho > 0 sufficiently large and for any fixed £ € [0, l/2 + min{#,i9i}) such that 9K in (5.19) satisfies (6.4.8), then we have, for any fixed operator the last condition estimate is still in force with t~_ £~_* Furthermore, if in addition deg[cj] Ho > < —£, /or aZZ k £ (0,K] and tn,ti 6 fife suc/i i/iai tj < tn, that i9i > 0 tn,ti 6 fife suc/i i/iai tj < n, that deg[cj]0 < —£, /or aZZ and the t k £ and (0,K] sufficiently large for and any fixed and £ € [0, l/2 + min{#,i9i}) n-l and such that n-l max

max where Yn is the solution of (5.19), (5.20). In the case £ = both estimates if one (5.20). adds tq'12 g+i/fc) to tq for q = n + 1 where Y n is the remain solutionvalid of (5.19), In log(£ the case 12 and q = n — I, where both estimates remain valid if one adds tq' log(£g+i/fc) £ = and to tq for q f 1/2 t/ tf
D and The interested .reader can state without difficulty an analogue of Theorem 6.16 in the situation when hypothesis hol'~^(...) replacedofbyThehy(XI - k&itn))The interested reader can state without difficulty an isanalogue pothesis hor*~)(...). orem 6.16 in the situation when hypothesis hol'~^(...) is replaced by hypothesis Remark hor*~)(...). 6.6. In our considerations carried out within an abstract framework, we are forced to accept from time to time some differently stated hyRemark potheses as independent ones,from although they not differently really suchstated in actual work, we are forced to accept time to timearesome hy6.6. practical applications. example, if A(t)they is thought as ansuch approximation potheses as independentForones, although are not ofreally in actual In our elliptic considerations carried out operator, within antheabstract frameto a multidimensional partialif differential above inequalpractical applications. For example, A(t) is thought of as an approximation ities (6.62) and (6.63) elliptic are akin, in fact, to assuming hypothesis h oinequall^(...). to a multidimensional partial differential operator, the above ities (6.62)6.7. andApplying (6.63) are akin, in fact, to assuming hypothesis of Theorems Remark Lemma 1.11 shows that the assertions 6.12 — 6 . 1 6 remain valid if instead of the seminorms ||(2l(t n ) h+o l ^ ( . ^ Remark 6.7. and ||(21*(i n ) + Horf* .\\mWe use ||(2l n + M o /)C . || and n ..). 6.12 ) +Theorems Applying Lemma 1.11 shows that the assertions of n and — ) + Horf* .\\mWe use ||(2l n + M o /)C . || and ||(21*(i 6 . 1 6 remain valid if instead of the seminorms

^ ||(2l(t

144

CHAPTER 6.6. ANALYSIS ANALYSIS OF OF STABILITY STABILITY CHAPTER

respectively, under under slightly slightly stronger strongerrestrictions restrictions on on££and and£*, £*,except exceptTheorem Theorem respectively, needs no no additional additional assumptions. assumptions. More Moreprecisely, precisely,one onethen thenhas hastoto 6.15 which needs addition that that £* £* << £o> £o>££<< £i£i inin the thestatements statements ofofTheorems Theorems6.13 6.13 require in addition that ££ << £o> £o> £* £* << £i £i in in the the statement statement ofof Theorem Theorem6.14, 6.14,and andthat that and 6.16, that min{$, 11 — 71} in the statement statement of of Theorem Theorem 6.12. 6.12. Also, Also, asasconcerns concerns £* < min{$, statement of of Theorem Theorem 6.12 6.12 in in the the particular particular case casewhere whereA(t) A(t) satisfies satisfies the statement hol[i?], itit should should in in addition addition be berequired requiredthat that££<<11++ hypothesis hol[i?], 6.8. The The results results of of Section Section 6.1 6.1 allow allowusus totoobtain obtainstability stabilityestiestiR e m a r k 6.8. ||, involving some some different different seminorms seminorms ininplace placeofof||(2l(£ ||(2l(£ mates involving ||, n) + jtxol)^

||2Jn(2l(*n)+Mo/)Hl, ||(2l*(i,)+Mo/)Hl*, and||3J?(a*(t,)+/xo/)HU. and||3J?(a*(t,)+/xo/)HU.More More ||2Jn(2l(*n)+Mo/)Hl, ||(2l*(i,)+Mo/)Hl*, 21 as concerns, concerns, for for instance, instance, the thefirst firsttwo twomentioned mentionedseminorms,' seminorms,' precisely, as ' let (£|
is aa {(p)-regular {(p)-regularrational rational function function that that isis under under the the condition condition where ip(z) is replaced by by the the condition condition deg[ip] < — £ if £ £ N degf^] < — [£J — 1, which is replaced a)-concordant pair pair with with A(t). A(t). Then, Then, seminorm $$forms forms aa(£\£\
substituted „ substituted

for || ( » ( * „ ) + M o / ) €

for | | 5 3 n ( 2 l ( t n ) + Mo-O ? || or for

II,

assertions of of Theorems Theorems 6.10, 6.10, 6.11, 6.11, 6.12, 6.12, 6.13, 6.13, 6.14, 6.14, and and6.16 6.16remain remain the assertions in addition, addition, ifif the the RK RK method method isis ofof class classSu Su ororofofclass class valid, assuming in deg[^] << —((w —((w ++ 1), 1), where whereww == — deg[a(-) — a(oo)] and the the Sin, that deg[^] last nonstrict nonstrict inequality inequality isis replaced replacedby by the the corresponding correspondingstrict strict one oneif if£ £isis an integer. Moreover, Moreover, ifif (6.21) (6.21) holds, holds, then thenfor for ££== 00ititisisenough enoughtotoaccept accept that deg^] << 00 instead instead of of all all the the above aboverestrictions restrictions on ondegfV']. degfV']. With Withthe the same substitution, substitution, the the assertion assertion ofof Theorem Theorem 6.15 6.15 remains remains valid validprovided provided that deg[^] << 00 ifif ££ == 00 and and deg[-i/>] deg[-i/>]<< 11— 2£ otherwise. otherwise. assertion is, is, in in fact, fact, aa consequence consequenceofofTheorems Theorems6.1, 6.1,6.3, 6.3,and and The above assertion 6.5.

in virtue virtue of of the the above above analysis analysis ofof concrete concrete families families ofofRK RK Note that in (see Section Section 5.2), 5.2), Theorems Theorems6.10, 6.10,6.12, 6.12,6.13, 6.13,and and6.14 6.14are areapplicable applicable methods (see Radau IA, IA, IIA IIA and and Lobatto Lobatto IIIC IIIC methods methods asas well wellasasfor forallallRK RK for all Radau whose Butcher Butcher array array isisspecified specified by by(5.26), (5.26),provided providedthat thatcondition condition methods whose holds. At At the the same same time time all all Gauss-Legendre Gauss-Legendre methods methodsofofodd oddorder order (5.27) holds. can serve as as examples examples of of application application ofofTheorems Theorems6.11, 6.11,6.15, 6.15,and and6.16. 6.16.Also, Also, Gauss-Legendre methods of of even even order order and and Lobatto LobattoIIIA, IIIA,IIIB IIIBmethods methodsofof Gauss-Legendre methods odd order lead lead to to rational rational approximations approximations which whichare areappropriate appropriatefor fordealing dealing with the assertion assertion of of Theorem Theorem 6.10 6.10 and and rational rational schemes schemesononthe thebasis basisofof 21

seminormscan can be be replaced replacedininaasimilar similarmanner. manner. The other two seminorms

6.4. COMMENTS AND AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL

REMARKS REMARKS

145 145

Lobatto IIIA, IIIB methods methods of of even even order order can can be be used used in in the the format format ofof Theorem 6.11.

6.4

Comments and bibliographical bibliographical remarks remarks

Section 6.1: As concerns concerns Theorems Theorems 6.1, 6.1, 6.3, 6.3, and and 6.5, 6.5, what what the the reader reader meets meets here is their maiden presentation, presentation, while while their their particular particular versions, versions, that that is, is, Theorems 6.2, 6.4, and 6.6 6.6 are are actually actually present present in in our our old old work work [16, [16, 18, 18,22]. 22]. The proof of Theorem 6.1 6.1 combines combines some some ideas ideas from from [37] [37] with with those those ememployed in [18] (see also [35]). [35]). Section 6.2: Apparently, Apparently, Theorems Theorems 6.7 6.7 and and 6.8 6.8 are are new. new. At At the the same same time time our earlier work [16, 19, 19, 25, 25, 26] 26] contains contains assertions assertions that that are are quite quite close close toto Theorem 6.9. This group group of of results results is, is, however, however, essentially essentially based based on on applyapplying the hypotheses like hoi® hoi® [i9; | \t], which is of limited use in practice. practice. (See (See also the comment to Section Section 6.3.) 6.3.) Section 6.3: In the situation situation when when A(t) A(t) == AA isis independent independent ofof tt22 under the assumption of >l(c^)-stability >l(c^)-stability and and under under the the condition condition |r(oo)| |r(oo)| << 1,1, aa stability estimate which which is is equivalent equivalent to to (6.49) (6.49) with with ££ — — 0,0, isis first first stated, stated, apapwe also also mention mention parently, in Brenner &: Thomee Thomee [52].23 (For later references, we Larsson, Thomee, and Wahlbin Wahlbin [100] [100] and and Lubich Lubich & & Nevanlinna Nevanlinna [104].) [104].) ItIt is worth noting, for fairness' fairness' sake, sake, that that the the same same achievement achievement (with (with ££== 0)0) is presented in a Hilbert Hilbert space space framework framework by by Le Le Roux Roux [102] [102] whose whose techtechniques are, nevertheless, nevertheless, extended extended to to the the Banach Banach case. case. ItIt turns turns out out that that the restriction |r(oo)| << 11 is is not not crucial, crucial, and and itit isis avoided avoided inin our our work work [10, 11, 12, 16, 18, 22], where where the the corresponding corresponding non-strict non-strict inequality inequality isis alallowed as well. Later, a similar similar result, result, for for ££ == 00 only, only, isis shown shown inin Crouzeix Crouzeix etet straightforward techniques. techniques. The The al. [61] and Palencia [127], by using more straightforward general case £ > 0 has been been examined examined by by the the author author in in [10, [10, 11, 11,12, 12, 16, 16,18, 18,22], 22], and Theorems 6.10 and and 6.11 6.11 arise arise from from there. there. Also, Also, we we mention mention two two recent recent papers of Hansbo [82, 83] 83] who who in in particular particular presents presents certain certain strong strong stabilstability type estimates. As As concerns concerns her her results results for for single single step step methods, methods, itit isis easy, however, to deduce deduce them them directly directly from from [16, [16, 18, 18, 22]. 22]. In In the the matter matter ofof the assertions collected collected in in the the present present chapter, chapter, itit isis seen seen that that ifif ££ >> 0,0, we we have to distinguish between between the the classes classes Si, Si, 5// 5 / / on on the the one one hand hand and and Sm Sm since for for the the last last class class of of methods methods additional additional damping damping on the other hand since same time, time, when when using using general general seminorms seminorms ofof the the form form is required. At the same 22

One actually handles then then the the rational rational method method (5.30). (5.30). [52] deals deals with with rational rational approximations approximations ofof general general (not (not necesnecesIn principle, the paper [52] sarily holomorphic) Co semigroups. semigroups. 23

146

CHAPTER 6. ANALYSIS OF STABILITY

(6.77), the role of a damping factor played by the operator ip(—kA(tn)) is distinguishable in a different manner for these classes (see Remark 6.8). Turning now to the history of the question in the situation with a variable operator, we mention, first of all, certain early results in Gudovich & Terteryan [78], Sobolevskii [149, 152], and Thomee & Wahlbin [163], which deal with concrete problems or concrete narrow families of discretization methods. For results of general character, we refer to our work [16, 19, 25, 26] and to Gonzalez & Palencia [73, 74]. We remark that the papers [73, 74] which have appeared later contain no improvements in the properly Holder case in comparison with our results.24 The assertions of the present Theorems 6.12 and 6.15 follow in essence our investigation in [16, 19, 25, 26] while Theorems 6.13, 6.14, and 6.16 are presented for the first time here. We remark that the assertions of Theorems 6.12 and 6.15 are of limited use in practice and the same can be said regarding all results of Gonzalez & Palencia [73, 74]. The point is that those results are based on putting to use Holder-continuity type hypotheses which cannot be verified for partial differential operators and their discrete approximations in some useful function spaces, for instance, in those with maximum norm, unless we are dealing with the one-dimensional case. At the same time the assertions of Theorems 6.13, 6.14, and 6.16 are free of that disadvantage.

24

In fact, the situation considered in [73, 74] is even restricted to the case where £ = 0 and the condition |r(oo)| < 1 is assumed.

Chapter 7

Convergence Estimates Estimates Since stability estimates estimates have have been been shown, shown, the question of convergence is easily settled if the input data of the starting differential problem problem (5.1) are sufficiently smooth. In applications connected with with PDE PDE problems, problems, we meet, however, the phenomenon of order reduction when the order of convergence certain compatibility compatibility conobserved is smaller than the classical one, unless certain ditions are imposed on the input data. Apparently, Apparently, Crouzeix Crouzeix [59] is the first to examine this phenomenon phenomenon in detail in the context of Hilbert spaces. possibilities, depending depending on the smoothness of the Below we study different possibilities, input data, for showing error estimates estimates by using Banach space techniques. techniques. data not In what follows we are particularly interested in dealing with input data being smooth enough, that that is, when the classical order of consistancy is not achieved. In this chapter, given numbers numbers 0 < T\ < TI and K G [1,OO], we denote by L K(ri,T2;3£) the Banach space of strongly measurable functions functions ip(s) : \T\ > T2] —> £ with the norm

esssup s e [ r i ) T 2 ] ||^(s)||

if

K = OO.

7.1 Preliminaries It is known (see Appendix B) that under certain conditions conditions there there exists exists a unique solution of problem (5.1), 1 where the operator A(i) depends on t. Sometimes it is also possible to point out conditions under which the solution :

B y solution we always mean classical solution solution (cf. Appendix B).

147

148

CHAPTER CHAPTER 7. 7. CONVERGENCE CONVERGENCE ESTIMATES ESTIMATES

has the desired order order of of regularity regularity (see, (see, e.g., e.g., Sobolevskii Sobolevskii [148]). [148]). However However for for the reasons explained explained in in Appendix Appendix B, B, we we never never use use such such conditions conditions below. below. Instead we posit the the unique unique solvability solvability of of the the starting starting problem problem (5.1) (5.1) asaswell well as the fact that the solution solution is is as as smooth smooth as as needed needed in in order order to to achieve achieve the the desired order of convergence. convergence. At At the the same same time time we we keep keep the the conditions conditions that that ensure the unique solvability solvability and and stability stability of of the the discrete discrete problem problem (5.19), (5.19), (5.20) so that our main main assumptions assumptions will will be be that that A{t) A{t) isis (ip, cr)-sectorial 2 for some

are associated with with the the quadrature quadrature formulas formulas

/ "

3=1

Jo

and / Jo

ip(s)ds^k}^a jiip(cik),

j = l,...,u,

(7.2)

1=1

where ip(s) may be thought thought of of as as an an (3£)-valued (3£)-valued function. function.3 It is well known (see, e.g., [53, 65]) that stating stating the the simplifying simplifying assumptions assumptions S(j>) S(j>) and and C(g) C(g) leads to the fact that that the the quadrature quadrature formulas formulas (7.1) (7.1) and and (7.2) (7.2) are areprecise precisefor for all polynomials of degree degree pp — 1 and q — 1, respectively. For subsequent subsequent use use we denote

and

(L

rtn+Cjk rt n+cjf

2

n

+ C,k)

,

j = l,...,K

Domj4(t) = Dam is independent of t, which is accepted throughout throughout We recall that Domj4(t) Part II. 3 It is just the case in what follows.

7.1. PRELIMINARIES

149 149

Lemma 7.1. Given p, qq > > 1, 1, for for any any ip(t) ip(t) :: [0,T] [0,T] —> X sufficiently smooth, the simplifying assumption assumption ®(p) ®(p) implies implies

(7.3) (7.3)

[ Jtn

while C(q) implies

\\Qnj(n\
[^ \\^{s)\\ds,

j = l,...,u.

(7.4)

Jtn

Proof. Clearly, it suffices to show show (7.3) (7.3) only. only. By By Taylor Taylor expansion expansion we we have have ds.

(7.5)

1=0

With this formula in mind, mind, (7.3) (7.3) easily easily follows follows noting noting that that the the value valueofofQQn(V)) remains unchanged when when substituting substituting ip(t) ip(t) — —X)?=o X)?=o ~i\ ~i\ V'^fa) V'^fa) f° f° r since the quadrature formula formula (7.1) (7.1) isis precise precise for for all all polynomials polynomials ofof degree degree p-1. Further, we denote

, r \i=x

Lemma 7.2. Lei, groen groen q>\, q>\, the the simplifying simplifying assumption assumption G(q) G(q) hold. Then, for any tp(t) : [0, T] —>>XX sufficiently sufficiently smooth, smooth,4 ds,

where 7j(s) is a bounded bounded Peano Peano kernel. kernel. Proof. In fact, the claim follows follows in in the the same same manner manner as as in in the the proof proof ofof the preceding lemma, using using (7.5) (7.5) with with qq ++ 11 in in place place of of p,p, noting noting that that the the quadrature formula (7.2) (7.2) is is precise precise for for all all polynomials polynomials of of degree degree qq— —1,1, and and

observing that Q nj(ipq) = k"+1Hqj^\tn), 4

Cf., e.g., Lubich & Ostermann Ostermann [105] [105]

where %{t) =

^-^q)(tn).

150

CHAPTER CHAPTER 7. 7. CONVERGENCE CONVERGENCE

ESTIMATES ESTIMATES

Lemma 7.3. Let the RK method method in in question question be be of of order order pp and and of of stage stage order q < p — 2. Then the order condition holds holds

This result can be established in in the the standard standard manner manner by by using using Taylor Taylor expansion (cf. Ostermann &; Roche Roche [123]). [123]). Now we derive suitable representations representations of of the the error, error, assuming assuming that that ££RR== I.5 When making this assumption, assumption, we we are are distinguishing distinguishing between between the the auautonomous and non-autonomous cases, cases, and and by by doing doing so, so, we we show show that that this this distinction is caused by the accepted accepted compromise compromise between between our our intentions intentions and and real possibilities. In the first (autonomous) (autonomous) case case we we show show error error estimates estimates in in terms of the data, which are realistic; realistic; however however similar similar estimates estimates in in terms terms of the solution are also possible. In In the the general general (non-autonomous) (non-autonomous) case case we we estimate the error only in terms of of the the solution, solution, and and we we cannot cannot satisfactosatisfactorily presently demonstrate anything anything different. different. In In both both cases, cases, autonomous autonomous and non-autonomous, we now assume assume that that problem problem (5.1) (5.1) possesses possesses aa unique unique solution y(t). Let A(t) = A be independent oft. oft. 6 Suppose that A € S(
Yn+1 = r(-kA)Y n + kJ2^j(-kA)

f(t nj) for tn G Clk, Yo = y°.

3=1

Also, for the solution y(t) of (5.1) at nodal nodal points points we we use use the the representation representation

y(tn+i) = r(-kA)y(tn)

+ k^Wj{-kA)

f(tnj) - „ for t n G &k,

3=1

where (f)n is the truncation error. Therefore, Therefore, for for the the global global error error EEn =

Yn-y(tn),

we have En+i = r(-kA)En + „,

which further gives, since EQ = 0, 0, n-l

£„ = J^r(-fcA) n - J '-V; for£ n efV 5

(7.6)

In this chapter more general expressions expressions for K will be used only in the very particular case where A(t) = A and /(£) = 0. 6 In this case the unique solvability of (5.1) will, in fact, follow from our subsequent suppositions.

7.1. PRELIMINARIES 7.1. PRELIMINARIES

151

151

Turning next tothe the generalcase case where A(t)depends dependson ont,t,we we assume Turning next to general where A(t) assume that the theconditions conditions ofTheorem Theorem5.1 5.1are aresatisfied satisfiedso sothat thatthe thediscrete discreteproblem problem that of (5.19), (5.20)is isuniquely uniquelysolvable. solvable. Fory(t) y(t) the solutionof of(5.1), (5.1),we wehave, have, (5.19), (5.20) For the solution usingthe theabove abovefunctionals functionals Qnn(-) (-)and and Qnj('), that that using Q Qnj('),

YY ^^

j

)

f(tnjnj)) )) -- QQn(yt) (yt) - -f(t

and and

Subtracting these equalities from the respective Runge-Kutta equations in respective Runge-Kutta equations in Subtracting these equalities from the (5.19)and and(5.20), (5.20),we weget, get, withE Enn just just the thesame sameas asabove aboveand andwith withE Ennj = = (5.19) with

En+1 =E Enn-kJ2 -kJ2 E n+1 =

bjA(t )Enj +Q Qnn(y (yt) bjA(t nj)E nj + nj

(7.7) (7.7)

and and V V

Enj =E Enn-k^ -k^aj iA{tnni)E i)Enl E nj = ajiA{t nl

Qnj (yt), ++ Q nj(yt),

l,...,v. j j==l,...,v.

(7.8) (7.8)

Therefore, solving (7.8) for = (E (Enn\,... \,... ,E ,E and assumingthat that the Therefore, solving (7.8) for E nnE= and assuming the nv) nv ) expressions(i (i++A;a'P A;a'P )^11Q Qnn(yt) (yt) and andT> T>n (i (i ++A:oX A:oX )~11Q Qnn(j/t) (j/t) are aredefined, defined, expressions ri)^ nn)~ ri Qnn (-) (-)= = (Qni(-), (Qni(-), Qnu{-)) Qnu{-)),,similarly similarlyto to(5.54) (5.54)we wefind find whereQ where Enn = = (i(i++kaV kaVnn)~ )~lee ®E ®E +(x (x+ +kaV kaVnn)~* )~* QQ (yt), ), E n + n(y n n whichfurther further gives, inserted into (7.7), which gives, inserted into (7.7), T T (I-kb Vnn{i+kaV {i+kaV )-11e)E e)Enn-kb -kbTTV Vnn(i+kaT> (i+kaT> )-lQ Qnn(y (yt)+Qn(yt) )+Qn(yt) (I-kb V n)n)n n

E = En+X n+X =

kbTTV Vnn(i (i + +kaVn^Qniyt) kaVn^Qniyt) + +Q Qnn{y {yt). ). n -- kb n Using this recurrence formula, since —0, 0,EQ followsthat that fortnnt £ £Q^, Q^, Using this recurrence formula, since EQ — ititfollows for n--ll n

Enn = =J2^n-i,j+i( J2^n-i,j+i( Vj{S kh ++ kaV^Qjiyt)+ Qj(yt)). + Qj(yt)). E ~ kh~TTVj{S kaV^Qjiyt) (7.9) 3=0 3=0

we werecall recallthe thewell-known well-known factfact (cf., (cf.,e.g., e.g., Sinestrari Sinestrari[146]) [146]) In In conclusion, conclusion, that that if ifA(t) A(t)GG S(ip, S(ip,a) a) for forsome some

(7.9)

152

CHAPTER CHAPTER 7. 7. CONVERGENCE CONVERGENCE ESTIMATES ESTIMATES

any fixed t £ [0,T], by using formula (B.2), a family of operators e s > 0, which possesses all the properties of holomorphic semigroups. semigroups.7 For further reference we now show a result of a preparatory character which which is involved with the operator e~sA^\ L e m m a 7.4. Let A(t) 6 S(tp, a) a) and and let let aa seminorm \ \t form a (£\ 0. Then, if T < oo, we have, with K > 0 sufficiently small, for all k € (0,-fiT], t,s e [0,T], andue X, ||,

(7.10) (7.10)

#

(( ii .. ll JJ .. ))

and, in particular, for any fixed r\ € N U {0}, t )

&

_

^ ^

^ ^

If o~ > 0 in addition, then the above holds in the case T — K — oo as well. Proof. Let Ys be the contour coinciding with with d(Y d(Y1{pWD{\/s)). Furthermore, Furthermore, it is is sufficient to going over to the shifted operator A(t) — al if a < 0, it consider the case a > 0 only. Assuming therefore therefore the last restriction and using next formula (B.2) with Ts in place of H, we have, for all s > 0, e-sA(t) =

J_

Now (7.10) follows when when integrating integrating by parts and acting further as in the 8 proof of Theorem 2.1. With regard to the second stated estimate (7.11), (7.11), it is the particular case of (7.10) where £ = r\ and | |t = ||-A(<)^ ||.

7.2

Autonomous equations equations

In this section we examine convergence in the case with a constant operator. (Therefore we write A instead of A(t).) As above, we then work with the extended solution of (5.19), (5.20) specified by (5.30). We begin with the homogeneous case f(t) = 0, for which the extended discrete solution is specified by Yn+1 = r(-kA)Y n 7

iovt neClk,

Yo = 9V\

(7.12)

Except strong continuity continuity at 0 if Dom A(t) is not dense in X In fact, this technique can be thought of as a slight modification of a well-known semigroups (cf., e.g., Reed & Simon [133, the argument in the theory of holomorphic semigroups proof of Theorem X.52]). 8

7.2. AUTONOMOUS EQUATIONS EQUATIONS

153 153

and it is compared to the the extended extended solution solution of of the the original original Cauchy Cauchy problem problem (5.1) given by y(t) — e~ tAy° for t > 0, where the operator operator e~ e~tA, t > 0, is denned by formula (B.2). (B.2). First we show an auxiliary auxiliary assertion. assertion. L e m m a 7.5. Let A G G S(
rn(z) = (r(z)n - enz) ( ^ J

,

(7-13)

it holds, with K > 0 sufficiently sufficiently small, small, for for all allkk G G(0,K] (0,K] and andttn G Qk> \\rn(-kA)\\ 0 one can can take take TT — —K K= = oo. oo.

Proof. Let r\(z) be the stability stability function function of of an an A-stable A-stable method method ofof order order > p, with the property ri(oo) ri(oo) == 0. 0. (For (For example, example, one one can can take take aa z/-stage z/-stage Radau IA method with v > > (p+ (p+ l)/2.) l)/2.) Then, Then, clearly, clearly, itit suffices suffices to to show showthe the desired estimate with r^ (—kA), (—kA), jj == 1,2, 1,2, in in place place of of rrn(—kA), where

Note that there exists a fixed fixed d\ d\ > > 11 such such that that with with some some qq>>1,1, ^1| —
T (v\\
fr>T oil v d. / 7 . "\ T n + T V^\U'\). r-r- ^-^^ ^-^^

Since |r(oo)| < 1, this estimate estimate further further implies implies that that |r(z) |r(z) n | < ec"l2l q and, thus, ^(z)] < Ce cn|z| ~ 9 , for all n > 0 and z $ IntD(d IntD(d x ). Also, in view of the fact that \r(—z)\ < 1 and |ri(—2)| < 1 for for zz G GS 1, that \rf\-.z)\ < Ce-cnW~" for all n > 0 and z G E for any fixed cpi G (0, ip). Next, fixing arbitrarily ip\ ip\ G G (0, (0,
— (r(y\

— r, r,

(S (S

as well as the estimates, for for some some dd >> 0, 0, \r(z) - n(z)\ < C\z\v+l

for all z G 2?(d),

154

CHAPTER CONVERGENCEESTIMATES ESTIMATES CHAPTER 7.7.CONVERGENCE H-z)\ H-z)\

Ce~c^ and and \n(-z)\ \n(-z)\ < Ce~ Ce~c^ < Ce~

for all all zz££EE¥ i n V{\), V{\), for

and taking taking the condition condition r(0) r(0) = = r'(0) r'(0)== ri(0) ri(0)==r'i(O) r'i(O) = 1 1 into account, account, it follows that follows that \r^\z)\ < < Cn Cn\z\ecnW for for all allzz €€V{d) V{d) \r^\z)\ and cn]zl |r£>(-z)| < Cn\z\eCn\z\e-cn]zl |r£>(-z)|

for all all**€€ for

EVlnV(l). nV(l). E

It is therefore therefore seen seenthat thatthe function function r\, (z) (z)satisfies satisfies the conditions conditions of Lemma Lemma C. which yields yieldsthat that Hr&^-fcA)!! Hr&^-fcA)!!< C. 5.6, which Similarly, can be beshown, shown, applying applyingLemma Lemma5.5, 5.5,that \\r\ (—kA)\\ (—kA)\\ < C, C, Similarly, it can and the proof proof is now nowcomplete. complete. convergence is as as follows. follows. The first first assertion assertionon convergence Theorem 7.1. 7.1. Let LetAA EES( > 00 sufficiently sufficiently small, small,for all allkk E E[0,K] [0,K]and andtnt G f2fc, holds, with |.

(7.14) (7.14)

case a > 00 one onecan cantake T = =K K — oo. In the case Proof. follows from from Theorems Theorems6.10 6.10 and and 6.11 6.11and andLemma Lemma 7.4 that with with Proof. It follows 0 sufficiently sufficiently small, small,for all all kk E E(0,K] (0,K]and andtnt E fife, fife, K >0 \H-kA)n\\ \\e~tnA\\ + \H-kA)

< C. C. <

(7.15) (7.15)

method is of of order order p, by by Taylor Taylor expansion, expansion, for any any Furthermore, since sincethe method Furthermore, fixed ipi E (0,(f) (0,(f) the the function function r{z) r{z) = = z~^~ z~^~1{r{z) {r{z) — ez), where where j comes comes from from fixed ipi statement of the the theorem, theorem, satisfies, satisfies,when whensubstituted substitutedfor w(z), w(z),the the conconthe statement ditions Lemma 5.4 and andthe thecondition condition f(oo) f(oo) = = 0. 0. ItIt therefore therefore follows follows that that ditions of Lemma uE EDom m^^+ 1 , for all u || (r(-kA) (r(-kA)

< ^ +1 ||F(-fcA)|| Ch>+1\\Aj+1u\\. - e-fcA) u|| < ||F(-fcA)|| | | |^^+ 1 u | | < Ch>

mind, since since Yn = r(—kA) r(—kA)ny°, using using (7.15) (7.15) and andthe the With this inequality inequalityin mind, With this identity identity Yn-y{tn)

= = (r(-kA) (-kA)n

- e-tnA) y° -tn-lAr(-kA) r(-kA)l-ly°,

(7.16) (7.16)

7.2. AUTONOMOUS AUTONOMOUS EQUATIONS EQUATIONS 7.2.

155155

a simple simple estimation estimationyields, yields, for all allttn G G fife, fife, for \\Yn - y(t y(tn)\\ which shows(7.14). (7.14). which shows for aa >>00isisobvious. obvious. The more precise precisewording wordingfor The more 7.1 can can be bewidely widely used usedsince sincethe the property property of of A ((^-stability ((^-stabilityis Theorem Theorem 7.1 established for very very many many discretization discretizationmethods. methods. For further further progress progresswe we established for For have to distinguish distinguish between betweenthe the above above introduced introducedclasses classesof methods. methods. have to First we we introduce introduce one one more more helpful helpful notion. notion. First Definition 7.1. We Wesay saythat that a rational rational function function LO(Z) LO(Z)approaches approaches a function function Definition 7.1. T](z) with order orderm ifif T](z) with m u(z) = = ri(z) ri(z)++OO(\z\ (\z\ as \z\ \z\-> ->0.0. u(z) ) as

Our next result resultconcerns concernsmethods methods of class class £/. £/. Our next of Theorem 7.2. Suppose Suppose that thatA G GS(ip, S(ip,a), a), the theRK RKmethod method is of of order order p and and Theorem 7.2. Ai[ip]-stable, and aa seminorm seminorm | --||t forms forms a (£\ip, (£\ip,a)-concordant a)-concordantpair pairwith withA, A, Ai[ip]-stable, and for some some p G G N, N, ip ipGG(0,TT/2), (0,TT/2), a G G R, R, and and££>>0.0. Suppose Suppose further, further, for for some non-negative non-negative integers integersj ,, ?, ?, and andmm subject subject to to jj < < pp and and££<<s s+ + some j ,j , that y° G G DomA? DomA? and and 9\ 9\ — —u(—kA), u(—kA), where whereu>(z) u>(z)is aa rational rational function function that y° (if)-regular, approaches approaches with order orderp, and andsatisfies satisfies the the condition condition which is (if)-regular, emz with which deg[o;] < — —q. Then, Then, if if TT < < oo, oo,for forYY specified by (7.12) and for y{t)the the deg[o;] specified by (7.12) and for y{t) n extended extended solution solution of of (5.1) (5.1) with with f(t) f(t) == 0, 0, we wehave, have, with with K > > 00 sufficiently sufficiently for all allkk GG(0, (0,K) K)and and G fifc fifc with with n > > 1, 1, that that small, small, for tn t G )[Y y(tn+m Ckn-^-j)\\A^y\ )[Yn- y(t )}[tn< Ckn-^\\A^y\ n+m)}[t

(7.17) (7.17)

= mm ++j j provided provided that, that, in in addition addition to to The result result remains remains valid valideven evenif ££ = The above, the above, the seminorm seminorm <, (£|£|,cr)-concordant cr)-concordantpair pairwith with A A ifif ££ >>11 and (6.21) is is satisfied satisfied with with}[-][t }[-][t substituted substituted for for || |t|t ifif££==0.0. InInthethe case and (6.21) case a > 0, 0, the theabove above statement statement holds holdsfor for TT == oo ooand andone onecan can then take take K == oo. oo. then Proof. Proof.Noting Noting that that all allthe theinvolved involved operators operators are are pairwise pairwise permutable, permutable, we we n\, have, with with n\n\ ==[n/2\ [n/2\and and n n\, have, n^n^ — n——

Yn-y(t -y(tn+m n+m)

n = r(-kA) r(-kA) u(-kA)y° = u(-kA)y°

=

(r(-kA)ni (r(-kA) +{r(-kA) +{r(-kA)n2

--

-*"iA ) -- e -*"i e-^A) -- e-^

+(co{-kA) - +(co{-kA) G\ ++ G2 G2++G3. G3. G\

tmA tmA

0 e'^+^y e'^+^y

r(-kA)n2Lu(-kA)y° Lu(-kA)y° r(-kA) ee tnA tnA

t

e'e' )e- y°

^

A

u (( °°

156

CHAPTER CHAPTER 7. 7. CONVERGENCE ESTIMATES

With rn(z) given by (7.13), G\ and Gi can also be represented by d = (-k)pAp~J'(/ +

kA)- pr(-kA)n2Lu(-kA)rm(-kA)Ajy°

and

G2 = (-k)pAp-j(I +

kA)- pe-tniAu(-kA)rn2(-kA)Ajy°.

It is obvious that the claim follows if we show that for all k G (0, K],tn G flk w i t h n > 1, and t G [0,T],

X

(7.18)

Observe that instead of estimating }[Gi][t as stated in (7.18) it suffices to

show that with G = (-k)pAp-i(I +

kA)- pr(-kA)nu(-kA)rn(-kA)A^y°,

for all k G [0,K], tn G Clk, and t G [0,T], $i)\\Aiy°\\.

(7.19) (7.19)

To that end, we note that since the method is A/(~pp(zy~puj(z), it follows from the above observations that the function uo(z) is (y)-regular and for any fixed tp\ G (Q,ip) t m s function is bounded in S ^ with LOO(Z) = UQ(OO) + Od^l" 1 ) as z\ —> oo, for z G E^j. In virtue of Lemma 5.4, the operator UIQ(—kA) is defined and uniformly bounded on X, that is, IIM-fcA)|| M - f c A ) | | < C for all k G {0,K}.

(7.20)

Now, recalling that 21 = Ap(-kA), applying Theorem Theorem 6.10 in the case /(£) = 0 and Remark 6.8, and using the fact that all the involved operators are pairwise permutable, we get, since £ < m + j , 9 with m and ni defined as above, for all t G [0, T] and t n G f2fc,

Ckpt-l1\\^p-ir{-kA)n2uo(-kA)rn(-kA)Ajy°\ pj) \\u0(-kA) 7n{-kA) Aiy% which further leads to (7.19) in view of (7.20) and Lemma 7.5. Finally, the needed estimate (7.18) for \G2\t and \G3\t follows by actually using the same argument, with the aid of Lemma 7.4 and of the estimate 9

Or £ = m + j and the extra conditions on |-][t are assumed as pointed out in the statement of the theorem.

7.2. AUTONOMOUS EQUATIONS EQUATIONS

\\p(-kAy-'\\ Z-P{W(Z) -

+ \\(kAy-3p(-kA)P-J\\ \\(kAy-3p(-kA)P-J\\ ++ \\Ul(-kA)\\

157 157 < C, 10 where UJX(Z) =

mz

e ).

It is also seen from the above above reasonings reasonings that that in in the the case case aa >> 00the the result result is still valid for T = oo, oo, with with no no restriction restriction on on kk from from above. above. For methods of class Su Su or or of of class class Sm Sm we we have have the the following following converconvergence result. Theorem 7.3. Let the the conditions conditions of of Theorem Theorem 7.2 7.2 be besatisfied satisfied with with the the difdifference that the RK method method is is now now Ajj(ip)Ajj(ip)- or or Ajjj(ip) Ajjj(ip) -stable -stable and and the the concondition £ < ? + j (or the the condition condition ££ << m m ++ jj with with the the corresponding corresponding extra extra requirements) is replaced replaced by by the the condition condition ££ << (<; + p)/{w + 1) — p + j , with w = — deg[a(-) — a(oo)], where the nonstrict nonstrict inequality inequality isis replaced replaced by by the the corresponding strict one one ifif ££ is is an an integer, integer, unless unless ££ == 00 and and (6.21) (6.21) holds. holds. Then, if T < oo ; for Yn and y(t) specified the same same as as in in the the conditions conditions of of Theorem 7.2, the error estimate estimate (7.17) (7.17) holds holds in in the the considered considered case case as as well, well, and the result is still valid valid for for TT == K K— —oo oo ifif aa >> 0. 0. The proof is similar to that that of of Theorem Theorem 7.2, 7.2, applying applying Theorems Theorems 6.10 6.10 and and 6.11 and Remark 6.8. Theorem 7.3 can be applied applied for for all all Gauss-Legendre Gauss-Legendre and and Lobatto Lobatto IIIA, IIIA, IIIB methods while Theorem Theorem 7.2 7.2 isis applicable applicable for for all all other other concrete concrete families families of methods discussed in in Section Section 5.2. 5.2. Further, we examine inhomogeneous inhomogeneous problems problems of of the the form form (5.1) (5.1) but but concontinuing in the case of a constant constant operator. operator. We We therefore therefore handle handle the the rational rational method (5.30) now. In In the the inhomogeneous inhomogeneous case case we we meet meet aa phenomenon phenomenon called order reduction. This This phenomenon phenomenon shows shows itself itself by by the the fact fact that that ifif the the discretization method is is of of high high order, order, we we have have to to impose impose certain certain compatibilcompatibility requirements on the solution solution or or on on the the data data 11 to achieve the optimal order order of accuracy otherwise aa lower lower order order only only isis observed. observed. Below Below we weconsider consider difdifferent situations, including including those those for for which which the the order order reduction reduction phenomenon phenomenon occurs. is of of type type Theorem 7.4. Suppose Suppose that that AA ££ S(tp, S(tp, cr) and the RK method is C(y>), of order p, and of strict strict order order q, q, for for some some ip ip GG (0,TT/2), a G M, and p, q G N subject to to pp > > q. q. Suppose Suppose further further that that 9K 9K== I,I, TT << oo, oo, problem (5.1) possesses possesses an an extended extended solution solution y(t) y(t) of of order order pp ++ I,I,12 and 10

Which follows by Lemma Lemma 5.4, 5.4, similarly similarly to to (7.20) (7.20) These restrictions are often often thought thought of of as as artificial. artificial. In In fact, fact, there there isisno nocommon common opinion opinion of what are artificial conditions. conditions. Many Many authors authors try try to to avoid avoid conditions conditions ofofthe theform form y"' y"' (t) (t)€€ Dom A* with j > 2 while while the the situation situation with with jj == 11 isis thought thought ofof as as quite quite acceptable. acceptable. 12 Which is simultaneously aa solution solution of of (5.1), (5.1), see see Appendix Appendix B. B. 11

158

CHAPTER CHAPTER 7. CONVERGENCE

ESTIMATES ESTIMATES

forl = q,...,p and t G [0,T], tuti/i some G L^O^-X) m — 0 , . . . ,p — q. Then, ifYn is the extended solution of (5.19), (5.20), we have, with y? p+D coming from formula formula (B.5) stated for p + 1 in place of p and with K > 0 sufficiently small, for all k € (0, K] and ttn G fijt, that 1^

\\Yn - y(tn)\\ < C^ n ||y° p + 1 ) || + Ckr\\(tn -

In the case a > 0, the above holds for T — oo as well and one can then take m— — 0. 0. K > 0 arbitrarily large but finite if m > 1 and K = oo ifif m Proof. Our argument will be based on using formula (7.6). An essential ingredient of this effort is finding a suitable estimate of the quantity ||^ n ||) appearing in (7.6). where n is the truncation error appearing - y{tn+\) + kwT(-kA)&n, where n is given Since (pn = r(-kA)y(t n) by (5.29), we have, by Taylor expansions for y(t) and f(t),

n = r(-kA) y(tn) - J2 -^y{l){tn) + kW0;n + Rn,

(7.22)

where n

j=l

'"

,)

f o r J = 0, . . . , p - l

1=3

and V

Rn

= ds

Now we introduce the functions

j=0

"Therefore, the classical order is observed in the case m = 0 only.

(7.23)

7.2. AUTONOMOUS

EQUATIONS EQUATIONS

159 159

and

It can easily be checked checked (cf., (cf., e.g., e.g., Brenner, Brenner, Crouzeix, Crouzeix, and and Thomee Thomee [50, [50, Lemma 2]) that hi(z) = 0 for Z = 0 , . . . , 9 - l (7.26) (7.26) and hi(z) = O(\z\p~l)

as z^O

iorl = q,...,p,

(7.27)

since the method is of order order pp and and of of strict strict order order q. q. With With the the aid aid ofof the the equality -Ay — y' + f, we we derive derive from from (7.22), (7.22), with with Wj Wj;n defined by (7.23),

=

kQQ(-kA)y'(t n)

P u - V -rri/W(t n) + feWi ;n -

kh Q(-kA)f(tn),

where we denote V

Qj(z)

= hj{z) + ] T cJjWj{z),

J =

0,...,p-l.

Applying repeatedly the the trick trick based based on on using using the the relationship relationship —Ay® —Ay® — — y('+i) _ j(i) for i = l ^ . ^ p j we finally obtain obtain aa new new representation representation for for 4>n — Rn as follows P

f>n-Rn

k 2Q{kA){A)'{t)Y

=

a

1=2

+kWx.n - kho(-kA)f(tn) W

= ^

^

+

1

\

(7.28)

Putting further for short short I'I' = = m&x.{0,p m&x.{0,p — —m m— —I}, I}, II == q,... q,... ,p,p and and denoting denoting hi(z) — z~ l hi(z) for I = q,... ,p, we we note note that that the the functions functions hi(z) hi(z) are are (<^)(<^)14 regular and subject to deg[/ij] deg[/ij] << 0, 0, in in view view of of (7.27). (7.27). Applying Applying therefore therefore Lemma 5.3, we find, for for all all kk G G (0, (0, K], K], that that = q,...,p. 14

We recall that tuj(z), jj == 1, 1,

, v, v, possess possess this this property. property.

(7.29) (7.29)

160

CHAPTER CHAPTER 7. 7. CONVERGENCE CONVERGENCE ESTIMATES ESTIMATES

Using now (7.28) instead of (7.22) yields, with the aid of (7.26),

(7.30)

which further implies, by (7.29), Un\\ < CkP +1~m J2 H^'/(0(*™)ll + Cfc P+1||2/(P+1)(*n)ll + \\Rnl

(7.31) (7.31)

At the same time since ||IUJ(—kA)\\ < C, a trivial estimation based based on using (7.24) gives

< CkP+1

max

\\y{p+l\s)\\

| | / « ( a ) | | + Ch? f ^

0<S
ds.

(7.32)

J tn

Next, since problem (5.1) possesses an extended solution of order p + 1, applying formula (B.5) with p + 1 in place of p and noting that ||e~ ij4 || < C for 0 < t < T, it follows that

||y (p+1) (t n )|| < C\\y°{p+1)\\ + C t

||/ ( P + 1 ) ( S )|| ds

(7.33)

J0

and

< Ck\\ y°{p+1)\\ + C ftn+\tn+1

- s)

+Ck [\\f^ +1\s)\\ds. Jo Combining (7.31), (7.32), (7.32), (7.33), (7.33), and (7.34) thus yields

(7.34) (7.34)

l=q

Ckp / Jtn

(t 4-1 — s) II f(p+1^(s) I ds + Ckp+1 I

\\fij'+1Hs)\\ds

(7.35)

7.2. AUTONOMOUS EQUATIONS EQUATIONS

161 161

Finally, by a trivial calculation we we have have n-l

n-1 n-1

ftj

j=0J°

j=l j=l

J

^-

"-1 ft,-

inequality and and by by inserting inserting (7.35) (7.35) into into (7.6) (7.6) the the claim claim With the aid of this inequality is shown because ||r(—kAy\\ ||r(—kAy\\ < > qq >> 11 (z) is (ip)the condition condition deg[o;] deg[o;] << — £(c7 + 1) in which < should should be be regular and satisfies the unless |r(oo)| |r(oo)| << 11 and and the the pair pair (][-|t; A) is (1|1| 0 sufficiently sufficiently small small (with (with K K — —oo oo ifif aa >> 0) 0) and and with with any any we K] and and ttn € fife, n > 1, fixed K € [1/(1 — £)i °°]> have, for all k € (0, K]

M-kA) (Yn - y(t n ))k l=q+l

15

oo ifif aa > > 0. 0. It is admissible that T = oo

162

CHAPTER CHAPTER

7. 7. CONVERGENCE CONVERGENCE

ESTIMATES ESTIMATES

is replaced by ££n := log(t n + i/fc) if where, unless £ = 1, the the factor factor tt n ~ K — 1/(1 — £), while, in in the the extreme extreme case case KK == oo oo and and ££ == 11 ; the factors and tnT are replaced tn replaced by by t\t\ and and ££n, respectively.

Proof. We demonstrate our argument argument only only for for ££ << 11 since since the the below below proof proof barely changes in the case case ££ == 1. 1. Having therefore assumed assumed that that ££ << 1, 1, for for any any scalar scalar positive-valued positive-valued function ip(s), s > 0, with the aid of aa simple simple calculation calculation itit follows follows that that with tpn(s) — (£„ - s)ip(s), for all t n € ftfc, n-l

rti

n-\

n-j n-j

n-1

and further, applying Holder's Holder's inequality inequality for for integrals integrals and and sums, sums, consecuconsecutively, n l

~

^ 3=0

nn

/-*3+i

n

~-' Jt Jt

3

~

~1

^ j=0 j=0

/

n

~^

rh+i

1/K

\

\Jt

I

\\Jtl

I

(E .

(7.38)

Also, under the accepted conditions, conditions, by by Theorems Theorems 6.10 6.10 and and 6.11 6.11 we we have, have, for all t e [0, T], « € X, and and 00 << jj << nn -- 1, 1, that that

l .|

(7.39)

Finally, the claim follows by by using using formula formula (7.6) (7.6) in in combination combination with with (7.35), (7.35), (7.37), (7.38), and (7.39). It appears that sometimes sometimes the the same same order order of of convergence convergence isis achieved achieved under reduced regularity requirements requirements in in comparison comparison with with what what isis stated stated above. We will give only only one one particular particular assertion assertion which, which, however, however, enables enables us to show the main idea. idea.

7.2. EQUATIONS 7.2. AUTONOMOUS AUTONOMOUS EQUATIONS

163

163

Theorem T h e o r e m 7.6. 7.6. Suppose Supposethatthat the the conditions conditions of of Theorem Theorem 7.4 7.4are are satisfied satisfied for for m ==00and forfor some g GgNG N subject m and some ip ipGG (0,7r/2), (0,7r/2), crcrGGR,R,p, p, subject to toqq<

0), thatthat itit isisnow T < ifooa if a > 0), with with the the difference difference now assumed assumedthatthat (lj) (lj) 99 11 pp qq (0) GG D ~ f^(-) f^(-) GG Loo(0,i;£) / (0) Doomm^ -^ - "-- instead instead of ofthe the condition conditionA A ~ Loo(0,i;£) for for 16 te[0,T}. Let te[0,T}.16 Let furthermore furthermore deg[h qq ] << deg[p], deg[p], deg[h

(7.40) (7.40)

where (z) isis given q. q. Then (7.21)(7.21) where hhqq (z) given by by (7.25) (7.25) for forI I—— Then the the error errorestimate estimate pp pp qq qq holds \\A ~ ~ f^ f^ \-)\\i \-)\\i oo ^Q ytn x) isis replaced in which which the the summand summandCk Ck tnn \\A replaced by by holds in oo ^Q ytn .tx) C can can substitute CPPppPP- «--«7 -^ 7) (^ ) ( 00 ) ||||. . Also, Also, ifq ifq ==p-l,p-l, one one substituteC WC W nn | ||//^^( (0 0) | )| | | Ck Ckpp tnn \\f^(-)\\ \\f^(-)\\ Loo(0ttn for the the summand summand for Loo(0ttn .x) x) . Proof. andand (7.30) Proof. Clearly, Clearly, by by(7.6) (7.6) (7.30) itit suffices suffices to to find find aa different differentestimate estimate for for the expression ~ :>~ >~ 11 4>j, 4>j, where the expression EEnn == Yl^Zo Yl^Zor(—kA) r(—kA)nn ~ where

fa fa Using Using now now (7.40) (7.40) and andthe the fact fact that that the the function function p(z)~ p(z)~l isis (
||^(-fcA)||
(7.41)

(7.41)

observing that that kAp(-kA) observing kAp(-kA) ==

a(—kA) a(—kA) — —I I — —r(—kA), r(—kA), and and using using the the identity identity n --l l n

En

==

j

^ ^ r r{ {- -

ll

3=0 3=0

= ZZnn -- r(-kA) = r(-kA)nn ZZ00 ++Y,r{ Y,r{ 3=0 3=0

we find, we find,with with the theaid aidof of (7.15) (7.15) and and (7.41), (7.41), n-\ n-\

\\E \\En\\ << wZnW wZnW ++ cwzoW cwzoW ++ J^ J^ 3=0 3=0

< Ck Ckpp ((

(

77

I,

++ 11

3=03=06

W ee therefore r e s t r i c t i o n s oonn/ / W t h e r e f o rleave e leave tthhee restrictions

((t )t ,) , II ==qq++ l,...,p— l,...,p—

1,1,a sa s ssttaat et de dbefore. before.

164

CHAPTER CHAPTER 7. 7. CONVERGENCE CONVERGENCE ESTIMATES ESTIMATES

Therefore the claim follows by combining this inequality inequality with with the estimate

Furthermore, note that that

ll/^OHwo*.*) < H/(P)(O)II + r \\&+1){s)\\ds. Jo

Using this result for an estimation in (7.31) and (7.32) and taking (7.36) into account, it is seen, with the aid of the argument used in the proof of Theorem 7.4, that the more precise wording for q = p — 1 is true as well. It is not hard to see that for methods of type C/ (
<-I

= deg[p],

l=

q,...,p.

In conclusion, we remark that all RK methods belonging to the concrete families discussed in Section 5.2 are of type C(
7.3

Non-autonomous equations equations

In the present section we show error estimates in terms of the solution for the case with a variable operator. We assume throughout that that T < oo, 17 problem (5.1) possesses a unique solution y(t) and that 91 = I in (5.19). In our consideration we concentrate on examining the situation where the solution of (5.1) is of low space regularity, and more precisely, it is assumed that at most y^(t) G Vom for t € [0,T] [0,T] and and for for some l>\. We start by showing the following result. Theorem 7.7. Suppose that A(t) G S(ip, a) and A(t) A(t) satisfies hypothesis lip, while the RK method is of type Vi(
Hereinafter y(t) has the same meaning.

7.3. NON-AUTONOMOUS EQUATIONS EQUATIONS

165 165

of the numerical solution by (5.19), (5.20), it holds, with K > 0 sufficiently small and pi\ > 0 sufficiently large, for all k £ (0, K) and t n G £}&,

|| L l ( o, i n ;£)-

(7-42)

Proof. Similarly to (5.43) we conclude that with ^i > 0 sufficiently large, the operator (D n - V{tn)) (i + fcaDn)" 1 (8) (A(t n) + Mi)"1 i s defined on X and18

||(D n - T)(tn)) (i + fca'Dn)"1 ® (A(tn) + MX)"1 || < Ck.

(7.43)

Using therefore the representation

) ) - 1 ^ - V(tn))(i + fcaDJ" 1 ® (A(t n) + MI)" 1 , (7.44) which is, in fact, an analogue of (5.44), it follows, with the aid of (5.35) and (7.43), that the operator D n (i + kaVn)~l ® (A(tn) + Mi)"1 is defined on X and (7.45) ||Dn(i + kaVn)- 1 ® (A(t n) + MI)" 1 II < C.

Also, it follows from (6.56) with £ = 0 that for £n,£j £ ^fc such that tj < tn,
(7.46) (7.46)

Now, by (7.9) and Lemma 7.2, we have En = E^] + E%] + E^\ where, w i t h H = (Hqi,...,

Hqu)

,

3=0

j=0

and

n-l

3=0 18

We recall that the operator A{t) is now Lipschitz-continuous.

166

CHAPTER CHAPTER 7. CONVERGENCE CONVERGENCEESTIMATES ESTIMATES

It will be well defined. thethe be seen seen from from below belowthat thatthese theseexpressions expressionsareare well defined.By By triangle inequality with \\En\\En ||, ||, triangle inequality itit thus thussuffices sufficestotoshow showthat that(7.42) (7.42)holds holds with j = 1, 2, 2, 3, 3, in in place placeofof ||.E n||. First of of of \\En' of all all we we observe observethat thatthis thisfact facteasily easilyfollows followsin inrespect respect \\En' since we we have have by by (7.3) (7.3)and and(7.46), (7.46),

"'"lly^OOIIda. "'"lly^OOIIda.

o

Further, using thethe assumed hypothesis Further, using (7.46) (7.46)and and(7.45) (7.45)and andnoting notingthat thatbyby assumed hypothesis of Lipschitz-continuity, Lipschitz-continuity,similarly similarlytoto(6.47), (6.47), -1\\
foraUt,ae[0,T], foraUt,ae[0,T],

(7.47) (7.47)

we find, find, since since A(t) A(t) isisclosed, closed,that that n-l j=0

This result result shows shows (7.42) (7.42)with with||^||^ 2 ) || substituted substituted for for \\E \\En\\. '\\.'\\.ToTothat wewe findfind from It thus thus remains remains toto estimate estimatesuitably suitably\\En \\En thatend, end, from { ] { ] ] where (7.44) that that EE n - E n + E£ , where

j=0

and

3=0

It follows follows from from (7.46), (7.46),(5.35), (5.35),and and(7.43) (7.43)that that \\(A(tj) 3=0

7.3. NON-AUTONOMOUS EQUATIONS EQUATIONS 7.3. NON-AUTONOMOUS

167167

However, by the thefollowing following identity identitystated stated for (X)-valued (X)-valued functions, functions, However, by for ftj+i ftj+i K*7/J I "" ll K*7/J

II

I 7/j I 7/j | c| I c I

I " I- i "-i- i -i SI 1/1SI I 1/1 Q II I QnII o

no

we have instead, instead,using using(7.47) (7.47) and the thefact fact that that A(t) A(t) isisclosed, closed, we have and *+ 2

^^

rtrt +i

k J 22 j=0 JtJtJ j=0

l=q+l

This fact further furthergives givesthat that This fact +2 +2

\\(A(0 \\(A(0 +2) \\{A{s)+ ml) ml) y^ y^+2) (s)\\ ds, ds, fn \\{A{s) (s)\\

o

and (7.42) holds holdswith with\\En \\En ||||substituted substitutedfor for ||£? ||£?n||. It It thus thus remains remainsto to show show and (7.42) 4) (7.42) when whensubstituting substituting||£n ||£n for ||E ||En ||. (7.42) || for Using now nowthe theidentity identity Using (-kA{tj)) wT(-kA{t

=bbT - A;A(^)b A;A(^)bTa(i a(i ++ =

kaV(tj))-\ kaV(tj))-\

and takingLemma Lemma7.3 7.3 into into account, account,we we find find and taking n - ll { ]

+2 +2

T = k^ k^ J2^n-i,j+ikA(t J2^n-i,j+ikA(t a(i En = j)b a(i

+1 1 ka'D(t HA(tj)y^ )y^+1 ++ka'D(t \tj). j))- HA(t

(7.48) (7.48)

Since deg[p] Since deg[p] = = —1, —1, p(z)"" p(z)""1 is is ((^)-regular, ((^)-regular, and and the the entries entries of of the the matrix matrix 1 a(t — — 2a)"" 2a)"" vanish vanish as as \z\ \z\ — — >>00, 00,by byLemma Lemma 5.3 5.3 itit follows follows that that with withr](z) r](z) == \r](-kA(t))\\
forkke e(0,K] (0,K]and and [0,T]. 6 for t 6t [0,T].

Next, using usingthe theequalities equalities Next, (-kA(tj))H ))H wT(-kA(t

p(-fcA(t,-))»7(-fcA(tj)) == p(-fcA(t,-))»7(-fcA(tj))

and and kA(tj) fc9l(tj) = I= - I ilj- -ilj k
(7.49) (7.49)

168

CHAPTER 7. CONVERGENCE

instead of (7.48) we have, with Z3 = kq+2r]{-kA(t3))

ESTIMATES

A(t3) y{g+l)(t3),

that

n-l n

3=0

where n-l

n-l

3=0

and n-l j=0

Since A(t) is under hypothesis lip, it follows from (6.61) that \\?Bj\\ < C, which further implies

3=0

Therefore \\En || is finally estimated just the same as \\En || and it remains to prove that (7.42) is fulfilled with ||£ n 6) || in place of ||E n ||. Applying again hypothesis lip for A(t), it holds

\\(A(tj)-A(tj+1))y^1Htj)\\
Using now the last two estimates together with (7.47), (7.49), and the above argument and applying the identity

+V+2V(-kA(tj+1))

7.3. NON-AUTONOMOUS NON-AUTONOMOUS EQUATIONS EQUATIONS

169169

simple estimation estimation gives gives a simple <

\\(A(s)+fi1I)y^+2\s)\\ds. \s)\\ds. \\(A(s)+

(7.50) (7.50)

-i

Therefore, noting that that by by(7.47), (7.47), Therefore, noting

Hs)\\ds I*" \\A(tn)y^+2Hs)\\ds

Jo


||(A(S) + M l 7)^^( 7)^^(S )||d S , ||(A(

Jo with the the aid aid of of (7.46), (7.46),(7.49), (7.49),and and(7.50) (7.50)it itfollows followsfrom from above representhethe above representation for for EEn that that

Jo n-l

J=0

This completes completes the the proof proofsince sincethe thelast lastterm termononthetheright right estimated is is estimated justjust same as as ||ij4 ||ij4 ||-||the same D D natural to to expect expectthat thatthe theorder orderofofconvergence convergence reduces under weaker It is natural reduces under weaker requirements of of space spaceregularity. regularity.InInthe thefollowing followingstatement statement space regurequirements nono space regurestrictions are areimposed imposedononhigher higherderivatives derivatives exact solution. larity restrictions of of thethe exact solution. Theorem 7.8. 7.8. Let Let the theoperator operatorA(t) A(t) and andthe theRK RKmethod method satisfy condiTheorem satisfy thethe condiof Theorem Theorem 7.7 7.7for for some sometptp6 6(0,TT/2), (0,TT/2), a G N, N, andp,q andp,q €€ NN such suchthat that tions of Let also also the the coefficient coefficientmatrix matrixa abebenon-singular. non-singular.Then, Then, forfor thethe q < p — 1. Let En of the the numerical numerical solution solutionofof(5.1) (5.1)byby(5.19), (5.19),(5.20), (5.20),it it holds, with error E holds, with £lk, sufficiently small, small,for forall allk kGG(0,(0, and K > 00 sufficiently K]K]and tn t € £lk, \\En\\ <

(O(O +1

+CF||^ +CF||^ )(.)|| L l (o, t t i ; £)Proof. Since oo is Proof. is non-singular, non-singular, we wehave have

(7-51) (7-51)

170

CHAPTER7. 7.CONVERGENCE CONVERGENCE ESTIMATES CHAPTER ESTIMATES

where Qj(-) Qj(-) == a^Q^-)a ^ Q ^ - ) -Also, Also,byby(5.43) (5.43) it follows where it follows thatthat \\CDj- V(tj)) V(tj)) (i(i++ kaVj^aW kaVj^aW <
n-l n-l

\\En\\ = | ||ffcc^^iilln _ 1 J ++ 11b r D ( i ,,))))((ii ++

1 fcaD(^))fcaD(^))aQJ(yi)ll

(7.52) (7.52)

This aim aim can canhowever howeverbebeachieved achieved using argument place of of ||jE ||jEn||. This in place byby using thethe argument employed in the theproof proofofofTheorem Theorem7.77.7 and applying equality employed in and by by applying the the equality aQ^yt) = aQ^yt) =

A^)) (-kA(tj))GT(-kA(t (-kA(tj))QJ(yt), A^)) P(-kA(t

— p(z)~ p(z)~1hT(x — za)~ za)~1a, since since where GT(z) — where G

Lemma 5.3. 5.3. by Lemma Next we we show show an analogue analogue of both both Theorem Theorem 7.7 7.7 and and Theorem Theorem 7.8 7.8 when when Next A(t) is is under under a different different Holder-continuity Holder-continuitytype typehypothesis. hypothesis. operator A(t) the operator Theorem 7.9. Let Let A(t) A(t) €€S(ip,cr) S(ip,cr) and and let let the the RK RK method method be both both of type type Theorem 7.9. Vj(ip) and of of type type Vf(, 7*|y>, a)-concordant a)-concordant m = 0, an A(t) and andthat that A{t) A{t) itself itself satisfies satisfies the the hypotheses hypotheses holS.Ji?o; holS.Ji?o;I It] an pairs with with A(t) pairs d lip(_4.(t; , ;;|| | m; t,m = = 0, 0,*0), *0), with with some some 70,7,7* 70,7,7*G G [0,1) [0,1) and and e (0,1]. (0,1]. lip(_4.(t; m;t,m (9+2) (p+1) Then, A(-)y(9+2) (0)2/(p+1) Li(0,T;3£), the the error error of the the numerical numerical Then, if A(-)y (0)2/ (') ^ Li(0,T;3£), solution of(5.1) (5.1)by by(5.19), (5.19),(5.20) (5.20) satisfies estimate (7.42). Furthermore, solution of satisfies thethe estimate (7.42). Furthermore, the coefficient coefficientmatrix matrixa aisisnon-singular, non-singular, then estimate (7.51) holds if the then thethe estimate (7.51) holds as as well, with with the the same sameregularity regularity requirements as the in the conditions well, requirements on on y(t)y(t) as in conditions of of 19 Theorem 7.5. 7.5. Theorem proof isis similar similartotothose thoseofofTheorems Theorems aid of The proof 7.77.7 andand 7.8,7.8, withwith the the aid of Theorem 6.13. 6.13. Theorem Also, Also, we give give some some related relatedconvergence convergenceresults. results. WeWe firstfirst show show howhow the the error error can can be be estimated estimatedininstronger strongernorms. norms. 3

can then then assume assume that thatq < < pp — 1 instead instead of the the above above restriction restrictionq

7.3. NON-AUTONOMOUS NON-AUTONOMOUS

EQUATIONS EQUATIONS

171 171

Theorem 7.10. Let the the operator operator A(t) A(t) and and the the RK RK method method satisfy satisfy the the conconditions of Theorem 7.7 for for some some tp tp G G (0,TT/2), a G N, andp,q G N such such that that q < p — 1. Also, suppose that aa seminorm seminorm $$ forms forms aa (£|1|<£>, (£|1|<£>, a)-concordant a)-concordant pair with the operator A(t) A(t) for for some some ££ G G (0,1] (0,1] and and that that aa rational rational funcfunction to(z) is (ip)-regular (ip)-regular and and satisfies satisfies the the condition condition deg[o>] deg[o>] << —1. —1. Then, Then, ifif ^(.)y(«+i)(.) i?/ (p+i)(.) e LK{0,T;X) for some K G G [1/(1 [1/(1 -- £),oo], the error En of the numerical solution solution of of (5.1) (5.1) by by (5.19), (5.19), (5.20) (5.20) satisfies satisfies the the estimate, estimate, with some [i\ > 0 and with with K K > > 00 sufficiently sufficiently small, small, for for all all kk GG(0, (0,K] K] and and

M-kA(t n))En]

tn

<

where the right-hand side side of of this this inequality inequality isis taken taken log(2 log(2++ l/fc) l/fc) times times inin the the case K = 1/(1 — £).

Proof. First of all it follows from from Theorem Theorem 6.12 6.12 and and Remark Remark 6.8 6.8 that that for for all all tn, tj G Q.k such that tj < < ttn and u e X ,

Using therefore this estimate estimate together together with with (7.45), (7.45), (7.3), (7.3), and and (7.4), (7.4), aa direct direct estimation on the basis of of (7.9) (7.9) shows shows the the claim, claim, with with the the aid aid of ofthe the argument argument leading to (7.38).

One more direction for for possible possible extension extension isis performed performed in in the the next next asassertion. Theorem 7.11. Let the the operator operator A(t) A(t) and and the the RK RK method method be beunder under the the conconditions of Theorem 7.7 for for some some tp tp G G (0,TT/2), a G N, andp,q G N such such that that q < p — 1, and let the coefficient coefficient matrix matrix aa be be non-singular. non-singular. Let Let there there further further be given seminorms \ j . and and \\ |*.;t of which the former one is independent independent of of t and has domain C X while while the the latter latter one one forms forms aa (*;"f*\(p, a)-concordant pair with A(t) for some some 7* 7* G G [0,1) [0,1) so so that that we we have, have, with with some some /ii /ii >> 00 sufficiently large, for all all tt G G [0,T], x G %*> and u £ Dom | |. ; that

|(X, A(t) (A(t) + fnIT'u)] < C\((A(t) C\((A(t) + +

niI)- 1yx\..-M-

Then ify {p+1){-) G Li(0,T;X) and and \yiq+1)(-)\. G LK{0,T;R) for some K G [1/(1 — 7*), 00], the error E n of the numerical solution solution satisfies satisfies the the estimate, estimate, with K > 0 sufficiently small, small, for for all all kk G G (0, (0,K] K] and and ttn G fife,

(7.53)

172

CHAPTER7. 7. CONVERGENCE CONVERGENCE CHAPTER

ESTIMATES ESTIMATES

where the the first first term term on on the the right-hand right-hand side sideof this this inequality inequality is taken taken log(2 log(2+ where 1/fc) times times in the the case case K = = 1/(1 1/(1 — 7*). 7*). 1/fc) Proof. As in in the the proof proof of Theorem Theorem 7.8, 7.8, using using (5.44), (5.44), (7.3), (7.3),and and (7.4), (7.4), it Proof. As suffices to show show the the claim claim with with \\E \\En\\ defined defined by by (7.52) (7.52) in place place of ||J5 ||J5n ||. By By suffices (7.53) we have, have, with with the the aid aid of ofthe theabove above argument, argument, (7.53) we

Ck^J2 \\En\\ < Ck^J2

sup \((A( \((A(tj) sup

j=0 llxll*=l llxll*=l j=0

apply Theorem Theorem6.12 6.12 (more (more precisely, precisely,the the stability stability estimate estimate thus remains remainsto apply It thus (6.58)), Remark Remark 6.8, 6.8, and andthe thereasoning reasoning leading leadingto (7.38). (7.38). (6.58)), the last last two two theorems, theorems, note notethat thatsimilar similarassertions assertions are esesAs concerns concerns the are tablished under underthe the Holder-continuity Holder-continuitytype typehypotheses hypotheses that are assumed assumed in tablished that are the statement statement of Theorem Theorem 7.9. 7.9. Further we present present some someerror errorestimates estimatesthat that are intended intended for for use use when when Further we are RK method method is of of class class 5 / ////.. We Wedo dono nomore more than than to show show two two separate separate the RK results. results. Theorem 7.12. Suppose Suppose that that the the operator operator A(t) A(t) is is under under hypothesis hypothesis lip lip Theorem 7.12. and A(t) A(t) G GS((p, S((p,a)a) while while the the RK RK method method satisfies satisfies the the condition condition deg[a(-) deg[a(-)— and a(oo)] = —1 —1and andisis both both of type type Vm(ip) Vm(ip) and and of of type type Vj Vjn{(p) {(p) as as well well as of of a(oo)] order p and and of of stage stage order order q, for for some some

> 00 sufficiently sufficiently small, small,for for all allkk GG(0,iT| (0,iT| and and ttn G Qk> Qk> estimate,

Proof. fact, the the result result can can be be shown shown in aa manner manner similar similarto that that used used in Proof. In fact, the proof proof of Theorem Theorem 7.8. 7.8. To To achieve achieve success successwe we also also use use the the estimates, estimates, for for < nn -- 1,1, 0 < jj < and and l

-^-^ <
of which which the the former former one one follows follows by by Theorem Theorem 6.15 6.15 while while the the latter latter one one can can be proved proved in the the same same way way as as (5.43). (5.43).

7.4. WORKING AGAINST AGAINST ORDER ORDER REDUCTION REDUCTION

173 173

Theorem 7.13. Suppose Suppose that that A(t) A(t) G GS((p, S((p, a) a) and and the the RK RK method method isis both bothofof type Vjjj(ip) and of type type Vf Vfn(ip) as well as of order order pp and and of of stage stage order order qq and satisfies the condition condition deg[a(-) deg[a(-) — a(oo)] = —1, for some some

, a)-, (*; (*; 1/2|
|t] and \ip(A(t;

, ; | | mm ; t,m = 0,*0), wt/i .4(£;-,-) a testing testing

functional for A(t) and and with with some some 70 70 G G [0,1) [0,1) and and $\ $\ GG (0,1]. Let also the conditions (6.62) and and (6.63) (6.63) hold hold with with 77 == 7* 7* == 1/2 1/2 and and with with some some G [0,1) such that 1?. + i9*. << 1. 1. TTien TTien i/ie i/ie error error estimate estimate (7.42) (7.42) holds under the same regularity regularity requirements requirements on on y(t) y(t) as as in in the the conditions conditions ofof Theorem 7.7. If in addition addition the the coefficient coefficient matrix matrix aa isis non-singular, non-singular, then then 0 the estimate (7.51) holds holds as as welP welP under the regularity requirements requirements on on y(t) y(t) as in the conditions of Theorem Theorem 7.8. 7.8. The proof is similar to those those of of Theorems Theorems 7.7 7.7 and and 7.8, 7.8, noting noting that that the the stability estimate (7.46) (7.46) is is in in force force under under the the accepted accepted conditions, conditions, as as itit follows from Theorem 6.16. 6.16. It is worth noting that the the conditions conditions of of Theorems Theorems 7.7, 7.7, 7.8, 7.8, 7.9, 7.9, 7.10, 7.10, and and 7.11 are fulfilled for all Radau Radau IA, IA, IIA IIA and and Lobatto Lobatto IIIC IIIC methods methods as as well well as as for all RK methods which which satisfy satisfy (5.26), (5.26), (5.27). (5.27). Besides, Besides, Theorems Theorems 7.12 7.12 and and 7.13 are applicable, for instance, instance, for for all all Gauss-Legendre Gauss-Legendre of of odd odd order. order. The The magnitudes of orders and and stage stage orders orders for for these these methods methods are are pointed pointed out out inin Section 5.2.

7.4

Working against order order reduction reduction

Now the reader sees that that the the classical classical order order of of Runge-Kutta Runge-Kutta methods methods isis not not in general achieved if the the operator operator in in question question isis unbounded unbounded and and the the data data are not sufficiently smooth. smooth. Therefore Therefore the the question question arises arises ifif there there really really are any means of avoiding avoiding the the phenomenon phenomenon of of order order reduction reduction or or atat least least of raising the order of convergence. convergence. In In the the present present section section we we discuss discuss some some possibilities along those those lines. lines. For For simplicity simplicity we we restrict restrict ourselves ourselves to to the the case case where A(t) G S(tp, a) with with aa > > 0, 0, which which yields yields that that the the operator operator A(t) A(t) has has aa bounded inverse with aa uniform uniform estimate estimate ^ ( i ) " 1 ! < C for t G [0,T]. We begin with a discussion discussion in in the the case case where where A(t) A(t) == AA isis independent independent of t. Let the RK method method be be of of order order pp and and of of strict strict order order qq << pp — —1.1. We We associate to problem (5.1) (5.1) the the following following auxiliary auxiliary problem, problem, with with jj — —pp — q — 1 B

We then assume that qq << pp — —1. 1.

174 174

CHAPTER 7. CONVERGENCE ESTIMATES CHAPTER 7. CONVERGENCE ESTIMATES

or with with jj = =p p— — q, q, {j j + Az Az = = (-l) (-l) Az' + A-jf{j \

0
1=1 1=1

The point pointis is that that the thesolutions solutionsof of (5.1) (5.1)and and(7.54) (7.54) are are connected connectedby by The

y{t) z(t)- f^i-lYA-'f^Ht). f^i-lYA-'f^Ht). y{t) = z(t)

(7.55) (7.55)

i=i i=i

Therefore, havingobtained obtained an approximation approximationfor for z(t) z(t) and andusing using next nextthe the Therefore, having an last relation,it itis easy easy to to get getan anapproximation approximationto to y(t). y(t). At Atthe the same time time last relation, same it is is essential essential that thatone one can canapproximately approximatelycalculate calculate z(t) by bya a suitable suitableRK RK z(t) method, achieving achievingthe theoptimal optimalorder orderof of accuracy accuracy (= (= p) p) under under reduced reducedspace space method, regularityrequirements requirements on the thetime time derivatives derivativesof of /(£). /(£).Acting Acting so, so, we wehave have on regularity the following. following. the Theorem 7.14. 7.14. Let LetT T= =oo, oo, j = =p p— —q or or j — —p p— —q — —1, 1, A A£ £ S( > 0, 0, and andp,p,q q€ €N N such such that that qq <

Then, assuming assuming in inaddition, addition, if ifjj = =pp— —q — —1, 1, that that f^ f^q\t) GDom.A Dom.A for for Then, G t G G [0, [0,T], T], the the error error Y Yn — —y(t y(tn) of of the thenumerical numerical solution solutionof of the the original original (5.1) satisfies satisfies the the estimate, estimate, with withK K> >0 0sufficiently sufficiently small, small, Cauchy problem problem(5.1) Cauchy allk kG G (0, (0,K] K] and andtn G G Qk, Qk, for for all \\Y \\Yn-y(t -y(tn)\\

< <

^^ 1=1 1=1

l=q l=q

where y? y? ,,^ comes comesfrom fromformula formula(B.5) (B.5)stated stated for for pp++11ininplace place of of p. p. where

7.4. WORKING AGAINST AGAINSTORDER ORDER REDUCTION 7.4. WORKING REDUCTION

175 175

Proof. Since Since problem problem(5.1) (5.1)possesses possessesan an extended extendedsolution solutionof of order order p p+ + 1, 1,itit Proof. not hard hard to to see seethat that the the same same is is true true for for problem problem (7.54) (7.54)and and is not

( = 11

which results resultsfrom from(7.55) (7.55) by using using (B.4). (B.4).The Theclaim claim thus thusfollows followsby by direct direct which by applicationof of Theorem Theorem7.4 7.4 to toproblem problem (7.54). (7.54). application theexpression expressionon on the theright-hand right-handside sideof of the thelast last error errorestimate estimate Note thatthe Note that term of of the theform form Cfc Cfcp||^4y^ ||^4y^p~1)(.)||£ )(.)||£oo 3e) if j — — — while contains contains aa term (0 t 3e) if j — p — q — 1 while oo 1 n; of A A appear appear in in the thecase case j — —p — — q. q. no positive positivepowers powersof no Further we we turn turn to to the thecase case with with aa variable variable operator. operator. We We will will show show Further only that, that, modifying modifyingthe the RK RKmethod method in in aa suitable suitable way, way,itit isispossible possible to to only raise the the order order of of converegence converegenceat at least least by by 1, 1, without without imposing imposingstronger stronger raise restrictions of space space regularity regularityon on the theexact exact solution solutionbut but requiring requiring instead instead restrictions of more regularity regularityin in time. time. Note Notethat thatby by further further extensions extensions of these these techniques techniques more of to get gethigher higher orders ordersof of convergence convergenceunder underthe theoriginal original requirements requirements in order order to of space spaceregularity regularitywe we are areled ledtotothe the necessityof of imposing imposingcertain certain(difficult (difficult necessity for verification)extra extraconditions conditions on the theoperator operator A(t). A(t). Anyway Anyway we we leave leave for verification) on our present present consideration. consideration. such extensionsbeyond beyond our such extensions theRK RKmethod methodin in question questionbe be of oforder orderp and andof ofstage stage order orderq q<
0
= jj//''--^^oor rVVCCOO z(0) = ) .) . z(0)

(7.56) (7.56) (7.57)(7.57)

For our our aims aims it it is is important important that thaty(i) y(i) can canbe beexpressed expressed by by y(t) y(t) — —z(t) z(t)+ + For A(t)~ f(t). We Westate state here hereonly onlyone oneresult result illustrating illustratingthe themain main idea. idea. Further Further A(t)~l f(t). canbe bestudied studiedon onthe thebasis basis of of the theassertions assertionspresented presented in Section Section possibilities in possibilities can 7.3. 7.3. Theorem 7.15. Let LetTT< > 0, 0, and andp,p,q q6 6NN such such that thatq q<
We recall recallthat thatA(t) A(t) 11 is is uniformly uniformlybounded, bounded, by our ourassumptions. assumptions. We by A(t) A(t) isisindependent independentof of t.t. particularthese theseassumptions assumptions mean In particular mean that that Dom Dom

22 22

176

CHAPTER 7. CONVERGENCE CONVERGENCE CHAPTER

ESTIMATES ESTIMATES

auxiliary problem problem (7.56) (7.56) and let let YYn = Zn + A'1 (tn) f (tn). Then it holds, the auxiliary sufficiently small, small, for all kk G G(0, (0,K] K] and andttn G Cl^, Cl^, with K > 0 sufficiently q+2

\\Yn-y(tn)\\

2

<

++

P+1

l=q+3

Proof. By Theorem Theorem 7.7, we we have23 \\Yn-v(t \\Yn-v(tn)\\ = \\Zn-z(tn)\\ < ;^)+C^||^+1 )(-)|| il (o,t n ;3e)+ W +2 ||A(.)^ +2 )(-)||L 1 (o,t n ;^)+C^||^

(7-58) (7-58)

with the the aid aid of of the the equalities equalities Now, with y(t) -- A(t)-lf(t) - - A ( i ))--yy (t) z(t) - y(t) and

-A{ty'A'(t)A{t)-1 (Ait)'1)1 = -A{ty'A'(t)A{t)as well as of the the estimate estimate24 l

\\
fori = l , . . . , pp ++ ll,,

that for m = = qq + + 1, 1,qq+ +2, 2, Leibniz' rule rule it follows that by Leibniz' m+l

\\A{t)z^\t)\\ = \\A(t) (A(t)-ly'(t)){m) \\ \\A{t)z^\t)\\ and p+2

(t)\\
Therefore combining combining the last two estimates estimates with with (7.58) (7.58) and observing observing that that Therefore [0,T], f o r t e [0,T], l(o,t;3E),

1=1

m = qq + + 2,p+l 2,p+lt

1=1 1=1

the desired desired result result follows. follows. 23

+ nil nil since A(s) 1 is uniformly uniformly bounded. bounded. One can write A(s) instead of A(s) + This estimate estimate obtains obtainsby the (p+ (p+ l)-tuple strongly stronglycontinuous continuousdifferentiability differentiabilityof A(t)

24

on "Dora.

7.5. COMMENTS AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL REMARKS REMARKS

177 177

Note that applying the RK RK method method directly directly to to problem problem (5.1) (5.1) and and using using Theorem 7.8 does not allow allow us us to to be be confident confident that that \\Y \\Yn - y{tn)\\ ~ O(k g+2) unless we make certain assumptions assumptions on on the the behaviour behaviour of of A{t)y^ A{t)y^q+2\t) (cf. Theorem 7.7).

7.5

Comments and bibliographical bibliographical remarks remarks

Section 7.2: The homogeneous homogeneous case case seems seems to to have have been been thoroughly thoroughly invesinvestigated by now even in the the framework framework of of Banach Banach spaces. spaces. Apparently, Apparently, Le Le Roux [102] is the first who who derived derived error error estimates estimates in in the the case case with with sectorial sectorial operator. It is remarkable remarkable that that her her result, result, although although presented presented for for Hilbert Hilbert spaces, admits an extension extension to to aa Banach Banach space space setting. setting. Such Such an an extension extensionisis stated in Brenner &; Thomee Thomee [52], [52],where where an an analogue analogue of ofthe thepresent present Theorem Theorem 7.1 for general Co semigroups semigroups isis also also contained. contained. Analogues Analogues ofof Theorem Theorem 7.2 7.2 in the case £ = 0, stated for for methods methods of of class class 5/ 5/ as as well, well, are are found found inin [102] [102] and [52]. Regarding the general general case case ££ >> 0, 0, the the results results related related to to Theorem Theorem 7.2 are presented in our work work [37]. [37]. Although Although the the paper paper [37] [37] deals deals with with the the case m — 0 only, its techniques techniques are are trivially trivially extended extended in in order order to to be be appliapplicable for m > 1 too. Apart Apart from from everything everything else, else, we we give give here here aa similar similar result for methods of the classes classes 5// 5// and and 5/// 5/// (see (see Theorem Theorem 7.3). 7.3). Note Note that that for both classes of methods methods the the presence presence of of aa damping damping factor, factor, whose whose role roleisis the initial initial conconplayed by the operator u>(—kA), is essential even for £ = 00 ifif the dition y° is not smooth enough. enough. The The idea idea of of damping, damping, when when deriving deriving error error estimates for non-smooth non-smooth data, data, goes goes back back to to the the work work ofof Luskin Luskin && RanRannacher [109] and Rannacher Rannacher [132] [132] carried carried out out for for Hilbert Hilbert spaces spaces and and some some special methods. Also, some some nonsmooth nonsmooth data data error error estimates estimates inin aa Banach Banach space setting are contained contained in in Hansbo Hansbo [82, [82, 83], 83], for for A-stable A-stable methods methods only only when dealing with the classes the classes Sn Sn and and 5///. 25 It is easily seen from the above considerations that that showing showing nonsmooth nonsmooth data data error error estimates estimates reduces, reduces, through applying Lemma Lemma 7.5, 7.5, to to making making use use of of certain certain strong strong stability stability type type estimates (cf., e.g., the proof proof of of Theorem Theorem 7.2) 7.2) while while the the role role ofof damping damping inin stability analysis has already already been been demonstrated demonstrated in in the the preceding preceding chapter. chapter. In the inhomogeneous case case one one should should expect expect extra extra difficulties difficulties which whichreresult from the fact that the the solution solution of of the the original original Cauchy Cauchy problem problem (5.1) (5.1) has has to satisfy certain compatibility compatibility conditions conditions in in order order to to obtain obtain the the optimal optimal order of converegence otherwise otherwise one one meets meets an an order order reduction. reduction. ItIt turns turns out out that a suitable choice of method method allows allows one one to to reduce reduce the the requirements requirements ofof 25

For methods of these classes, classes, the the requirement requirement of of ^-stability ^-stability implies implies the the condition condition deg[a(-) - a(oo)} = - 1 .

178

CHAPTER7. 7.CONVERGENCE CONVERGENCEESTIMATES ESTIMATES CHAPTER

space regularity,which whichare arevery veryoften oftenthought thought artificial. This possibility space regularity, of of as as artificial. This possibility is studied studied in indetail detailfor forthe thefirst firsttime timebyby Crouzeix in the context of Hilbert Crouzeix [59][59] in the context of Hilbert spaces; unfortunately, unfortunately, his histechniques techniquescannot cannotbebe extended Banach case. spaces; extended to to thethe Banach case. For equations equations inin aaBanach Banachspace, space,a arelated relatedresult result is first given in Brenner, is first given in Brenner, Crouzeix, and and Thomee Thomee[50]. [50].Their Theirachievement achievement been further extended Crouzeix, hashas been further extended in in Ostermann & &Roche Roche[123, [123,124] 124]and andininLubich Lubich Ostermann [105]. It follows follows [105]. Ostermann &; &; Ostermann their work workthat thatsometimes sometimesit itis is natural expect even fractional orders from their natural to to expect even fractional orders convergence. The Theproofs proofsofofTheorems Theorems7.47.4 and follow in essence of convergence. and 7.67.6 follow in essence the the paper [50]. [50]. paper Section 7.3: 7.3: The The question questionofofconvergence convergence Runge-Kutta methods, Section of of Runge-Kutta methods, ap- applied to to linear linear non-autonomous non-autonomousdifferential differentialequations equations a Banach space, plied in in a Banach space, not seem seem to tobe bestudied studiedthoroughly thoroughlyupup now. mention however does not to to now. WeWe mention however a a paper of of Gonzalez Gonzalez &&Ostermann Ostermann[71] [71]which which contains results close paper contains results thatthat are are close the assertion assertion ofofthe theabove aboveTheorem Theorem7.7. 7.7.AtAt same time Theorem to the thethe same time Theorem 7.7 7.7 stated under under somewhat somewhatless lessrestrictive restrictiverequirements requirements than in [71]. Some is stated than in [71]. Some further developments developments are aregiven givenininthe thestatements statements Theorems further of of Theorems 7.8,7.8, 7.10,7.10, 7.11. Theorem Theorem7.12 7.12presents presentsa aconvergence convergence result methods of class and 7.11. result for for methods of class as well. well. The The mentioned mentionedresults, results,including including that paper Si a as that of of thethe paper [71],[71], are are helpful for for applications applicationsconnected connectedwith withmultidimensional multidimensional problems not helpful problems in in spaces with with maximum maximumnorm normwhile whilethe theassertions assertions Theorems spaces of of Theorems 7.9 7.9 andand 7.137.13 intended for for use useininsuch suchsituations. situations. are intended Section 7.4: 7.4: The The question questionofofworking workingagainst againstorder order reduction Section reduction has has beenbeen extensively discussed discussedininthe theliterature literature(see, (see, e.g., Abarbanel, Gottlieb, extensively e.g., Abarbanel, Gottlieb, and and Carpenter [1], [1], Calvo Calvo kk Palencia Palencia[55], [55],Carpenter Carpenter [57], Keeling Carpenter et et al. al. [57], Keeling [92],[92], [130], and Sanz-Serna, Sanz-Serna, Verwer, Verwer,and andHundsdorfer Hundsdorfer[140]). [140]).OurOur conPathria [130], conPathria sideration in in the the present presentsection sectionhas hasnonopretensions pretensions being a complete sideration to to being a complete exposition of of the the problem. problem. exposition

Chapter 8

Variable Stepsize Approximations Our aim here is to extend some of the previous results to the case of nonuniform time partitions. partitions. In this chapter we never speak about discrete discrete semigroups since the transition operator depends depends essentially essentially on the corresponding time moment. distinguish moment. In our analysis below we nevertheless distinguish between two cases: commutative commutative and non-commutative. In the commutative case, although the transition operator varies varies with with time, time, it is always a (varying) function of one fixed operator. This circumstance circumstance allows allows us to use the operator calculus methods. methods. In this sense the situation is similar to that covered by autonomous equations and uniform grids. In the non-commutative case we make use of certain perturbation techniques. techniques. Throughout this chapter chapter any set of nodal points {io,*i> , £JV}J N > 1, such that to = 0 < t\ < . . . < % = T is called a non-uniform grid (non-uniform partition) in the segment [0, T] and is denoted by 15^- The situation when T — oo is also covered by our consideration; but in this case Uk consists of infinitely many nodal points. points. The parameter k is now not a number but a sequence (finite or infinite), and more precisely, k is defined as k = {ko,..., fcjv-i}, where kn = tn+\ — tn is the variable stepsize. Hereinafter, given a grid 15 & with N + 1 nodal points, it is always accepted that k]y — /cyv-i and ijv+i = % As above, we distinguish any objects that may depend on discrete time tn by n as a subscript. Also, we denote &fc = Uk \ {tN} and dZn = k-1(Zn+1-Zn)

for any discrete function Zn : 15^ —* -£ 179

for t nel5k,

(8.1)

180

CHAPTER 8.8. VARIABLE VARIABLESTEPSIZE STEPSIZE APPROXIMATIONS APPROXIMATIONS CHAPTER

let the the above aboveRK RKmethod method Further, let let there therebe beaanon-uniform non-uniformgrid grid 15k and let Further, (5.19), (5.20) (5.20) be be used usedfor fordiscretization discretizationofofproblem problem (5.1) 15k, with with 9t 9t == I.I.1 (5.19), (5.1) on on 15k, Since Runge-Kuttadiscretization discretizationapplies applies locally, that separately at each Since Runge-Kutta locally, that is, is, separately at each moment, we wethen thenhandle handlea adiscrete discreteproblem problem which takes, in fact, time moment, which takes, in fact, the the same form form as as the the above abovediscrete discreteproblem problem(5.19), (5.19),(5.20), (5.20), with minor same with the the minor difference that kk isisreplaced replacedbybyknk and, and, in in particular, particular,tntj are are now now given givenby by difference that is not not hard hard to tosee seethat thatthe thestatement statementof of main theorem tnj = ttn + Cjkn. It is thethe main theorem representabilityofofthe thestarting startingproblem problem (5.19), (5.20) in the form (l.l)2 is (l.l) on representability (5.19), (5.20) in the form place of of k,k, provided providedthat thatdYdY (1.1) isis now now understood understood force with with kkn in place in force n in (1.1) the new new sense, sense, asasspecified specifiedbyby(8.1). (8.1).AtAtthethe same time, if A(t) in the same time, if A(t) = A= isA is independent ofof t,t, similarly similarlytotothe thecase casewith witha constant a constant stepsize, independent stepsize, we we thenthen consider problem problem (1.1) (1.1)with with consider k-1a(-knA), fHn = k-

Qn = I, I, Q

(8.2) (8.2)

and V

T

A))//^^--)), , Fn = w (-fc n ^)* n = J ^ - ffccnnA

(8.3) (8.3)

as a generalized generalized form formofof(5.19), (5.19),(5.20). (5.20).AsAs Chapter 5, we then in in Chapter 5, we then dealdeal withwith substituted for fork,k,which whichdefines definesuniquely uniquely the rational rational method method(5.30), (5.30),with withknk substituted a discrete discrete function function YYn : 15 k — — This function functionisiscalled calledthe theextended extended solution >> X. This solution the autonomous autonomouscase. case.Regardless Regardlessofof way by the the RK RK method methodon on15 k in the thethe way in in which 2l 2ln appears, appears, we we always alwaysassume assumethat that2l^v 2l^v — 21JV-I> 21JV-I> where where iV iV++ 11isisthe the which number number of of nodes nodesofof15 kBelow we Below we derive derivestability stabilityand anderror errorestimates estimates forfor thethe solution solution or iforA(t) if A(t) = = A then then for for the theextended extendedsolution solutionofof the the Runge-Kutta Runge-Kutta procedure procedure (5.19), (5.19), (5.20) (5.20) in the the situation situationwith withaavariable variablestepsize. stepsize.InIn fact, fact, wewe take take thethe same same conditions conditions on the the operator operator A(t) A(t) asasininthe thecase casewith witha constant a constant stepsize. stepsize. At At the the same same time, using using the the assumption assumptionthat thatA(t) A(t)€ €S(ip,a), S(ip,a),wewe assume in addition time, assume in addition thatthat a > 0, 0, for for simplicity. simplicity. Also, Also,we weare arenot notconcerned concerned with of the above with all all of the above SJJJ, dealing dealing with with class class5/5/only. only.InIn introduced classesofofmethods methods5/, 5/,SJI, SJI, SJJJ, introduced classes the case case with with aa time-variable, time-variable,Holder-continuous Holder-continuous operator, some operator, we we taketake some extra restrictions restrictions on onthe theused usedfamilies familiesof of non-uniform grids. extra non-uniform grids. ^ o r the sake of simplicity, simplicity, no correction correction of the the initial initial condition conditiony° is is introduced introduced in this chapter. chapter. 2 of Theorem Theorem 5.1 That is, of

8.1. THE THE COMMUTATIVE CASE 8.1. COMMUTATIVE CASE

181 181

8.1 The commutative 8.1 The commutative case case Here weconsider considerthe thecase casewhere where theoperator operatorA(t) A(t)== doesnot notdepend depend on Here we the AA does on thissituation, situation, forany anytnn,ti ,ti G GUk Uk the corresponding transition operators t. In Inthis for the corresponding transition operators t. andit/ it/commute, commute, andone one can therefore usean anoperator operatorcalculus calculus formula iin and and can therefore use formula iin themain maintool tool ofanalysis. analysis. as of as the ofall allwe we givesome some useful comments. Letthe the RK methodbe beAj(ip)Aj(ip)First First of give useful comments. Let RK method \ y>\ be beany anyfixed fixednumber number ipi G G stable with some

>00 sufficientlysmall small andd\d\> > sufficiently 0 large the thatfunction functionr(z) r(z) sufficiently and 0 sufficiently large such such that the 33 hasno nopoles polesin inV(d) V(d) andoutside outsideof ofInt£>(di), Int£>(di),and andthe the followinginequalities inequalities has and following hold hold (8.4) | rr((zz) )| <| < ee (8.4) forallzeX>(d), forallzeX>(d), e^11**11 e^

forall all G zS S for zG

(8.5) (8.5)

and and max{(1 {(1+ + |z|) |z|)11, ||rr((zz) |)}| <}e< efor forall all z z max

(8.6) (8.6)

with some G(|r(oo)|, (|r(oo)|,1). 1). Note Notethat that atthe the expenseof ofchoosing choosingd\ d\ suffisuffiee G at expense with some can takenas asclose closeto to|r(oo)| |r(oo)|as asdesired. desired. ciently large, ee can bebe taken ciently large, Next weshow showcertain certain auxiliary results. Inthe the corresponding statements Next we auxiliary results. In corresponding statements letthe the operatorA(t) A(t)depend dependupon upon forthe the reasonthat that assumption we let we operator tt for reason thisthis assumption beneeded neededfor forour our subsequent purposes. will subsequent purposes. will be L 8 . 1 .Let LetA(t) A(t) € S(tp,cr)and andlet let the methodbe be Aj(ip)-stable, Aj(ip)-stable, L eemmmm a a8.1. € S(tp,cr) the RKRK method >0.0.Then, Then,for forany any finitesequence sequence ofpositive positive withsome some

Proof. Withoutlossloss ofgenerality generality it can can assumedthat that > > hi. hi. Proof. Without of it bebe assumed hhnn > > Accepting therefore thelast last condition, condition, wefurther furtheruse usethe thefollowing followingidentity identity Accepting therefore the we (8.7) (8.7) where where i --ii

= (r(-hjA{t)) (r(-hjA{t)) r(oo)/) = -- r(oo)/) Clearly, Clearly,the thesame sameis istrue truefor forthe the function functiona(z). a(z).

(8.8) (8.8)

182

CHAPTER CHAPTER 8. 8. VARIABLE VARIABLE STEPSIZE STEPSIZE APPROXIMATIONS APPROXIMATIONS

Denoting next next

1=1

we let F ^ be be the the contour contour coinciding coincidingwith withthe theboundary boundaryofofthe theset setT>(r^ T>(r^ld) U E Vl , where d is the same as in (8.4) and ip\ ip\ comes comes from from the the statement statement now with with R = 0 of Lemma A.I. A.I. Note Note that that this this lemma lemma can can be applied now since a > 0. Clearly, F ^ can be decomposed into into two two parts parts as follows l } T(j) = rU) u r C;)) w h e r e TT(J) = dV{drr ) n ( - E^_ V1) and i f = { z £ C : argz = dzipi, T~ ld < \z\ < oo}. Using then then the the operator operator calculus calculus formula formula (A.10) with A(t) A(t) in place of £ and with F ^ in place of ER+£, we get, for 1 < 3 < n, jj ll

1 Gj(t) = j)

j) { j G[ = G[ = +G+G 2\

where G\

{j

— (2vrz)~ 1 fa)

\

(8.10) (8.10) / = 1,2. Since A(t) € S(tp,o),

using Lemma

A.I yields, with the aid of (8.4), that that

HG^II < C f e^lzl^ldzl

< C.

(8.11) (8.11)

Further, applying applying again again Lemma Lemma A.I A.I and and taking taking into intoaccount account(8.5), (8.5),(8.6), (8.6), and the obvious obvious inequality inequality

r(-z) -r(oo)| -r(oo)| < C(l + l^l)"1 for all zeH Vl,

we obtain d\h~

/

=:

poo poo e-^j-ixx-idx

+c c

(l + + hjx^x^dx hjx^x^dx

h + h-

(8-12) (8-12)

It is not hard to see that for hj < Tj-i, /

hj d\

e'crixx'1 dx
roo roo b

e'^x' e'^x'1

dx = C,

(8.13) (8.13)

8.2. THECOMMUTATIVE COMMUTATIVE 8.2. THE CASECASE

183 183

whilefor forhj hj>>Tj_i, Tj_i, while

h
x'11 dx dx <
(8.14) (8.14)

Jd/2 Jd/2

andusing usingthe theobvious obviousfact factthat that 1% <
D D

8.2.Let Let the conditionsof ofLemma Lemma 8.1 8.1be be satisfiedfor forsome some

> Thenfor forany any non-decreasing finite sequence of positive positive (0,TT/2) and 0.0.Then non-decreasing finite sequence of 0 <
a(-hiA(t))JJJJr(-hjA{t)) r(-hjA{t)) a(-hiA(t))
whereee comes comesfrom from(8.6) (8.6)and and given is by (8.9). (8.9). TjTj is given by where Proof. Sincedeg[p] deg[p] = —1, —1, it it follows followsfrom from Lemma 5.3that that Proof. Since = Lemma 5.3 + hA(t))p(-hA(t))\\ hA(t))p(-hA(t))\\ | | ((// +

h>0, <0,

which further yields, forall allh h>> 0 and which further yields, for 0 and u EuX,E X,

\\a(-hA{t))u\\ \\a(-hA{t))u\\ == l

u\\. u\\.

(8.15) (8.15)

Let now nowFFbebe the contourcoinciding coinciding Using ^ therefore theoperoperLet the contour withwith d E ^ d..44EUsing therefore the ator calculus calculus formula (A.in 10) in whichthe thepath path of ofintegration integrationis isreduced reducedto to ator formula (A. 10) which F since sincethe theintegrand integrandis isbounded boundedat at 0, 0,we we have F have

hiA(t) hiA(t)

(8.16) (8.16) \<j
4

As fromfrom As above, above,ii comes the thestatement statementof ofLemma LemmaA.I. A.I. Also, Also, recall recall that that the theestimate estimate inthis this statement statement holds holds withwith R R= =00since sincea a >>0.0. (A.I) (A.I) in

CHAPTER 8. VARIABLE VARIABLE STEPSIZE STEPSIZE APPROXIMATIONS APPROXIMATIONS

184 where

{zl -

dz.

(8.17) F 2 = dV(d xh~{1) n 00 << |z| Let next r i = {z G C :: argz argz == |z| << d^ d^1}, (— ETT-^J) and let F = T\ UF2 be be oriented oriented clockwise. clockwise. Since Since the the norm norm of of the the 2 integrand on the right-hand right-hand side side of of (8.17) (8.17) isis O(|z|~ O(|z|~ ) as \z\ —* oo, after a tolerable deformation of of the the path path of of integration integration we we use use formula formula (8.17) (8.17) with with T in place of F. Therefore, Therefore, taking taking into into account account that that Jf(zl Jf(zl — —A(t))"" A(t))""1 dz = 0 and applying Cauchy's Theorem Theorem we we obtain, obtain, instead instead of of (8.17), (8.17),

G (0 = A

2m

/" z(i + htz)-1 TT r(-hjz)(zl Jr t,,

dz.

(8.18)

Denote next by G 1 and G 2 the right-hand side of of this this equality equality with with Fi Fi and F2, respectively, in in place place of of F. F. With With the the aid aid of of Lemma Lemma A.I, A.I, (8.5), (8.5), and and (8.6) we get

< Chi

'n~jX dx

j=0

j=0 j=0

At the same time by Lemma Lemma A.I A.I and and (8.6) (8.6) itit follows follows that that

G2l)\\ < Chi

\r(-hjz)\\dz

n Chiee n ff \dz\ < 77r

(8.20)

It thus remains to combine combine (8.15), (8.15), (8.16), (8.16), (8.18), (8.18), (8.19), (8.19), and and (8.20). (8.20). We further extend both both Lemma Lemma 8.1 8.1 and and Lemma Lemma 8.2 8.2 in in the the following following way. way.

8.1. THE COMMUTATIVE COMMUTATIVE CASE CASE

185 185

L e m m a 8.3. Let Let the the conditions conditions of Lemma 8.1 be satisfied for some tp G (0,TT/2) > 0. 0. Let Let also seminorms seminorms \ \t and and || |* |*;t form respectively respectively (0,TT/2) and a > (£|1|, (*;£*|l|, a)-concordant a)-concordant pairs pairs with with A(t), for for some £,£* G [0,1], and let w(z) w(z) be beaa rational function function which which is (ip)-regular (ip)-regular and satisfies max{£,£*}. Then Then for any any non-decreasing non-decreasing finite finite the condition condition deg[w] deg[w] < — max{£,£*}. numbers 0 < hi < < holds, for all all tt G G [0,T] [0,T] < hhn, it holds, sequence of positive numbers and I = 1 , . . . , n , n-l

sup \oj(-hiA(t)) \oj(-hiA(t)) JJJJ r(-hjA(t))u\ r(-hjA(t))u\t < C ^ ee j T ^ + Ch^\r(oo)\ Ch^\r(oo)\n and

sup |a;(-M(t)r |a;(-M(t)r I I ri-

n-l

from (8.6) and and TJ is given by (8.9). where e comes from The proof proof is similar to that of Lemma 8.1 and of of Lemma 8.2 in the the cases and ££ G G(0,1], respectively. respectively. £ = 0 and introduce some some notions notions which which are needed to state the below Next we introduce families of grids. restrictions on the used families restrictions Definition 8.1. With any finite sequence sequence K,n — (kj)^=1, we associate associate the Definition reordered sequence 7in = (/ij)^ =1 given by hj = kkL^, where the bisection bisection t(-) reordered sequence is chosen so that h\ <


and 1) whenever whenever hj — hj+\. L(J) < i(j + 1) finite sequence sequence Tin will then then be called the ordered sequence sequence for /Cn. The finite Definition Let 77 be beaafamily of grids in [0,T]. The family 7 is called Definition 8.2. Let if there exists exists a constant C > 0 such exponentially balanced balanced with with basis basisQ > 1 if exponentially points5 and for for any any indices I, n that for any grid 15k G T with N + 1 nodal points subject to 0 < I < < nn < < N, N,

3=1 5

recall tthhaatt k = W e recall

{k0,...,

186

CHAPTER CHAPTER 8. VARIABLE STEPSIZE STEPSIZE

where the finite sequence W n _; + i = (hj)^~^ +l P*

/;

APPROXIMATIONS APPROXIMATIONS

is the ordered sequence for

\n—Z+l

K-n-l+1 = Kkj+l-l)j=l

with range range Definition 8.3. A family of grids 7 is called locally balanced with or l Q > 1 if Q" kn+i f all Uk 6 T. Let T be a locally balanced family family of grids with range Q > 1. Given a grid Ofc € T with^JV + 1 nodal points and given indices /, n such that 0 < I < n < N, let /Cn_,+i = (fc j+ i_ 1 )^-{ +1 and let ftn_m = (hj)]!1^1 be the ordered sequence for /C n _j + i. Then, for any m — 1 , . . . , n — I + 1, j < (n - f + 1 - m)K-i+u

(8-21) (8-21)

range Q, while since T is locally balanced with range ^ 3=1

1

^

> Q- {n-l+l-m)K_l+l.

(8.22)

3=1 3=1

Combining (8.21) and (8.22) therefore yields, yields, with with any fixed Q\ > Q, for m = 1 , . . . , n — I + 1,

3=1

33== 1

33== 1

which means that T is, in fact, exponentially balanced balanced with with basis basis Q\ > Q, where Q\ can be chosen as close to Q as desired. As a simple example, we note that the family of grids defined by kn — Qnko, n = 0 , 1 , . . . , Q — const > 1, where &o > 0 is a free parameter, is obviously locally balanced balanced with with range range Q. Yan [166] defines the so-called quasi-quasiuniform quasi-quasiuniform assumption assumption on the variable stepsize which is as follows. T is called a family of quasi-quasiuniform grids if, whenever tn,tn+\ G U^, for all 15k € T, kn < kn+1

(8.23) (8.23)

and ckn+i
Ctn+1/(n

+ 1).

(8.24)

However it turns out that any family of increasing quasi-quasiuniform quasi-quasiuniform grids grids is, in fact, exponentially balanced balanced with with basis basis Q > 1, where Q can be chosen as close to 1 as desired.

8.1. THE COMMUTATIVE CASE CASE

187 187

To see this fact, we note first first that, that, as as itit follows follows from from (8.23), (8.23),for for all allUk Uk££ 77 and ti,tn £ Uk with 1 < I < n, ., i-i

nn

j=o

j=i j=i

which further implies t n + i/(n + 1) < (<„+! - ti)/(n ti)/(n -l-l + + l). l). Obviously, the last inequality inequality holds holds for for II == 00 as as well. well. Therefore, Therefore, instead insteadofof n (8.24) we actually have, whenever whenever t/,£ t/,£n+i 6 Uk & d U < *n> for all Uk £ T, cfcn+i
{tn+i ~ ti)/(n ti)/(n -- // ++ 1). 1).

With this inequality in mind, mind, we we obtain, obtain, whenever whenever t;t; << ttn and 1 < m < n-l + 1, n-l

n

k

n-l

kj <J2 J 3=1

+ C

'kn-1 < (1 + C/( n-l))

E fc j

3=1

n-l

<


n + C/g) ^ fc : < ( — l + l m i —I 7='

i+m-1

and, clearly, 5^?=/ % admits admits the the same same bound bound for for // == nn as as well. well. Also, Also, letting letting Q be any fixed number >> 1, 1, itit holds, holds, with with any any fixed fixed dd>>0,0, < CQn-l+1~m.

(8.26) (8.26)

m

Finally, combining (8.25), (8.25), (8.26), (8.26), and and the the fact fact that that any any Uk Uk€€ TTisisan anincreasincreasing grid shows the claim. Definition 8.4. T is called calledaa family family of of quasi-uniform quasi-uniform grids grids ifif there thereexists existsaa constant C such that max kn < C min k n 6 u

for all Uk £ 7.

If T is a family of quasi-uniform quasi-uniform grids, grids, then, then, for for any any 13k £ 7, ti,t n £ with / < n, and l<m
— —ll + + l,l, letting, letting, as as above, above, TC TCn-i+i

— (hj)^l[

188 188

CHAPTER VARIABLE STEPSIZE APPROXIMATIONS CHAPTER 8. 8. VARIABLE STEPSIZE APPROXIMATIONS

be theordered orderedsequence sequence for K Kn-i+\ be the for

+1 +1

(kj+ == (kj

,

m m

x x

3=1 3=1

m m Using this thisestimate estimate andthe theparticular particularcase caseof of (8.26), (8.26), where wherethe the exponent exponent Using and on the theleft left equals equals1, 1, we weobtain obtain that thatany anyfamily family of of quasi-uniform quasi-uniformgrids grids is is d on exponentiallybalanced balanced with Q >> 1,1,where where Q Q can canbe betaken taken as as close closeto to exponentially with basisbasis Q as required. required. 1 as The above abovediscussion discussion andour oursubsequent subsequentconsideration consideration in this this chapter chapter The and in shows that thatit it suffices suffices to to deal deal only onlywith withexponentially exponentially balanced families of shows balanced families of grids. grids. Now we westate state aa stability stabilityresult resultfor for problems problemswith withaa constant constant operator. operator. Now Letthe theoperator operatorA(t) A(t) == AAbebe independent of of t,t, let letAA€ € independent T hheeoorreemm 88..11. .Let S(ip, a), a), and andlet letthe the RK method be beof oftype type Ci(ip), Ci(ip),for for some some

0. 0. Let Letalso also T T be bean anarbitrary arbitraryfamily familyof ofgrids grids and andlet letTT==oo. oo.Then, Then, for for a > all grids gridsUk Uk€€ 7,7,for forYnY the the extended extendedsolution solutionby bythe theRK RKmethod method on onUk, Uk,itit all holds holds \\Yn\\ <
*n€€ UUk. f°f°r *n

(8.27) (8.27)

Furthermore, if if in in addition addition 77 is is an anexponentially exponentiallybalanced balanced family of grids grids Furthermore, family of Q such such that that with basis basisQ with Q<|r(oo)|-\ (8-28) Q<|r(oo)|-\ (8-28) then holds, holds,for for any anyfixed fixed£ £ 66 [0,1], [0,1], for for all allUk Uk€ €7 7and and G t Uk, Uk, it then forfor all tall n G n-ln-l

/ n ||

i


\

+Y Ytkl(tn+ i-ti+1 )-tmax + n+i-ti +1)-tmax i=o i=o

\\f(tij)\\).. \\f(tij)\\)

\

(8.29) (8.29)

Proof.First First of of all allby byLemma Lemma5.3 5.3we weobtain obtain since sincedeg[u>j] deg[u>j] < 0, 0, Proof. < C, jj ==l,...,u, l,...,u, < C,

(8.30) (8.30)

uniformly withrespect respect to to kk >>0.0.Therefore, Therefore,if if TT isisan anarbitrary arbitraryfamily family of of uniformly with grids, grids, (8.27) (8.27) follows followsby by applying applyingLemma Lemma 8.1 8.1and and(8.30) (8.30) for for an anestimation estimationon on

8.1. THE COMMUTATIVE

CASE CASE

189 189

the basis of the representation, for tn G Uk, n-l

Yn = r(-fc n_iA)

r(-koA)y° + J^ kr(-k n^A)

r(-kl+1 A)Ft. r(-k l+1A)Ft.

(8.31)

1=0

Let now 7 be an exponentially balanced balanced family family of grids with basis Q satisfying (8.28). We then have, for all Uk G 7 and tn,ti G Uk such that | r ( o o ) r / + 1 < Q - > - ; + 1 ) < Ckn(tn+1

- U)- 1

(8.32)

and

(

\ ~l

n-l+l-j

Y, m=l

h

m)


j = 0,...,n-l,

(8.33)

/

where (^m)^] 1 " 1 ^s * n e ordered sequence for (km+i-i)^L.f1. Select further 1 6 some s G (^(oo)!,^" ). Applying then Lemma Lemma 8.2 and taking (8.32) and (8.33) into account yields yields that that for ti < t n, n-l ) 3=1

ti)-1.

(8.34) (8.34)

(A.18), we Therefore, by Lemma 8.1, (8.34), and the convexity inequality (A.18), get, for any fixed £ G [0,1], for fy
3=1

Using now this inequality for an estimation on the basis of (8.31), with both sides multiplied from the left by 2l|, leads to (8.29). and let let the the operator A(t) = A and the RK RK Remark 8.1. Let T = oo and method satisfy the conditions of Lemma 8.1 for some

0. Let also a seminorm }{-][t form a (£,\l\ip,a)-concordant pair pair with with A for some £ G [0,1] and let u(z) be beaa rational function which which is ((p)-regular and satisfies the condition deg[w] < —£. Then the estimate (8.27) holds with with ][Yn][tn substituted for \\Yn\\ if £ — 0 and if 7 is an arbitrary family of grids, 6

We recall that e in the statement of Lemma 8.2 can be chosen as close to |r(oo)| as desired.

190

CHAPTER 8. VARIABLE VARIABLE STEPSIZE STEPSIZE APPROXIMATIONS APPROXIMATIONS CHAPTER

and the estimate estimate (8.29) (8.29) holds holdswith with}[u(—k }[u(—k substituted for for nA)Yn][tn substituted (0,1] and and ifif 77 isis an an exponentially exponentiallybalanced balanced family of of grids grids with withbasis basis if £ £ (0,1] (8.28). satisfying (8.28). Q satisfying result follows follows from from Lemma Lemma8.3. 8.3. This result

non-commutative case case 8.2 The The non-commutative In this section section we we proceed proceed from fromthe thesupposition suppositionthat thatthe theoperator operatorA(t) A(t)may may on t.t. Throughout Throughout the thesection sectionititisisassumed assumedthat thatT T< >0,0,for forallall13 k £ 7 and satisfying kn
(8.35) (8.35)

and Ckn+i. kn < Ckn+i.

(8.36) (8.36)

Let further further zznj and w wnj be nonnegative nonnegativesolutions solutionsofofthe therespective respectiveinequaliinequali7 ties, for all all 13k £ 7 and and ti,t ti,tn £ 13k with t\ < < ttn, n-\ Cl

M++ii((VVii - f 0~ C2 zn,i < C(tn+1 - i,)~ + C j ] f cc,,ZZM

(8-37) (8-37)

q=l

and n-l q=l

with some some Ci,C2 Ci,C2 ££ [0)l)[0)l)- Then, Then,ififKK isissufficiently sufficientlysmall, small,wewehave, have, forfor allall 13k £ 7 and and ti,t ti,tn £ 13k with ti
(8-38) (8-38)

^r C l -

(8.39) (8.39)

and In the particular particular case case(j(j == 0,0, condition condition(8.36) (8.36)isisnot not needed. needed. 7

Actually, both zn,i and w wn,i depend on 15kActually, both

8.2. THE NON-COMMUTATIVE NON-COMMUTATIVE

CASE CASE

191 191

Proof. We show only the former former stated stated estimate estimate (8.38), (8.38), noting noting that that (8.39) (8.39) is proved by using the same same reasoning. reasoning. from (8.37) (8.37) that that for for all all Uk Uk GG TT Denoting z n^ = {tn+\ —tfi^Znj, it follows from and ti,tn G Uk with t\
^n,l

—^

' ^ v n~Hl

I) I)

//

^qx^n-hl

^ q)

*"n, q+\\^q-\-l

^1}

yo.^uj

q=i

since \tn+l ~ tq+l) X ^ C\tn-\-\ ~ t q) 1 f° r ^g+1 ^ ^ni by (8.36). Let further T\ T\ G G (0, (0, T/2] T/2] be be aa fixed fixed number number which which will willbe be specified specified later and let K > 0 be subject subject to to K K < < T\. T\. Given Given ttn G 13k, we put, for j > 1, 1, = maxi"^, maxi"^, Tj = [max{0,t n + i - j T i } , t n + i - (j - l)7i) and Zj = IJtTj

and denote by t', -, and £'/-, £'/-, the the largest largest and and smallest smallest nodes, nodes, respectively, respectively, ofof ^5fcU{£n+i} such that TJ C [£ , unless Tj is void. By (8.40) we therefore therefore have, for m > 1,

\

x rnax

( i n + 1 - t;) Cl

.

(8.41)

Taking (8.35) into account, account, recalling recalling that that KK << T\, T\, and and noting noting that that titi calculation now now gives, gives, with with AAXo t'o < t'f-s < t n+\ if ti ^ Tj, a simple calculation U< < ttn, [x0, X0 + 3Ti], for j > 1 and U < I

(*n+l - ^)" ^ ) " C l ((^^ -

'9+lS^

max

/

(t n+i

-

x06[0,r-2T1] 7[t,,t B+i]nAX0

Combining this estimate estimate with with (8.41), (8.41), we we obtain, obtain, for for mm >> 1,1, Zm
l

+ T-,

192 192

CHAPTER 8. VARIABLE VARIABLESTEPSIZE STEPSIZE APPROXIMATIONS CHAPTER 8. APPROXIMATIONS

noconstant constantC C depends dependson onT\. T\.Taking Takingnow nowT\T\sufficiently sufficientlysmall, small, it whereno where it follows that follows that 1 1 —I— ~ f ,, —I— ~

III III

XX

£j £j

which further implies Zm < CCsince sincem m <<\T/T\\ \T/T\\ + + 1.1. Clearly, Clearly,this this which further implies that that Z m < bythe thedefinition definitionof ofZj Zjand and ~znnj.. j.. inequalityshows shows (8.38), by ~z inequality (8.38), In conclusion, note applying thesame sameargument, argument, it is isnot not necessary In conclusion, note that,that, applying the it necessary use(8.36) (8.36)if if Ci Ci==0.0. to use to Now weshow showstability stability inthe thecase casewith withaa variable variableoperator. operator. It isisconveconveNow we in It nientto todenote, denote,given given gridUk, Uk,for forany any tG[0,T] [0,T]and and ti,tnn €€Uk, Uk, nient aa grid tG ti,t t), t),

i(t)=U (t)---Ui(t), iXiX n>i(t)=U n(t)---Ui(t), n> n

dealingwith with theRK RK method(5.19), (5.19), (5.20) applied to (5.1) (5.1) Uk, dealing the method (5.20) applied to onon Uk, andand

8 8 dealing with problem (1.1)considered consideredon onUkUkUsingthe theabove abovenotation, notation, dealing with problem (1.1) Using to (1.4) (1.4) the solutionY Ynn of of (1.1) (1.1) the non-uniform case isreprerepresimilarly similarly to the solution inin the non-uniform case is sented as sented as

n --ll n

o Ynn = =U Unn.. ..lfi yo +YY/kUn-i,i+iFi. kUn-i,i+iFi. Y + lfiy

(8.42) (8.42)

1=0 1=0

Also, forsubsequent subsequent usewe we pointout outthe the identity,for forttnn,ti ,ti 6615k 15k such suchthat that Also, for use point identity,

U< U
iln_l,l = Un-l,l(*l) Un-l,l(*l) + Yl Yl kkqq&n-l, &n-l,q+ l{%{tl) -- 2t,)U,_i,j(*i). 2t,)U,_i,j(*i). iln_l,l = + q+l{%{tl)

(8-43) (8-43)

q=l q=l

Henceforth, Henceforth,given given aa non-uniform non-uniform gridgrid Uk, Uk,by bythe the discrete discretesolution, solution, for for brevity,we brevity, wemean meanthe thesolution solutionby bythe theRK RK method method(5.19), (5.19), (5.20) (5.20) applied applied to to problem (5.1)on on Uk- Note Notethat that allthe thebelow belowstatements statements contain assumpproblem (5.1) Ukall contain assumptionsthat thatensure ensure the therepresentability representability of ofthe the starting startingRK RKproblem problemin inthe theform form tions in which which2l 2lnn, Q3 Q3nn,and andFF arespecified specifiedby by(5.38), (5.38),(5.39), (5.39), and(5.41), (5.41), (1.1) (1.1) in and n n are with kknn substituted substituted for fork.k.InIn what whatfollows follows we wealways alwaysassume assume thatthat 2l 2lnn, Q3 Q3nn, with and F and Fnn are aredefined definedas asspecified specifiedand andwe we use use thethe representation representation (8.42) (8.42) for forthe the discrete solution discrete solution Y Ynn without withoutexplicit explicit mention. mention. Our Our first firstresult result on onstability stabilityis isas asfollows. follows. 8 8

isassumed assumedthat t h aiitnn,i{t) ,i{t) ii — —lln,i lln,i = = // ififn n< 2. < 2. It It is

8.2. THE NON-COMMUTATIVE NON-COMMUTATIVE CASE CASE

193 193

Theorem 8.2. Suppose Suppose A(t) A(t) G GS((p, S((p, a), a), aa seminorm seminorm \\ \\t forms a (71 \ip, a)concordant pair with A(t), A(t), and and A(t) A(t) saisfies saisfies hypothesis hypothesis hol®[i?; hol®[i?;\-\t], \-\t],for for some some

0, 71 G [0,1), and and 1? 1? G G (0,1]. Let further the RK method method exponentially balanced balancedfamily family of of grids grids with with be of type Vj ( 1 satisfying (8.28). (8.28). Then, Then, with with KK >> 00 sufficiently sufficiently small small and and with with any fixed £ G [0,1), for all all grids grids Uk Uk G G77 such such that that max.t max.tnevk kn < K and for all tn G Uk, the discrete solution solution YYn satisfies the estimate 9 n-l 0

||2ln(tn)^n|| < Ct^y \\

+ C *£ k(t n+1 - tl+1)~t max ||/(ty)||. (8-44) -- 3 -

1=0

v

Proof. Similarly to (6.61) and (6.59) (6.59) we we have, have, with with some some fig fig >> 0,0, for for all all Uk G T, t n G Uk, and u G X,

_

(8.45) (8.45)

and | | F n | | < C m a x ||/(t nj -)||.

(8.46) (8.46)

Next, by the convexity inequality inequality (A.18) (A.18) and and Lemma Lemma 8.1 8.1itit holds, holds,for fort\t\ << ttn,

= . (8.47)

Since T is an exponentially exponentially balanced balanced family family of of grids gridswith with basis basisQQ<< by Lemma 8.2 as in the proof proof of of Theorem Theorem 8.1, 8.1, itit follows follows that that < C(tn+l - i j ) " 1 .

(8.48)

Using also Theorem 6.9 and and Lemmas Lemmas 8.2 8.2 and and 8.3, 8.3, itit isis not not hard hard to to see see that that

(8.49)

Therefore, combining (8.47), (8.47), (8.48), (8.48), and and (8.49) (8.49) yields, yields, for for ttt < tn,

9

The fractional powers 2l 2l n (t n ) ? , ^ > 0, are defined and bounded bounded on on X X since since aa >> 0. 0.

194

CHAPTER 8. VARIABLE STEPSIZE APPROXIMATIONS APPROXIMATIONS

\t Further, by Theorem 6.1 it follows that the seminorm X^=i \vj(~kA(t)) forms a Aj (7i)-concordant pair with with k~ k~1a(—kA{t)). Hence applying applying Lemma Lemma 1.9 shows that with some fj-o > 0 and 72 € (71,1), for all u £ X, V

u||.

(8.50)

Now, with the aid of (8.45), (8.49), (8.50), and the above argument, we obtain, with 73 = max{l — i9,72}, for ti < t n,

Using this inequality together together with with (8.48) (8.48) for an estimation on the basis of (8.43), with both sides multiplied multiplied from from the left by 2l n (i n )^, the reader will have no difficulty in seeing that with znj = ||2ln(£re)^.Un_ij/||, for ti < t n, n-l

Zn,l < C{tn+l - tt)-t + CY/kqZn,q+l(tq+1

-

U)^.

q=l

Noting that 73 < 1, observing that (8.36) holds, holds, by the fact that the family T is exponentially balanced balanced with with basis basis Q, and applying then Lemma 8.4, we find, for ti
Theorem 8.3. Let the conditions of Theorem 8.2 be satisfied with the difference that the operator A(t) fulfils hypothesis hol[i9] hol[i9] instead instead of h6\®[d; \-\t\. Then the stability estimate (8.44) (8.44) holds holds for any fixed £ G [0,1]. Proof. Clearly, it suffices to show the result for £ = 1 only. To that end, we put, for t\ < t n, Let p be the smallest integer such that that I < p < n + 1 and t n + 1 - tp < - ( t n + i - **)

8.2. THE NON-COMMUTATIVE

CASE CASE

195 195

Observe that p > I + 1. If p = n + 1, 1, we we have tn+i — U < 2k n, which gives, applying Theorem 8.2 with £ = 0, Lemma 1.1, and Theorem 6.2, Wn,l < 2||fc n 2l n (t n )|| \\iin-l,l\\ < C Assume further that p < n. We then use the identity, for t\
(tn+i

n-1

+ (*„+! - t j ) J2 *?2ln(*n)^-l,«+l(*n)(a,(t *?2ln(*n)^-l,«+l(*n)(a,(t n )-2l,)ll,-l,J.

(8.52)

q=p-l

Under the accepted Holder-continuity Holder-continuity type type hypothesis, hypothesis, instead instead of (8.45) it holds, for all u G X, ||®n«|| < Cki{pin{tn)u\\

+ \\u\\).

(8.53)

With this estimate in mind and applying Theorems 6.9 and 8.2 (the latter one with £ = 0), we find, for ti
k% + {tn - t qf) < c(tn+1 - tqy ((t g+1 - t{)-lwqyl + I) . Using this estimate, Theorem Theorem 8.2 with £ = 0, and the argument employed in its proof for an estimation on the basis of (8.52), we now get, for t\ < t n,

- v-i n-1 v \ "* h (+ -^ / r°q\''n+l q=p-l

+ ,, + ,\~^(+ ,\~^(+ ''q+lj y^n+l ''q+lj

+ + \® \®((+ ((+ ,, ^q) \\ l>q+l

i,\~^in i 4- l\ w H) q,l '

(R ^4^ \p.Ot)

The fact that T is an exponentially balanced balanced family family of grids with basis Q implies (8.36). Therefore Therefore it holds (tn+i ~ tq+l)~l

< C{tn+l - tq)' 1

for t q < tn-

(8.55)

Also, by the definition of p we have (tn+\ — tp-i)~

< 2(i n + i — ti)~ and (t p — ti)~ < 2(tn+i — ti)~ . (8.56)

196

CHAPTER CHAPTER 8. 8. VARIABLE VARIABLE STEPSIZE STEPSIZE

APPROXIMATIONS APPROXIMATIONS

By (8.55) and (8.56) it then then follows follows from from (8.54) (8.54) that that for for ttt < tn, n-l

Wn,l
and further, since p > II + + 1, 1, n-l

wnii
k q(tn+i -

t q)~(l~^wqtl.

q=l

Finally, using Lemma 8.4 estimate 8.4 and and recalling recalling that that 11 — i9 < 1, the last estimate yields that w nj < C which shows the the claim, claim, with with the the aid aid of of (8.42) (8.42) and and ' (8.46). Remark 8.2. Let the conditions conditions of of Theorem Theorem 8.1 8.1 or or of of Theorem Theorem 8.2 8.2 be besatissatisform aa fied for some (p G (0,TT/2), a > 0, and£ G [0,1], 10 let a seminorm][-}[t form (£\l\ip, a)-concordant pair pair with with the the operator operator A(t), A(t), and and let let UJ(Z) be a rational function which is ((p)-regular ((p)-regular and and satisfies satisfies the the condition condition deg[w] deg[w] << —£. —£. Then Then the stability estimate (8-44) (8-44) holds holds with with ][w(—k ][w(—knA(tn)) Yn][tn substituted for This assertion can be established established with with the the aid aid of of the the above above argument, argument, G Dom Dom A(t), A(t), using, if £ G (0,1], the fact that for all u G

lui^CMAW + VufM1-*. Note that for £ = 1 the the last last inequality inequality isis aa trivial trivial consequence consequence ofof the the can (£|1|<£,cr)-concordance of of the the pair pair (][-][t; (][-][t; A(t)), while for 0 < £ < 11 itit can be shown in a manner similar similar to to that that used used for for proving proving the the convexity convexity ininequality (cf., e.g., Krein Krein [96, [96, The The proof proof of of Theorem Theorem 5.2]). 5.2]). It turns out that if A(t) A(t) is is under under hypothesis hypothesis lip, lip, in in the the case case ££ == 00 itit isis not necessary to impose impose any any restrictions restrictions on on the the family family of of grids. grids. Theorem 8.4. Let the the operator operator A(t) A(t) and and the the RK RK method method satisfy satisfy the the conconditions of Theorem 8.3 8.3 with with the the difference difference that that A(t) A(t) now now fulfils fulfils hypothesis hypothesis lip 11 and T is an arbitrary family family of of grids. grids. Then Then the the stability stability estimate estimate (8.44) (8.44) holds with £ — 0.

The proof of the theorem theorem will will be be based based on on the the following. following. 10 n

The case £ = 1 is included included under under the the conditions conditions of of Theorem Theorem 8.2 8.2 only. only. That T h a t is, one formally takes takes i? i? == 1. 1.

8.2. THE THE NON-COMMUTATIVE NON-COMMUTATIVE CASE CASE

197 197

Lemma 8.5. Under Underthe theconditions conditionsofofTheorem Theorem 8.4, Lemma 8.5. 8.4, forfor all all Uk Uk G T,GtT, n,ti G [0,T], and t,s t,s GG[0,T], Uk, and

Proof. Given tn,ti G Ufc Ufc such such that that t\t\ <
where Gj(t) isisgiven givenby by(8.8). (8.8).Let Letfurther furtherthethe contour ^ just be just same where Gj(t) contour F ^F be the the same as in the the proof proof ofofLemma Lemma8.1. 8.1.By Bythe theoperator operator calculus formula calculus formula (A.(A. 10),10), afterafter a deformation deformation ofofthe thepath pathofofintegration, integration,wewe have, 1, have, forfor j =j 1=, .1. .,,. n. . ,—nI + 1,

Jw

JJ ww

2TT* 2TT*

i-i

z)((zl - Ait))' Ait))'1 - (zl (zl -- A(s))~ A(s))~l) dz. (8.58) (8.58) x J J r(-hqz)((zl 9=1

By the the accepted acceptedLipschitz-continuity Lipschitz-continuitytype type hypothesis resolvent idenhypothesis andand thethe resolvent identity (6.45) (6.45) with with kk== 1,1,we wefind, find,forforz G zG

||(zJ -- A(i))A(i))-1 !! ||(A(t) ||(A(t) -- A( A(5)) (zl (zl -- A{s)) A{s))-l\ < ||(zJ

With With this this estimate estimateininmind, mind,the thenorm normof of thethe integral integral on on thethe right-hand right-hand sideside of (8.58) (8.58) can can be bebounded boundedbybythe theamount amount

r

C\t-s\ C\t-s\

/ \r(-h \r(-hiz)-r(oo)\T\\r(-h z)-r(oo)\T\\r(-hqz)\\z\z)\\z\-1\dz\ Jrti) *J{ *J{

r r

++C\t-s\ C\t-s\

/ \z\\z\-2\dz\. Jrti) Jrti)

Clearly, the is bounded by by C and the the same Clearly, the second secondintegral integralininthis thisexpression expression is bounded C and same thethe reasoning employed in the can be be shown shown for for the thefirst firstintegral, integral,following following reasoning employed in the proof of Gj(s)|| << C\t C\t — s\. Using Using this this fact fact proof of Lemma Lemma 8.1. 8.1.Therefore Therefore||Gj(t) ||Gj(t)— Gj(s)|| for an an estimation estimation on onthe thebasis basisofof(8.57) (8.57)shows shows thethe claim. claim.

198

CHAPTER CHAPTER 8.8. VARIABLE VARIABLESTEPSIZE STEPSIZE APPROXIMATIONS APPROXIMATIONS

Proof of for for ti ti
iinitn) iinitn)

-Hf(*z) (t,+l) - iin, -Hf(*z)==Un,l(ti) Un,l(ti)++^(iln.g+1 ^(iln.g+1 (t,+l) - iin,q+l(tq))ilq(tq) q=l

and applying a simple calculation shows that that applying Lemmas Lemmas8.18.1and and8.5, 8.5, a simple calculation shows n- l q=l

Therefore, Therefore, by byLemma Lemma8.4, 8.4, \\iin(tn)---iXi(ti)\\
(8.59) (8.59)

Furthermore, by i9 =i9 1= and Lemma 1.1 it1.1follows that that Furthermore, byvirtue virtueofof(8.53) (8.53)with with 1 and Lemma it follows II®n|| << C. (8.59) for for an estimation on the II®n|| C. Using Usingthis thisfact facttogether togetherwith with (8.59) an estimation on the basis of basis of the the identity, identity,forfort\ t\
more leads us to desired result, with with and applying applyingnext nextLemma Lemma8.48.4once once more leads us the to the desired result, the aid aid ofof the the representation representation(8.42). (8.42). forfor practical applications. The following following result resultisisalso alsoofofinterest interest practical applications. Theorem A(t) thethe RKRK method be under the conTheorem 8.5. 8.5. Let Letthe theoperator operator A(t)andand method be under the conditions ditions of of Theorem Theorem6.13 6.13forforsome some

0, 0, 7®, 7®,7,7i,7*,i9«,i?*« 7,7i,7*,i9«,i?*«£ £ < 1, + i9*. < 1?.< 1?. [0,1) [0,1) and and i9i9 e , i?, i? i?x e (0,1] (0,1] such suchthat that77++7*7*= =1, 1,7i7i+ +7*7* < i9. 1, i9. + i9*. Let further further 77 bebeananexponentially exponentially balanced balanced family family ofof grids gridswith withbasis basisQQ> >1 1 satisfying condition thatthat a seminorm }[-][t forms forms aa satisfying condition (8.28). (8.28).Finally, Finally,suppose suppose a seminorm }[-][t (£|1|(/?, £ ££ [0,1] andand thatthat u>(z) (£|1|(/?, a)-concordant a)-concordant pair pairwith withA(t) A(t)forforsome some £ [0,1] u>(z) is aa rarational function andand satisfies the the condition deg[w] < —£. tional function which whichisis(ip)-regular (ip)-regular satisfies condition deg[w] < —£. Then, if ££ << 11 — 7* ++ min{i?,i?i}, Then, min{i?,i?i},with withsome someK K> >0 sufficiently 0 sufficiently small, small, for for all grids grids 15^ 15^GGTTsuch suchthat thatmax max( £ g fc fcn < K, K, f/ie f/ie discrete discretesolution solutionY Yn satsatisfies the both as as it isit and withwith \u(—k isfies the stability stabilityestimate estimate(8-44) (8-44) both is and \u(—knA(tn)) Y n][tn substituted substituted for for ||2l£ ||2l£YYn \\. Furthermore, Furthermore, ififininaddition additionA(t) A(t)satisfies satisfies hypothesis hypothesis if 7if is7 an lip(.A(i; ,, ;;| | | |m;t,m = 0, lip(.A(i; 0, *0), *0), then then (8.44) (8.44)holds holdswith with£ =£ =0 even 0 even is an arbitrary family arbitrary family ofofgrids. grids.

8.3. ERROR ESTIMATES

199 199

This result can be shown in a manner similar to that used in the proofs of Theorems 8.2, 8.3, and 8.4, with the aid of Theorem 6.7 and Lemma 8.3, using as well some reasonings employed employed in the proof of Theorem 6.13. ; | |m;t, m — 0, *0), the Also, note that if A{t) is under hypothesis \ip(A(t; assertion of Lemma 8.5 is still valid, which allows one to use the argument applied in the proof of Theorem 8.4. The reader will have no difficulty in stating an analogue to Theorem 6.14 in the case of non-uniform grids.

8.3

Error estimates

In this section we present two types of error estimates, namely, a priori and a posteriori estimates. Our reasons are that both types of estimates are of great importance for practical applications in the case of nonuniform time partitions. In particular, a posteriori error analysis is widely used as a basis for applying adaptive stepsize stepsize control. control. Starting by showing certain a priori error estimates, we remark that the analysis of convergence in Chapter 7, carried out in the case of uniform time partitions, is based on a step-by-step estimation of the local error and on subsequent summing up the contribution to the total error coming from all steps of the numerical process. Clearly, Clearly, error error estimates estimates of RK methods on non-uniform grids are derived derived in the same way provided that one one has has already already shown the necessary stability stability estimates. estimates. Below Belowwe wegive givesome someanalogues analogues of the convergence results of Chapter 7, dealing with families of non-uniform grids. Actually, we show, as an example, error estimates of the standard form, that is, in the norm || ||. Note nevertheless that weighted weighted error error estimates estimates can be derived as well by analogy with what is done in Chapter 7. We first show an analogue of Theorem 7.4 for non-uniform grids, considering, for simplicity, only the case m = 0. Theorem 8.6. Let T = oo, let the the operator A{t) and the RK method be under the conditions of Theorem 7.4, assuming that that a > 0 in addition, and let 7 be an arbitrary family of grids. Then instead of (7.21) it holds \\Yn-y{tn)\\

<

3=0

200

CHAPTER CHAPTER 8. 8. VARIABLE VARIABLE STEPSIZE STEPSIZE

APPROXIMATIONS APPROXIMATIONS

for all Uk € 7 and t n 6 Ufc.

The proof of this result result follows follows that that of of Theorem Theorem 7.4, 7.4, using using Lemma Lemma 8.1 8.1 as well. Next we state a convergence convergence result result in in the the non-autonomous non-autonomous case. case. Theorem 8.7. Let T < < oo, oo, let let the the operator operator A(t) A(t) and and the the RK RK method method be be under the conditions of one one of of the the above above Theorems Theorems 7.7, 7.7, 7.8, 7.8, or or 7.9, 7.9, assuming assuming that a > 0 in addition, and and let let 77 be be an an arbitrary arbitrary family family of of grids. grids. Then, Then, with with K > 0 sufficiently small, small, for for all all 13k £ 7 with max ( g kn < K and for all tn € 13k, it holds, instead instead of of (7.1^2), (7.1^2), \\Yn -

y(tn)\\

+ CJ2k9j+2 E j=Q

/

l=q+l Jti

\\A(s)y^(s)\\d S + Cj2kPj j=0 j=0

Jt

i

and, instead of (7.51), \\Yn - y(tn)\\ <

This assertion follows by by analogy analogy with with the the proofs proofs of of Theorems Theorems 7.7, 7.7, 7.8, 7.8, and 7.9, using Theorems Theorems 8.4 8.4 and and 8.5. 8.5. Further, turning to giving giving an an aa posteriori posteriori error error estimate estimate on on nonuniform nonuniform grids, we consider such estimates, estimates, for for simplicity, simplicity, only only in in the the case case where where the the backward Euler method method is is applied applied to to problem problem (5.1) (5.1) with with A{t) A{t) == AA indeindependent of t. Noting that that under under our our below below conditions conditions on on AA the the operator operator bounded with with respect respect to to kk >> 0, 0, the the extended extended solusolu(/ + kA)~ l is uniformly bounded tion by the backward Euler Euler method method applied applied to to (5.1) (5.1) isis represented represented by by the the recurrence formula, with with !!„ !!„ == (/ (/ ++ knA)~l,

iovtneijk,

Y 0 = y°.

(8.60)

The next assertion gives gives an an aa posteriori posteriori error error estimate estimate for for this this method. method. Theorem 8.8. Let T = = oo, oo, let let AA ££ S( 0, and let 7 be a family family of of grids grids with with the the property: property: kkn < Ckn+\ when7. Then, Then, ifif f(t) f(t) isis strongly strongly continuously continuously ever tn,tn+i € 15k, for any 13k G 7. on [0,oo), forYn specified by (8.60) and for for y(t) y(t) the the extended extended

ERROR ESTIMATES ESTIMATES 8.3. ERROR

201 201

solution problem (5.1) in in the theconsidered consideredcase caseA(t) = = A, A, itit holds, for all all solution of problem with n > 1, 1, Uk G 7 and tn G Uk with \\Yn-y(tn)\\

C ^1 ^1++log ^ < C

j=o

Proof. accepted conditions conditionswe have, for any anyUk £ 7 and andttn £ Uk, Uk, Proof. Under the accepted y(tn+1) = e-k»Ay(tn) + kn I "

f ( ) ds.

Jtn tn

equation in (8.60), (8.60), we obtain, obtain, with with En = Subtracting this equality equalityfrom fromthe equation Subtracting this > 0, 0, Yn -y(tn), for nn > En+i - e~knAEn + i? n ,

(8.61) (8.61)

residual Rn is given by where the residual

Rn =

f(f( Jtn

=

e-fc"A) yn +fcnHn/(tn+l) e'^^-^fis) (iln -- efcnHn/(tn+l)" / ^ e'^^-^fis)

ds. ds.

Jtn

observing that that ^l" ^l"1 is bounded, bounded, by a > >0, 0,itit follows follows that that Next, observing s)A -(t -s) e(t n+1 f(s) n+1 tn

Also, since the equation equation in (8.60) can be be equivalently equivalently rewritten rewritten in the the form AiXnYn - iln/CVn), iln/CVn), dYn + AiXnYn

(8.62) (8.62)

we get, using using the thefact fact that thatall the involved involved operators operatorsare arepairwise pairwisepermutable, permutable, (iln - e~k"A) r n = ( / -- (( //++ kknA) e-k«A) (UnA-1f(tn+1)

-

A~ldYn),

whence whence - e-fc"A) Y n

= ((/ + knA) e' e'k«A - I) A-xdYn.

202

CHAPTER CHAPTER 8. VARIABLE STEPSIZE STEPSIZE

APPROXIMATIONS APPROXIMATIONS

Therefore, with the aid of these reasonings, the above representation for Rn is rewritten as follows Rn = ((I + KA) e~ k"A - I) A' ldYn

+ f " + 1 e - ^ + 1 - s ^ ( / ( * n + i - /(«)) ds. Jtn (8.63) Further, since the functions UJI(Z) — z~l((l — z)e z — 1), I = 1,2, are bounded in a neighbourhood of the origin, applying Lemma Lemma 5.4 gives, for all k > 0, ||u;i(-fci4)|| + ||a;2(-fci4.)|| < C. This estimate implies, for I = 1,2, \\((I + kA) e~ kA - I) u\\ < Ckl\\Alu\\

for all k > 0 and it ee Dom Dom A*. A*.

Using the last estimate next shows shows that that kn A l \\({i + k n^A)e~ -' -1) A- dYn-t\\
and, taking as well (7.11) for r\ — 1 into account, for tj+i < tn, ||((7 + kjA) e~ kiA - I) A^e-^-^+^dYjW

< <

Ck]\\AeCk]\\Ae-{tn'^

Also, by (7.11) with 77 = 0, we find, for tj+\ < tn, e

-i 33

Therefore combining the last three estimates and taking (8.63) into account account shows that \\f'(s)\\ds -i

Ck^

- i i + 1 ) - 1 | | y i + i - y^i if o < tj < ttn-i.

Now, since EQ = 0, similarly to (7.6) it follows from (8.61) that that n-l 3=0

8.3. ERROR ERROR ESTIMATES ESTIMATES

203 203

Using this this formula formula asaswell wellasas(8.64), (8.64),a asimple simple estimation yields Using estimation yields

(

n-2

\\

/_]fcj(in — fy-i-i)fy-i-i)-1 J ^ j n aaxx 1 + /_]fcj(i

n

||^j+i — —Yj\ ||^j+i

1=0

To conclude conclude the the proof, proof, it suffices suffices to to note notethat thatsince sinceknk < Ck Ckn+i, we we have, have, 1, that that for nn >> 1, n-2

n-2 n-2

,,

1=0

1=0 1=0

^^

nn

~

D

It is also also seen seen from from the theabove aboveproof proofthat thatthethe second term rightsecond term on on thethe righthand side of of the the aaposteriori posterioriestimate estimatecan canbebe replaced expression hand side replaced by by thethe expression ;

When applying applying the the ooposteriori posterioriestimate estimate Theorem to be When of of Theorem 8.8,8.8, oneone hashas to be certain of of the the behaviour behaviourofofthe thequantity quantity\\Y\\Y Y^-iH- By By (8.62) (8.62) itit follows follows certain n — Y^-iHthat

where 2l 2ln_i = AiXn-\. AiXn-\. Therefore, Therefore,assuming assumingininaddition addition that problem (5.1) where that problem (5.1) possesses an an extended extended solution solutionofoffirst firstorder orderand and that family of grids possesses that thethe family of grids is exponentially exponentiallybalanced balancedwith withbasis basis after a slight modification used is QQ >> 1, 1, after a slight modification and ttn € Uk Uk such such that that the proof proof of ofTheorem Theorem8.1, 8.1,we weobtain, obtain,forfor 15k S T and of the allall 1, n > 1,

n-l

<

Ck Ckn.1(\\Ay°\\+ (\\Ay°\\+

which shows which shows that that the theabove abovea aposteriori posteriori estimate estimate cancan be be used used for for constructconstructing adaptive adaptive algorithms. algorithms.

204

8.4

CHAPTER CHAPTER8.8. VARIABLE VARIABLESTEPSIZE STEPSIZEAPPROXIMATIONS APPROXIMATIONS

Comments Comments and andbibliographical bibliographicalremarks remarks

Section 8.1: weighted stability estimates for variSection 8.1: Apparently, Apparently,standard standardand and weighted stability estimates for variautonomousparabolic parabolicequations equations in a Banach Banach able stepsize stepsizeapproximations approximations to autonomous space setting obtained in our our work [44], [44], dealing dealingwith with the class Sj space setting are first obtained and quasi-uniform further shows thatthat the the stability quasi-uniform partitions. partitions.Palencia Palencia[127] [127] further shows stability standard sense sense is still preserved preserved for such partitions partitions when whenemploying employing in the standard methods of the classes classes SJI SJIand and5/// 5/// as well. The The case case of arbitrary arbitrary time timepartipartimethods tions is discussed discussed in our papers papers [30, [30,31, 31,36], 36], in [30, 36] 36] for methods methods of class 5/ that thethe stability es- esand in [31] for methods methods of ofthe theclasses classes5// 5//and and5///. 5///.Note Note that stability timate in [31] contains contains an additional additional factor factor log log(max (maxt g kkn/ min ( e g fc n). timate Theorem Theorem 8.1 is is shown shown first first in our our paper paper [30], [30],12 and andit it is is repeated repeated later later by Palencia further mention Palencia [128] [128] who whouses usesdifferent differenttechniques. techniques.WeWe further mention a paper paper working with families of Yan Yan [166] [166] who whoshows showssome somestability stabilityestimates estimates working with families of increasing quasi-quasiuniform 8.1). increasing quasi-quasiuniformgrids grids(see (seeSection Section 8.1). It appears, appears, however, however, using the above families above observations observationsconcerning concerninglocally locallybalanced balanced families of grids grids using quasi-quasiuniform grids, grids,that that the stability stability results results of [166] and families families of quasi-quasiuniform can easily easily be obtained obtained as well from from our earlier earlier work work [36], [36],where whereexponenexponengrids are in in use. Also, Also, it is is worth worth noting noting that that tially balanced balanced families families of grids only the the operator operatoriX^o iX^o is under under estimation estimation in [166] while while the thepaper paper[36] [36]deals deals estimation of i i ^ , 0 < < II <
proof of this result, result, which which is given given in [30] and and also reproduced reproduced in We follow follow here here the proof [36].

8.4. COMMENTS AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL REMARKS REMARKS

205 205

8.5 which then is useful and and itit isis stated stated here here for for the the first first time. time. Section 8.3: An a priori priori error error analysis analysis isis carried carried out out in in Gonzalez Gonzalez &; &;OsOstermann [71] under the hypothesis hypothesis of of quasi-uniformity quasi-uniformity and and some some additional additional restrictions on the used family family of of grids. grids. In In Section Section 8.3, 8.3, when when dealing dealing with with a priori error estimates, we we consider consider more more general general time time partitions, partitions, and and itit isis seen that these results are are simply simply compiled compiled from from those those achieved achieved inin Chapter Chapter 7 and in Sections 8.1 and and 8.2. 8.2. Also, Also, we we mention mention aa paper paper of of Saito Saito [138] [138]who who presents smooth and non-smooth non-smooth error error estimates estimates in in the the homogeneous homogeneous case, case, assuming no restrictions on on the the stepsize stepsize but but acting acting under under some some conditions conditions on the operator and method method in in question. question. Certain Certain aa posteriori posteriorierror error estiestimates for parabolic equations equations have have been been studied, studied, for for example, example, inin Akrivis, Akrivis, Makridakis, and Nochetto Nochetto [8], [8], Eriksson, Eriksson, Johnson, Johnson, and and Larsson Larsson [68], [68], JohnJohnson, Nie, and Thomee [88], [88], and and Nochetto, Nochetto, Savare, Savare, and and Verdi Verdi [120], as a rule, in Hilbert space settings. The The argument argument employed employed in in the the proof proof ofof Theorem Theorem 8.8 is close to that demonstrated demonstrated in in [8]. [8]. Note Note that that for for practical practical applications applications of Theorem 8.8 we have to to control control the the size size of of the the constant constant CC ininits its assertion. assertion. In principle, it is possible, possible, with with certain certain provisos, provisos, but but itit isis easier easier done done inin the the Hilbert case.

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Part III III Part

OTHERDISCRETIZATION DISCRETIZATION OTHER METHODS METHODS

This Page is Intentionally Left Blank

209

In this part of the book we we examine examine other other methods methods which which are are widely widely used in practice for discretization discretization of of abstract abstract parabolic parabolic problems. problems. As As such such we consider the 0-method, 0-method, certain certain operator operator splitting splitting schemes, schemes, and and linear linear multistep methods. The methods methods selected selected here here for for analysis analysis are are united united by by the purpose of demonstrating demonstrating wide wide possibilities possibilities of of the the approach approach presented presented in Part I. Unfortunately through our our limited limited opportunities, opportunities, many many interesting interesting disdiscretization methods are left left out out of of consideration. consideration. Nevertheless, Nevertheless, we we would would like to comment briefly on on some some of of them. them. First of all we mention the the discontinuous discontinuous Galerkin Galerkin method method which which has has been examined in the context context of of parabolic parabolic equations equations in in Jamet Jamet [87], [87],EriksEriksson, Johnson, and Larsson Larsson [68], [68], Eriksson, Eriksson, Johnson, Johnson, and and Thomee Thomee [69], [69], and and Thomee [162]. The virtue of this method method is is the the suitability suitability for for dealing dealing with with variable time stepsizes and and with with solutions solutions of of low low regularity. regularity. We Weexpect expect that that the discontinuous Galerkin Galerkin method, method, developed developed exclusively exclusively by by using using Hilbert Hilbert space techniques, can be be studied studied in in the the near near future future within within Banach Banach space space settings as well. At least, least, this this possibility possibility seems seems to to be be realistic. realistic. Also, in the context of parabolic parabolic equations equations using using variable variable stepsize stepsize mulmultistep methods is important. important. Unfortunately, Unfortunately, many many stability stability results results shown shown on those lines are not directly directly adapted adapted for for applications applications connected connected with with stiff stiff problems and, in particular, particular, with with parabolic parabolic problems problems in in themselves themselves (see, (see, e.g., Crouzeix & Lisbona [63], [63], Gear Gear & & Tu Tu [70], [70],Grigorieff Grigorieff [76], [76],and and Guglielmi Guglielmi & Zennaro [79]). At the same same time time we we point point out out the the work work ofof Becker Becker [46], [46], Calvo & Grigorieff [54], and and Grigorieff Grigorieff [77], [77], where where the the time time discretizations discretizations of parabolic problems by by the the variable variable 2-step 2-step and and 3-step 3-step BDF BDF methods methods are are examined from the viewpoint viewpoint of of stability stability and and convergence. convergence. In order to concentrate on on the the most most essential essential features features of of the the methods methods considered below, we deal deal only only with with stability stability and and carry carry out out our our investigation investigation working not with the most most general general case. case. As As above, above, we we let let flk flk be be aa uniform uniform grid in the segment [0,T] [0,T] with with stepsize stepsize kk >> 0. 0. In In what what follows follows we we keep keep the notation introduced in in Part Part II II and and which which isis connected connected with with the the use useofof uniform grids.

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Chapter 9

The ^-Method In this chapter we discuss discuss the the 0-method 0-method which which seems seems to to be be the the simplest simplest inin realization among all discretization discretization methods. methods. In In principle, principle, the the 0-method 0-method can be thought of as a particular particular case case of of the the linear linear multistep multistep method method whose whose analysis is carried out in Chapter Chapter 11. 11. In In the the autonomous autonomous case case A(t) A(t) == A, A, the ^-method can be examined examined within within the the study study of of Runge-Kutta Runge-Kutta methods methods expounded in Part II. However, However, in in the the case case with with aa constant constant operator operator there there are some interesting features features of of the the ^-method ^-method which which are are studied studied inin the the litliterature, as a rule, within Hilbert Hilbert space space settings settings (see, (see, e.g., e.g., Samarskii Samarskii [139] [139]1). Our main aim in the present present chapter chapter isis to to use use similar similar developments developments inin the the Banach case. Throughout the chapter the the operator operator A(t) A(t) isis assumed assumed to to be be independent independent of t. At the same time itit is is always always assumed assumed that that the the operator operator inin question question depends on a (scalar or vector) vector) parameter parameter hh €€ !H. !H. So So we we write write A^ A^ instead instead of A{t). Since we have as as an an aim aim certain certain applications, applications, we we assume assume Ah Ah toto be be a bounded operator on X X for for any any fixed fixed hh ££"K. "K. The (9-method is usually specified specified by by using using the the following following discrete discrete probprob2 lem, with 6 = 9(k, h) E R, R, dYn + Ah({l - 6)Y n + 0Y n+1) = (l-e)f{t n)+0f(tn+1)

for i n e

fife,

Y Q = Viyo.

(9.1)

The term (1 - 6)f(t n) + df{t n+x) can be replaced by /((I /((I -- 6)t 6)tn + 6tn+i) or in some other way as well. well. For For such such modifications modifications our our analysis analysis below below 1

called the the weighted weighted difference difference scheme. scheme. In [139] this method is called we introduce introduce an an operator operator 9t 9t 66 B(X) B(X) to to correct correct the the initial initial Unlike the general use, we 9^ may may depend depend on on kk and and hh as as well. well. condition. The operator 9^ 2

211

212

CHAPTER CHAPTER 9. THE 6-METHOD

changes in minor details, so subsequently we look only at problem (9.1). It is seen from (9.1) that the 0-method may be thought of as a 2-stage RK method3 whose Butcher array is 0 0 0 1 1- 9 9 1 —9 9

Clearly, in the case where 9 does not depend on k and h, this method is covered by the theory developed in Part II. At the same time it is well known (see, e.g., Samarskii [139]) that the order of accuracy for the 9method is raised at the expense of a suitable choice of 9 as a function of k and h. Therefore we must admit that 9 depends on k and h, which is in the case below.

9.1

Stability results

In the considerations below we require require that that at least Q > 9 = 9(k, h) > 0 and Ah G S( 0 and h G"K,with some 6 > 0, ip G (0,TT/2), and u G l . This assumption yields yields in particular that the operator (/ + k9Afl)~1 is defined and bounded on X for any fixed k G (0, K] and h £ "K. Therefore problem (9.1) is equivalently represented in the form (1.1) with 4 (9.2) which further implies that that (9.1) (9.1) possesses possesses a unique solution. In what follows we use this fact without explicitly explicitly mentioning mentioning it on each occasion. Now let, given (p G (0, n/2) and r > 0, the set Z(ip, r) be defined by (see Figure 9.1) ire . T \ pip ( i 2sin(/>'2sin(/?7 ' ' \2sinip' 2simpJ ' In this chapter together with with hypothesis hypothesis S(tp, S(tp, a) we use certain stronger numbers ip ip G G (0, (0, TT/2) and R > 0 hypotheses. As such, we introduce, given numbers and a positive-valued function ip{h), ip{h), the following. 3

It may be thought of as a 1-stage RK method if the term (1 — 6)f(tn) + Of(tn+\) in (9.1) is replaced by f(tn+i). 4 We recall that the operator Ah is bounded.

9.1. STABILITY RESULTS RESULTS

213

Figure 9.1: Z(ip,r) F o r a11

h G "K and A

> R,

and S2{
all /i G IK and A g IntZ( V(/i))> |A| |A| >> R, R,

We are now in a position position to to state state our our results results on on stability stability for for the the 0-method. 0-method. Theorem 9.1. Suppose Suppose that that 66 >> 00 -- 0(k,h) 0(k,h) >> 1/2 1/2 and and AAh £ S( 0 and h G IK, tmi/i tmi/i some some 66 >> 1/2, 1/2, ip ip G G (0,TT/2), and a G R. Xei a/so i/ie operator 9\ satisfy satisfy (6.4-8). (6.4-8). Then, Then, ifT< ifT< oo, oo, f/ie f/ie solution solution YYn of (9.1) satisfies the following stability stability estimate, estimate, with with KK >> 00 sufficiently sufficiently small, small, for for and t n G Qk, all k G (0,K\, heJi, \\Yn\\
(9.3)

If cr > 0 in addition, the above above holds holds for for TT == 00 == oo oo and and one one can can then then take take K = oo. Proof. For the sake of simplicity simplicity we we show show the the claim claim in in the the particular particular case case a > 0 only. In the general general case case aa G G R, R, the the below below argument argument isis slightly slightly modified with the aid of of reasonings reasonings similar similar to to those those used used in in Part Part II. II. 5 Having therefore assumed assumed that that aa > > 0, 0, we we recall recall that that for for all all kk >> 00 and and h G "K, the operator (I + + k9Ah)~ k9Ah)~l is denned and bounded bounded on on 3£. 3£. Moreover, Moreover, it obviously holds (9.4) ^WKC. (9.4) 5

Cf., e.g., the proof of Theorem Theorem 6.1 6.1

214

CHAPTER CHAPTER 9. 9. THE THE 9-METHOD 9-METHOD

Further, using the identity identity (Xl-kVl)-1

=

(XI - kA h(I + kOAh)- 1)

e

'

X

A

x x

~

kAh

leads us to the estimate, estimate, for for all all kk > > 00 and and hh G GOi, Oi,

IKAJ-fcSl)-1!! < | A - 0 - x

A

[—2

. (9.5)

i-ex

Clearly, the mapping a(—z) a(—z) = = z(l z(l ++ 9z) 9z) l takes the sector E v to the set Therefore, since A h G S(ip,a) and A/(1 - 6X) Z(v,0~l). 6X) ££ IntS^, IntS^, for for A A ££ that for for all all kk > > 00 and and A Agg Int Int Z(y>, Z(y>,6*" 6*"1), Int Z(
fl- 1!"1

which further implies, since since 66 >> 1/2, 1/2, that that for for all all kk >> 00 and and AA^^ Int IntZ(y, Z(y,2), 2), !! < C (|A|-X + |A - 2I" 1 ).

1

(9.6)

Observe that there exists exists aa set set TT of of A2-configuration, A2-configuration, with with singular singular points points Ci = 0 and (2 = 2, such such that that Z(ip,2) Z(ip,2) CC T. T. The The desired desired result result thus thus follows follows from Theorem 2.17 applied applied with with ££ — —00 to to problem problem (1.1), (1.1), (9.2), (9.2), taking taking as as well the estimates (9-4) (9-4) and and (9.6) (9.6) into into account. account. T h e o r e m 9.2. Suppose Suppose that that 66 >> 66 — — 6(k,h) 6(k,h) >> OQ and Ah G S(ip,a) for all k > 0 and h G "K, with with some some © © >> 990 > 1/2, (p G (0,TT/2), and cr G R. Lei ako (6.48) hold. Then, ifT< oo, i/ie solution solution YYn of (9.1) satisfies the stability estimate, with some some K K > > 00 sufficiently sufficiently small, small, with with any any fixed fixed [1Q > 0 sufficiently large, and with with any any fixed fixed ££ >> 0, 0, /or /or aH aH AA;;GG (O,JFST]; h £"K, and tn G fifc, +

i=0

)

(9-7) /

If o~ > 0 in addition, the above above holds holds for for TT == G G == 00 00 provided provided that that /^o /^o== 0,0, and one can then take K K == 00. 00. The proof is similar to that that of of Theorem Theorem 9.1, 9.1, applying applying as as well well Theorem Theorem 2.17 and using the fact that that the the pair pair (|](2l (|](2l ++ //o-O^ //o-O^' ' II; II;2l) 2l) ii s Aj(£)-concordant (Ai(£)-concordant if cr >> 00 and and /uo /uo == 0). 0). Now we show that the the above above results results can can be be improved improved ifif we we suppose suppose that Ah G S\(ip,R\tl>(-)) S\(ip,R\tl>(-)) or or Ah Ah G G S2( S2(lp,R','4>(-)), instead of the the condition condition Ah G S(tp, a) previously previously used. used.

9.1. STABILITY RESULTS RESULTS 9.1. STABILITY

215 215

Theorem 9.3. Suppose Suppose that thatAh Ah €€ S2( 00 and andhhGG"K "K are concordant in in the thesense sense that that are concordant k> 9(k,h)>^h)>^0 - 9(k,

(k^h))-1 (k^h))-

0, >> 0,

(9.8) (9.8)

with some (p (p€€ (0,TT/2), (0,TT/2), R R > > 0, 0, and and0 0> >0 0and and with some some positive-valued positive-valued with some with function ip(h). ip(h). Also, Also,let let (6.48) (6.48) hold. hold. Then, Then, if if TT << 00, 00,the thesolution solution Y Yn function of (9.1) (9.1) satisfies satisfies the the stability stability estimate estimate(9.3) (9.3) under under the the additional additional condition condition < K, K, with with K K >> 00sufficiently sufficiently small. small.If If R R ==00ininaddition, addition, then thenthe the above above k< holds for for TT — —© © == 00 00and andone onecan can take K K ==00. 00. holds take Proof. We Weshow show the the result result for for RR ==00only, only, remarking remarkingthat thatthe the below below proof proofis is Proof. slightly modified modifiedin in the thegeneral general case caseR R> >00(cf. (cf.the theremark remark at at the thebeginning beginning slightly the proof proof of of Theorem Theorem9.1). 9.1). of the Havingtherefore thereforeassumed assumed that R == 0, 0,we weobserve observethat thatby by spectral spectral theory theory Having that R (see, e.g, e.g, Hille Hille & & Phillips Phillips [84]) [84])the thespectrum spectrumof of the theoperator operatorfc2l fc2lbelongs belongs to (see, to the set Int Int£.(
and and Bh = =

Ah(I-A (I-Ah)-\ )-\ A

Our further argument argumentwill will be based based on on making making use useof ofthe therepresentation representation Our further be Bh)~x (A/ - B (A/

- 11

--

(I(I- - AA (\I -- (A (A++1)A 1)A h) (\I h ^ ))

ll

(9.9) (9.9)

By the the accepted acceptedconditions, conditions, the spectrum spectrumof of the theoperator operator A^ A^ belongs belongsto to the the By the set Z Z= = Int IntZ(ip, Z(ip, {0} U {1} and, all set 1)1)UU {0} U {1} and, for for all A gA Z,g Z,

which in in particular particular implies implies which

Uh\\ Uh\\< c. It thus thus follows followsfrom from(9.9), (9.9), using using as as well well the the last last two twoestimates, estimates,that thatwhenever whenever

||(A/ ||(A/ - B Bh)-'\\ )-'\\ < < C\\ C\\ ++l\l\-l\\ (j^I (j^I

1 -- I,)" I,)" !] < <

216

CHAPTER 9. THE 6-METHOD

Since the conditions A/(A + 1) ^ lntZ(ip, 1) and A £ IntE^ are equivalent, it follows from the last estimate that for all A ^ IntE^, \\(\I-Bh)'l\\
In other words, we have Bh E S(tp, 0) and, thus, k^Bh E S(ip, 0). With this fact in mind, it remains to apply Theorem 9.1 with k~lBh in place Ah and + (^(/i))"1)^)"1 with 9 + (kipih))-1 in place of 6 since fc2l = Bh(l+(6 l and 9 + (kiP(h))~ > \. D Theorem 9.4. Let Ah E Si( 0 and h E"Kbe subject to condition (9.8), with some (p E (0, TT/2), R > 0, and G > 0 and with some positive-valued function tp(h). Let also the operator 9\ be given by + 9kAh)-x.


Then, if T < oo and £ 6 [0,1], the solution Yn of (9.1) satisfies the stability estimate (9.7) under the additional condition k < K, with K > 0 sufficiently small. If R — 0 and HQ = 0 in addition, then the above holds forT — © = oo and one can take K = oo.

Proof. The operator / — |A;2l can be represented as follows I-l-m={I+(6-

l

-)kAh)

(I + OkAh)'1.

(9.10)

£ IntZ( 0 sufficiently small, for all k E (0, K) and hE"K, \\{I+{p-\)kAh)-l\\
(9.11)

Note that one can take K = 0 = oo if R — 0. It now follows from (9.10) and (9.11) that 6 \\(I + 6kAh

l

\\
(9.12)

It is seen in the proof of Theorem 9.3 that the operator 21 satisfies the A|condition, in particular, the A2-condition if R = 0, with the singular points 6

By (9.10) and (9.11) the operator (/ - \k%)~1 is bounded on X for any fixed he'K since the operator Ah has this property.

9.2. COMMENTS AND AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL

REMARKS REMARKS

217 217

Ci = 0 and ^2 = 2. Taking Taking then then (9.12) (9.12) into into account, account, noting noting that that the the operator operator / - ^fc2t takes the form form (1.39) (1.39) with with JJ == 2, 2, £2 £2 == 2, 2, 777== 1,1, and and 21 21inin place placeofof £ n , and using the representation representation (1.4) (1.4) with with both both sides sides multiplied multiplied from from the the left by (21 + /io-0^ (where (where fio fio — — 00 ifif RR == 0), 0), the the claim claim follows follows by by applying applying Theorem 2.3 together with with Remark Remark 2.2. 2.2.

9.2

Comments and bibliographical bibliographical remarks remarks

Section 9.1: The above above Theorem Theorem 9.3 9.3 actually actually arises arises from from our our work work [23, [23,24] 24] where a slightly different different terminology terminology isis used. used. Also, Also, itit follows follows from from [23, [23,24] 24] that the hypotheses S\(ip, S\(ip, R;ip(-)) R;ip(-)) and and S2{
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Chapter 10

Methods with Splitting Splitting Operator Here we consider approximations to problem (5.1), which employ employ certain certain constructions with splitting splitting operator. operator. In fact, in the general case where the operator in question is time-dependent, we examine only one operator discretization, which which is acsplitting method based on the backward Euler discretization, curate to the first order. However, completing completing this this chapter, chapter, in the special case A(t) = A and f(t) = 0, 0, we we discuss splitting constructions constructions which which may may lead to second order solvers. Note Note that that our below assertions are obtained on the basis of the general theory of operator splitting methods methods developed developed in Chapter 3. This is possible only under the condition T < oo which is therefore assumed throughout. throughout. In this chapter the main distinctive feature is that the operator A(t) in (5.1) is split by A(t) = Ai(t) + A 2(t) for t € [0,T], where Aj(t), j = 1,2, are closed linear operators operators acting acting in 3£, for which DomA,-(t) 2 Domi(i) for any t £ [0,T]. So, given a uniform grid flk m the interval [0,T], we consider the following splitting version of the backward Euler method method1 i

= (/ + feA1(tn+1))-1yn,

Yn+l

kA2(tn+1))-1f(tn+l)

for t n £ nk,

Yo = y°. x

In this chapter we only deal with the situation when 9\ = I.

219

220

CHAPTER 10. 10. METHODS METHODS WITH WITH SPLITTING SPLITTING OPERATOR OPERATOR

Eliminating Yn+i, we rewrite this discrete problem problem as a single step method, that is, Yn+i_ = (I + kA2(tn+l))-1(I Yo

+ kA1(tn+1))-1Yn

+k(I + kA2(tn+1))-1f(tn+i) = y°.

for t n G Qk, (10.1) (10.1)

Now we intend to apply the theory of Chapter 3 in order to examine the question of stability of problem (10.1). One of the most important points is then to show that the conditions (3.2) and (3.3) can be verified.2 In our subsequent consideration we use the following hypotheses which which are stated for some /xi > 0 and K G [0,1). coml[/t]: For any j = 1,2 and t G G [0,T] [0,T]the the operator Aj(t) has dense domain3 and it holds

Qj(t) := (A3_j(t) + Mi/)" 1 Dom A,-(t) C Dom Aj(t). Furthermore, for all j = 1, 2, t G [0,T], u G Qj(t), and \ € Dom

(10.2) A^^t),

+ tuiyxUxWl^KMt) + fill) fill) u

HI1"". com2[«;]: For any j = 1,2 and t G G[0,T] [0,T]the the operator Aj(t) has dense domain and satisfies (10.2) as well as Q*{t) := (A^it) + iiiIT1 Dom^(t) C DomA;(t). DomA;(t).

Furthermore, for all j = 1, 2, t G [0,T], u G Qj(t), and x G Q*(t),

Lemma 10.1. Suppose Ai(t),A2(t) G S((p,a) and hypothesis coralj/t] is satisfied, for some ip G (0,7r/2), a G R, and K; G [0,1). Then condition (3.2) is fulfilled with %Ljtn = Aj(£n+i) (-f + ^^(^n+i))" 1 ; J = 1,2. // in addition hypothesis com2[ft;] is satisfied, then (3.3) holds with the same 2lj;n2

(10.1) takes takes the form (1.1), (3.1). It is obvious that problem (10.1) This fact implies in particular that each of the operators Aj(t), j = 1,2, has an adjoint. 3

221

Proof. We show the claim in the the case case where where Aj(t) Aj(t) == Aj, Aj, jj == 1,2, 1,2, are are inindependent of t for the reason reason that, that, on on the the one one hand, hand, the the arguments arguments do do not not alter4 and, on the other hand, notation notation becomes becomes shorter shorter in in the the autonomous autonomous case. In particular, we write 2lj 2lj and and Qj Qj instead instead of of 2l 2lj;n and Qj(t), respectively.5 Next observe that, by Lemma Lemma 5.3, 5.3, the the operators operators 2tj, 2tj, jj — —1,2, 1,2, are are defined defined and bounded on X for any any fixed fixed kk G G (0,K), with K > 0 sufficiently small. small. Moreover, by Theorem 6.2 6.2 they they both both satisfy satisfy the the Af-condition. Af-condition. hypothesis comljfo] comljfo] we we Denoting now Qj = (I + + kAj)~x, j = 1,2, by hypothesis have Q2U G Qi for any u G Domii. Domii. Then, Then, for for all all uu €€ DomAi DomAi and and xx GGX*, X*, (x,Com (A l\g2)u)

- Q2)U) = ((/ - Q*2)X,A1Q2U)-(Q*2X,AI(I = k(A 2Qlx, AIQ2U) - k(g*2x, A1A2Q2U),

thus, with any /ii > 0, (X, {Ai + ii\I)Q2u)

= (x, (x, Q-i {Ai + ml) u) +k{A2g*2x, AIQ2U) - k(g 2x,AiA2g2u) (10.3) ) + V.

By hypothesis c o m l ^ ] we we further further obtain, obtain, for for all all uu G GDomAi DomAi and and xx GG 3£*, 3£*,

+Ck\\(A2 + mi)*Q 2x\\*\\(M + mi) Q2u\\K\\u\\1-K + mi)Q2u\\ + ll^i ^ ll^i ++ mi) mi) Clearly since A\ G S(ip,a), S(ip,a), the the last last estimate estimate isis in in force force with with any any fifi2 > 0 sufficiently large in place of of m m while while no no constant constant CC depends dependson on fj,2. Therefore we get instead <

C

{k1-^

{{(A! +/x 2 I)- 1 || 1 - K ) \\(Al + ^I) Q2U

< c {k 1-* + vK2~l) IK^i + Mi) e*A Applying this inequality in in an an estimation estimation on on the the basis basis of of (10.3), (10.3),we wefind find that that for all k G (0, K] and u G G X, X, u\\ < C\\(Ax + Ml) u\\ 4

(10.4)

In fact, all the estimates estimates retain retain their their shapes, shapes, becoming becoming uniform uniform with with respect respect toto t tn. We recall that Q(t) is specified specified by by (10.2). (10.2).

5

222

CHAPTER 10. 10. METHODS METHODS WITH WITH SPLITTING SPLITTING

OPERATOR OPERATOR

if K > 0 is sufficiently small small and and /i/i 2 > 0 is sufficiently large. large. Next, Next, with with the the aid of the following identity, identity, for for all all uu G GXX and and xx SS X*, X*,

= fc"1 (x, Qi(I - Q2) u) = (X, 420201^) + M2<X» M2<X»

+ employing hypothesis cc oo rr aall[[44 and and taking taking (10.4) (10.4) into into account account leads, leads, as as above, to the following estimate estimate 11(42 + toiy&QiXlU < C\\(A C\\(A2 + V2l)*Q2X\U,

(10-5) (10-5)

for all all xx G G X*. X*. Furthermore, Furthermore, itit isis with any fixed fi 2 > 0 sufficiently large, for not hard to see that for for all all uu G GXX and and xxG GX X**,, (X, Com (2lx|2l 2) u) = k~2(X, Com ((/ 2

= k~ (x,Com

6l),

(I - g 2)) u)

(QI\Q2)U)

so that we find, with the the aid aid of of hypothesis hypothesis COIII1[K] and of the above arguargument, with any fixed [i [i2 > 0 sufficiently large, for for all all kk €€ (0, (0, K], K], uu G GX, X, and and

|(x,Com (2l!|2l 2 ) U )| + C\\(A 2 + /x 2/)*020iXll*ll(^i + Ml) Q2Qiu\\ Q2Qiu\\K\\u\\1-K.

(10.6)

Therefore combining (10.4), (10.4), (10.5), (10.5), and and (10.6) (10.6) gives gives

)| < C||(A2 + M 2/r^x||^||xll^ +C\\(A2 + H2iy&x\\4{Ai

+ M2/) M2/)Qiu\\ Qiu\\K

So since both 2li and 2I2 2I2 satisfy satisfy the the Af-condition, Af-condition, (3.2) (3.2) with with jj == 22 follows follows from the last estimate in in view view of of Lemma Lemma 1.5, 1.5, and and (3.2) (3.2) with with jj — —11isis estabestablished by symmetry. If in in addition addition hypothesis hypothesis com2[«;] com2[«;] isis satisfied, satisfied, itit can can be be D shown in a similar way that that (3.3) (3.3) holds holds with with 2lj 2lj substituted substituted for for 2l 2l j;n . Now we state a stability stability result result for for the the splitting splitting method method (10.1). (10.1).

223

seminorms \\ jx jx;* and \ \^t Theorem 10.1. Suppose Suppose A\(t), A2(t) G S(tp,o), seminorms form (7i|y,a)-concordant (7i|y,a)-concordant pairs pairs with with A\{t) A\{t) and and A2{t), A2{t), respectively, respectively, each eachofof Aj(t) satisfies the respective respective hypothesis hypothesis hol®[i9;| hol®[i9;| \j-t], \j-t], jj — — 1,2, 1,2, and and the the hypotheses coml[/t] and and COITI2[K] are fulfilled, for some tp tp G G (0,TT/2), a G M, Then, ifif K K > > 00 isis sufficiently sufficiently small, small, for for the the d G (0,1], and 71,« € [0,1). Then, solution Y n of problem (10.1), we we have, have, with with any any fixed fixed /xo /xo >> 00 sufficiently sufficiently large and with any fixed fixed ££ G G [0,1), [0,1), for for all all kk G G(0, (0,K] K] and and ttn G Clk, that

||(2ln + Mo/fell + J ] ||(2lj;ri + X > i l l ,

(10.7)

where the operators Wj^n, Wj^n, jj == 1,2, 1,2, are are defined defined as as above above and and 2l 2ln is specified by (3.1). In the case where A\(t) A\(t) and and A^{t) A^{t) do do not not depend depend on on t,t, the the above above holds for any fixed £ G G [0,1] [0,1] while while the the first first term term on on the the left-hand left-hand side side ofof (10.7) is estimated as stated stated for for any any fixed fixed ££ >> 0, 0, even even ifif we we do do not not assume assume hypothesis com2[«;]. Proof. We consider the general general case case where where Aj(i), j — 1,2, generally depend depend on t. Since each of the the operators operators 2l 2lJ;n, j — 1,2, satisfies the the A^-condition A^-condition because of Theorem 6.2, 6.2, using using the the fact fact that that each each of of the the operators operators Aj(t), j = 1,2, satisfies the respective respective hypothesis hypothesis hol®[i?; hol®[i?; || \j-tt] and applying Theorem 6.9, it follows that the Holder-continuity Holder-continuity type type condition condition (3.40) (3.40) holds. holds. Since Since uniformly bounded bounded with with the operator (/ + kA2(t))~ kA2(t))~1 is, by Lemma 5.3, uniformly respect to k and t, it thus thus remains remains to to apply apply Theorem Theorem 3.4 3.4 with with 55j ;n = 0, j — 1,2, taking Lemma 10.1 10.1 into into account. account. As concerns the more precise precise wording wording in in the the particular particular case case where where Aj Aj (t), (t), j = 1,2, are both independent independent of of t,t, the the claim claim follows follows by by Theorem Theorem 3.3, 3.3, by by using Lemma 10.1 again. again. Note that Theorem 10.1 10.1 remains remains valid valid for for splitting splitting constructions constructions genergenerated by arbitrary RK methods methods of of type type Vj(ip), Vj(ip), but but ifif the the operators operators -Ai(t) -Ai(t) and and A2(t) are not permutable, permutable, approximations approximations of of that that type type would would be be only only first first order and of little practical practical use. use. At At the the same same time, time, using using the the fact fact that that the the stability function of the the backward backward Euler Euler method method vanishes vanishes atat 00 00 allows allows us us easily to obtain (10.7) with with Aj{t Aj{tn+\) substituted for 2lj ;n . Now, in the particular case case where where A(t) A(t) == AA and and f(t) f(t) == 0,0, we we examine examine an operator splitting method method which which leads leads to to second second order order solvers. solvers. In In order order to construct such a method, method, let let the the operator operator AA be be decomposed decomposed as as above, above,

224 224

CHAPTER CHAPTER 10.10. METHODS METHODS WITH WITH SPLITTING SPLITTING OPERATOR OPERATOR

with Aj(t) with Aj(t) ==Aj, Aj,j j= =1, 2, 1, 2, and and let let r(z)r(z) be the bestability stability the function functionof of some someRK RK method. We method. Weconsider considerthe thefollowing followingprocedure procedure ^

^^

^^

n n

for fortnt G G

titik,

YY0 = = y°. y°. (10.8) (10.8)

The The next nextresult resultshows shows the the stability stabilityof of the thesplitting splittingmethod method(10.8). (10.8).

10.2.Let LetA\(t),A2(t) A\(t),A2(t) 66 S( >00sufficiently sufficiently small, small,with with any anyfixed fixed > 00 sufficiently sufficiently large, large,and andwith with any any fixed fixed £ £> > 0, 0,for forall all k k6 6 (0,(0, K) K) andand Ho Ho > Qk>that that tn €€ Qk> where where 21 21== 2lj 2lj++2l2l ~~ ~kfyflii, kfyflii, 2lj 2lj == k~ k~la{—kAj), a{—kAj), jj == 1,2, 1,2,and, and, asas above, above, 2~ = 11— — r(z). r(z). IfIfininaddition addition hypothesis hypothesiscom2[/j] com2[/j] is is satisfied, satisfied, then thenthe the a(z) a(z) = quantity ]>Zj=i||(2l?;n ||(2l?;n + Mo-O^i'nll Mo-O^i'nll admits admitsthe the same same estimate estimatefor for any anyfixed fixed quantity ]>Zj=i +

£ ee[[oo,,i i]]. . Proof. By Proof. By virtue virtue of of Theorem Theorem6.2 6.2 each each of of the theoperators operators2li,2l2 2li,2l2satisfies satisfies the the Aj-condition. Therefore Aj-condition. Therefore the the claim claim will willfollow followby by Theorem Theorem 3.3 3.3 ifif we wehave have proved provedthat thatthe thehypotheses hypothesescoml[/c] coml[/c] and andcom2[«;] com2[«;]imply implythe theconditions conditions(3.2) (3.2) and (3.3), and (3.3),respectively, respectively, with with 2lj 2lj;n = = 2lj, 2lj,j j — —1,2. 1,2. The Thelast last fact fact is is however however in aa manner manner similar similarto to that that used usedin in the theproof proof of of Lemma Lemma10.1. 10.1. established establishedin In fact, In fact, by by the thecondition conditiondeg[p] deg[p]= = —1, —1, where, where,as as above, above,p(z) p(z) — —z~ z~1a(z), a(z), we we have, with have, with some someb\ b\q € C, C, II==1 ,,......, ,lo, lo,q q= = 11,,......, ,qi, qi,

1=1 9=1 1=1 9=1

where zi, where zi, II==1 ,,......, ,lo, lo,are arethe thepoles poles of of p(—z) p(—z) and andqi qidenotes denotesthe the multiplicity multiplicity of a pole pole zi. zi. By Bythe theAj((^)-stability Aj((^)-stability it it follows followsthat thatz\ z\ ££ S^,, S^,,and and the the operators operators x —1,2, 1,2,/ /= =l,...,lj, l,...,lj, are aredefined defined and and bounded bounded on on XX for for (kAj (kAj — —z;J)~ z;J)~, j — k€ € (0, (0,K] K] and andt tGG[0, [0,T], T],with withsome some K K> >00sufficiently sufficientlysmall. small.Using Using therefore therefore to the therepresentation representation (10.9) (10.9) leads leadsto

p(-kAj) == itt p(-kAj) itt Now, with Now, withthe the aid aidofofthis this expansion expansionand andof ofpermutation permutationtechniques techniques similar similar to those those employed employedin in the theproof proof of of Lemma Lemma 10.1, 10.1,it it is is readily readily seen seenthat thatthis this the thenewly newly defined defined2lj, 2lj,jj ==1,1,2.2. lemma lemma remains remainsvalid valid with with

225

we remark remark that that some some related related results resultsare arepresented presentedininOsOsIn conclusion, we under essentially essentially stronger stronger conditions conditionson onthe thesplitting splittingcompocompotermann [122] under nents.

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Chapter 11

Linear Multistep Methods Methods In this chapter we consider one more field for application of the theory developed in Part I. More precisely, here we examine examine linear linear multistep multistep methods methods applied to the Cauchy problem (5.1). We restrict ourselves to dealing with such methods for the sake of simplicity and for the reason that they are most important for practice. In principle, we could study even general multistep multistep methods, which would lead, lead, however, however, to a more involved analysis. It is remarkable that the investigation of linear multistep methods, methods, in comparison with the one of Runge-Kutta methods, possesses possesses its own distinctive features. Also, it is worth noting that the stability characteristics of the most popular linear multistep methods, methods, among among those those considered considered below, below, are well adapted for their application to parabolic problems.

11.1 Linear multistep multistep discretization discretization Let there be given numbers m € N and aj,bj G C, j = 0, ...,m. Let further f^ = {t n : tn = nk for n — 0 , 1 , . . . , N} be a uniform grid in the segment [0,T], with stepsize k,1 and let ft^ — {t n : tn = nk for n = 0 , 1 , . . . , N - m + 1} and fij:m) = {tn : tn = nk for n = 0,1,...

,N - m).

Now we associate to problem (5.1) the following discrete problem problem -l ~ f{tn+m-l)))

= 0 for t n G

1=0

Yo = y°, Y x = y\ ..., F m _i = ym-\ :

(11.1)

As above, k and N are connected by T = Nk, with the standard proviso in the case T = oo. 227

228

CHAPTER 11. 11. LINEAR LINEAR MULTISTEP MULTISTEP METHODS METHODS CHAPTER

assume that that the the initial initial with some y°,y 1,... ,ym~l £ X. Hereinafter we assume 1 m 1 (11.1) is is just just the the linear linear multistep multistep values y°, y ,..., y ~ are given. Problem (11.1) method applied formally to to (5.1). (5.1). However However this this form form of of writing writing isistypical typicalfor for Unfortunately, in in our our case case the the fact fact that that the the domain domain application to ODEs. Unfortunately, generally coincide coincide with with XX implies implies that that some some terms terms ofof the the of A(t) does not generally undefined. We We therefore therefore have have form A(t n+m-{) y(tn+m-i) in (11.1) may be undefined. to specify what we mean by by aa solution solution of of problem problem (11.1). (11.1). First of all note that, as above, above, our our main main assumption assumption in in this this chapter chapter isis that A(t) £ S(ip, a) with some some tp tp££ (0,7r/2) (0,7r/2) and and aa ££ M. M. IfIf inin addition addition to tothis this condition we assume as well well the the following following restriction restriction2 aobo 7^ 0 and | arg(ao/6o)| < TT - ip,

(H-2) (H-2)

kboA(t))~x is sufficiently small, small,3 the operator (aol + kboA(t))~ then, with K > 0 sufficiently bounded on X for k € (0, (0,K] K] and and tt ££ [0, [0,T] T] and, and, ifif the the entry entry inin (11.1) (11.1) makes makes sense, we proceed instead instead from from the the following following recursion recursion procedure procedure m

Yn+m = - (aol + kboA^n+m))' kboA^n+m))'1 Y](a;I + kbiA(t n+m^i)) Yn+m_i + k(aol + kbaA{tn+m))-lY,hf{tn+m-i)

for

t n £ f^m),

(11.3)

1=0

in (11.1). (11.1). Before Before taking taking further further with YQ, Y\,..., Ym-\ specified the same as in distinguish between between the the cases cases where where A(t) A(t) does does not not steps we now have to distinguish does depend depend on on t.t. depend and where it does where A{t) A{t) == AA isis independent independent of of t,t,4 observe that In the particular case where under the above assumptions assumptions the the operators operators (ail (ail ++ kbiA) kbiA)(aol (aol ++ kboA)' kboA)'1, I — 1,... ,m, are defined defined and and bounded bounded on on X. X. We We therefore therefore come come toto the the following definition. autonomous case, the discrete function Y n, tn £ Clk, Definition 11.1. In the autonomous recursion as defined by the recursion

1=1

m

+ k(aol + kboA)' 1 ^ bif(t n+m-i) 1=0 2

Which is present in all below below assertions assertions With K = oo if a > 0 4 necessarily dense dense But Dom A is not necessarily 3

for t n £

11.1. LINEAR MULTISTEP MULTISTEPDISCRETIZATION DISCRETIZATION 11.1. LINEAR

229 229

with YQ,Y\, ... ... ,Y ,Ym-i specified specifiedjust just the the same same as in in (11.1), (11.1), is called called the the CCwith YQ,Y\, extended solution solution of the the discrete discrete problem problem (11.1) (11.1)(or (or of of (11.3)). (11.3)). extended the general general case, case, we assume assume in addition addition that that CMJdora CMJdoraA(t)) A(t)) == XX for for In the G [0,T] [0,T] and andthat that the the operator operator (XI (XI — A(i))~ A(i))~1 A(s) A(s) extends extends to aa linear linear any t G the whole whole X for for any anyfixed fixed A fifi (Int (IntEEv U {0}) {0}) ++ aa and and bounded operatoron the bounded operator G [0,T]. [0,T].5 However, However, unlike unlikePart PartII, we we do do not not require require that that Dora DoraA(t) A(t) is is t,s G necessarily independent independentof t.t. necessarily Definition 11.2. 11.2. In In the thenon-autonomous non-autonomous case, case,under underthe the accepted accepted restricrestricDefinition the F-extended F-extended solution solutionYn, tn G fife of problem problem (11.1) (11.1) (or (or of of (11.3)) (11.3)) tions, tions, the the recursion recursion defined by the is defined -i)) Yn+m n+m-i

1=1 1=1

+ k(a k(aol + kboA(t kboA(tn+m n+m))

bif(tn+m 22 ^^ bif(t n+m-i)

for ttn G Cl Clk , for

1=0

with YQ,Y\, YQ,Y\, with

i^m-i specified specifiedjust just the the same same as as in in (11.1). (11.1). i^m-i

employ again againthe the product product space space3£(m) which which is defined defined In what what follows followswe employ same as in in Section Section 5.3, 5.3, with with m in in place place of v. v. As As above, above, the the norm norm in the same 3*(m) i s given given by | | O ii,,......,,uum ) r | | ( m ) = max max

IIUJH IIUJH

for for any any ( n j ,,...... ,um)T G G X X{m).

The symbol symbol || ||( ||(m) is is used used to denote denote as well well the the induced induced operator operatornorm normin m m Next, by X^* X^* we wedenote denote the the dual dual space space for X^ X^ \ For For our our subsequent subsequent £( ). Next, needs recall recall that that any any member member of X^* X^* is, is, inin fact, fact, a vector vector of the the form form needs T m T m Xm) with with Xj Xj ££ X*, X*,jj — —11, .,....,.m ,m Thenorm norm in in £( £( )* )* is is given given by by (Xi> ..Xm) . .The

3=1

m and the the duality duality pairing pairingbetween between3f(3f( and j£( j£(m) is expressed expressed as follows follows )* and

.

Xm)TT, (Ul,..., (Ul,..., U Umm)TT)(mm) == 55 33((XXJJ, ,««JJ) ). . Xm)

observed if, for for instance, instance, Domv4*(t) Domv4*(t)is independent independent of t.t. last fact fact is observed The last

(11.4) (11.4)

CHAPTER 11. LINEAR MULTISTEP MULTISTEP METHODS METHODS

230

Also, by I we denote the identity operator both in X^ and in X^*. Now we show that if Yn is the F-extended or C-extended solution of problem (11.3), the vector * n — \*m

(11.5)

i

solves, in fact, a discrete problem of the form (1.1) in the space that end, we put

n

\ To (11.6)

j-

= n

),

Q>m—j-L

J — 0 , . . . , 771,

and

F n = (0,..., 0, Fnf,

(n-7)

where Fn = B^)n J^ &j/(*n+m-01=0

by

Also, we introduce the operator 2ln : X^ I

I

-JO O

0

I

V Bm.nBo-n

0 0

-I

Bm.nBi-n

(11.8)

+ Bm.nBm-\-n

/

if J4(£) generally depends on i, and by 0 _/

/

2ln = 21 = A; - l

0

\ am(-kA)

0 0

a m-i(-kA)

) (11.9) with aj(z) = (aj — bjz)/{a,Q — boz), j = 1,..., m, if A(t) = A is independent of t. It is then easily seen under the accepted conditions 6 that if Yn is the ^-extended or C-extended solution of (11.3), the vector Yn solves the problem Y o = y°,

or, what is actually the same, Yo — y , 6

Which are different in the autonomous and non-autonomous cases

(11.10)

11.1. LINEAR MULTISTEP MULTISTEP DISCRETIZATION DISCRETIZATION

231 231

which takes the form (l.l). (l.l). 7 We therefore attempt attempt to to apply apply the the general general results of Part I. Throughout this chapter chapter itit is is assumed, assumed, without without explicit explicit mention mention on on each each occasion, that it holds k = 1, 1=0

(11-11) (11-11)

1=0 1=0

since this condition is satisfied satisfied by by all all methods methods being being of of practical practical value. value. For our below needs we we now now give give aa series series of of further further definitions. definitions. D e f i n i t i o n 1 1 . 3 . With With CCJ(Z), j = l,...,m, matrix /I

V am(z)

- 1

just the the same same as as above, above, the the

00

a m-i(z)

00

\\

l + ai(z) J

is called the amplification amplification matrix matrix of of the the linear linear multistep multistep method. Definition 11.4. The linear linear multistep multistep method method isis called called of of class class M/ M/ if,if, with with equation A(z) the amplification matrix matrix of of the the method, all the roots of the equation det(Ai - A(Q)) = 0,

(11.12) (11.12)

except the single simple root root Aj(O) Aj(O) == 0, 0, and and all all the the roots roots of of the the equation equation det(Ai - .A(oo)) = 0

(11.13) (11.13)

lie in the open disk I n t PP((ll;; I). I). 8 Definition 11.5. The linear linear multistep multistep method method isis called calledo/type o/type LM(ip) LM(ip)d for some if G (0, TT/2) if

(i) condition (11.2) holds; holds; and (ii) all the roots of the equation equation (solved (solved for for \)\)

det(Ai-.A(-z)) = 0

(11.14) (11.14)

belong to the closed disk disk V(l; V(l; 1) 1) whenever whenever ||argz| argz| << (p. (p. 7

In the particular case Q n = SRn = / This definition is easily modified modified in in order order to to be be applied applied to to the the general general multistep multistepmethmethods. That is why we use M in in the the abbreviation. abbreviation. 9 This abbreviation is relevant relevant when when dealing dealing with with linear linear multistep multistep methods. methods. 8

232

CHAPTER CHAPTER 11. 11. LINEAR LINEAR MULTISTEP MULTISTEP

METHODS METHODS

Definition 11.6. The linear linear multistep multistep method method isis called called of of type type LMj(ip) LMj(ip) ifif it is both of type LM((p) LM((p) and and of of class class M/. M/. 1 0

In this chapter, in the the general general case case where where the the operator operator A(t) A(t) depends depends certain Holder-continuity Holder-continuity type type conditions conditions on on A(t). A(t). In In this this on t, we employ certain modify slightly slightly the the concept concept of of testing testing functional functional ininconnection we need to modify 11 following definition definition will will be be valid valid troduced in Section 5.1. Therefore the following hereinafter. Definition 11.7. Let Cl(Dom.A(£)) Cl(Dom.A(£)) == XX and and let let V* V* and and VV be belinear linear manmanifolds (independent of t) t) such such that that T>ovaA*(t) T>ovaA*(t)CC V* V* CC X* X* and and Dom DomA(t) A(t) CC V C X fort G [0,T]. A bilinear functional A(t;-,-) A(t;-,-) with with Dom Dom A(t; A(t; == V* xV is then called aa testing testing functional functional for for A(t) A(t) if, if, for for any any tt GG [0,T], [0,T], (x,A(t)u)

— A(t;Xi u)

whenever x S "P* and u G DomA(t),

(A*(t)x,u)

= A(t\Xi u)

whenever x £ DomA*(t) DomA*(t) and and uu G G

and

Apart from everything else, else, one one more more hypothesis hypothesis of of Holder-continuity Holder-continuity isis in use below. More precisely, precisely, we we introduce, introduce, with with some some \i\ \i\ >> 00 and and i?i?GG [0,1], the following. hol/fcs [#]: The domain of A(t) A(t) is is dense dense in in XX for for any any tt G G [0, [0,T), T), the the domain domain of A*(t) is independent of of t,t, and and itit holds, holds, for for all all kk GG(0, (0,K], K], t,t, ss GG[0, [0,T] T] such such that \t - s| < k, and x G G 3t*, 3t*,

\\(A*(t) - A*(s)) X\U < Ck»\\(A*(t) + ^I)x\UAs a matter of fact we deal deal with with linear linear multistep multistep methods methods of oftype type LMj LMj () only and if the dependence dependence on on tt isis generally generally present, present, with with operators operators A(t) A(t) satisfying the hypotheses hypotheses holj^[#] holj^[#] and and h6l($; h6l($; A(t; A(t; ,, ;;|| || m; t,m = 0,*0) 12 when showing below the the assertions assertions on on stability stability with with the the help help ofof the the general general Part I.I. To To create create premises premises for for application application of ofthis this theory, theory, theory developed in Part give statements statements of of preparatory preparatory character. character. in the next section we give 10

complete classification classification of of the the linear linear multistep multistep methods methods owing owingtotothe the We do not give a complete fact that our consideration consideration in in the the present present chapter chapter isis of of confined confined character. character. 11 The point is that the analysis analysis in in Part Part IIII isis carried carried out out under under the the assumption assumption that that Dom A(t) is independent independent of of t,t, which which isis reasonable reasonable when when dealing dealing with with RK RK methods. methods. For For the aims of the present chapter chapter itit isis natural natural to to assume assume instead instead that that Cl(Dom Cl(Dom>!(£)) >!(£)) == XX for t 6 [0, T\. It turns out out that that we we then then have have to to modify modify the the concept concept ofof testing testing functional, functional, previously defined, in order order to to make make our our subsequent subsequent argument argument correct. correct. 12 With the newly defined concept concept of of testing testing functional functional the the latter latter hypothesis hypothesis isis stated stated just the same as in Section Section 5.1. 5.1.

11.2. AUXILIARY AUXILIARY RESULTS RESULTS

233 233

11.2 Auxiliary Auxiliary results results We begin begin by showing showing a certain certain resolvent resolvent estimate. estimate.

Theorem 11.1. multistep method method is of type LMj{ip) LMj{ip) Theorem 11.1. Suppose Supposethat that the linear multistep S(ip,a), for some ip ipG G (0, (0, TT/2) and a G R. Then there there exist exist and that that A(t) €€ S(ip,a), numbers numbers K > 0, 0, 55 >> 00 and andaaset setTT ofof Ax-configuration Ax-configuration such such that that for all all ke {0,K}, {0,K}, te [0,T], [0,T], and A <£Int <£Int (T UV(5k)), UV(5k)),

where A(z) is is the the amplification amplification matrix. matrix. If a > > 00 in in addition, addition, we then take take and 55 = = 0. 0. K — oo and Proof. First of all note that that the operator operator .A(—fc.A(i)) .A(—fc.A(i)) is defined defined and bounded bounded the accepted accepted conditions, conditions, the operators operators oij{—kA{t)), oij{—kA{t)), j — on £( m ) since, by the 1,..., 1 , . . . , m, are are defined defined and bounded bounded on X for for kk G G (0, K] and and tt G G [0, T], owing owing to Lemma Lemma 5.3. Further, Further, letting letting ipi G G (0, (0, tp) and R R > > 00 be be just the same as in the the 13 assertion assertion of Lemma Lemma A.I stated for A(t) A(t) in in place of £, we select some some d G (0, |ao|/|26o|)|ao|/|26o|)-14 Moreover, Moreover, for our our further needs needs we choose d > 00 to to be sufficiently sufficiently small small so that for z G G T>(d) all the the roots Xj(—z) Xj(—z) of equation equation (11.14), except the single simple (11.14), simple root root Xi(—z), Xi(—z), are bounded bounded away away from from 0. The The last is feasible feasible by Reed & Simon [134, Theorem Theorem XII. XII. 1] since 0 is a simple simple (11.12) and, hence, a simple root root of (11.14) (11.14) when when z = 0. 0. Also, we root of (11.12) set K < d/R d/R ifif aa < <00and andKK ==oo ooififaa>>0.0. By the assumptions assumptions it follows from from perturbation perturbationtheory theory (see [134, Section Section XII. 1]) that that there there exists exists a set T' T' of of Ai-configuration Ai-configuration such such that that all the the roots (11.14) belong belong to T' for for zz 66 EE^^.. Note also also that that there there are no no Xj(—z) of (11.14) poles of ctj(-z), j = l , . . . , m m,, lying in T, ipi since |arg(ao/fro)| |arg(ao/fro)| < TC — (p. Next, it is evident evident that that there there exists exists a set T T of of Ai-configuration Ai-configuration such such that that T ' \ { 0 }} ccllnnttTT.. We now show how to to estimate estimate the quantity quantity ||(AI ||(AI — A(—fc-^C*)))" A(—fc-^C*)))"1!!^) f° r A ^ T, |A| |A| < < Ao, Ao, with some some Ao > 00 which will will be specified specified later. later. Observe Observe that condition condition (11.11) (11.11) implies impliesthat that c\z\ < \Xi(-z)\ \Xi(-z)\ <
(11.15) (11.15)

Using this sufficiently small small to have this inequality inequality we select a fixed do > 0 sufficiently do\Xi(-z)\ < -\z\ do\Xi(-z)\ -\z\

for for all allzz G V(d),

"Observe that "Observe that if a > 0, 0, we wecan can take R = 0. 0. 14 Therefore the functions functions otj{z), j = 1,..., m, have no poles in T){d). Therefore

(11.16) (11.16)

234

CHAPTER 11. LINEAR MULTISTEP METHODS

and set Ao = d/do- Let further F\ be the contour coinciding with the boundary of the set "D(do|A|) U E^j. Applying then the operator calculus formula (A.10) componentwise, we obtain, with 6 — d$1R, for all k G (0,K] and A g Int(T U V(8k)), |A| < Ao,

(AI - A(-kA(t))yl

= (Ai - ^(oo))" 1 !

+ W~ I ((Ai - M-*))'1

- (^ - -A(oo))-1) ® (zl - kA{t))-ldz, (11.17)

where, as above, i stands for the identity (m x m)-matrix. For our needs below we further denote, given an (m x m)-matrix p = (pji), \Pji\-

In order to evaluate the integrand in (11.17), we first observe that ^ (Ai - yiC-a:))-1 - (Ai - yi(oo))-1 * < C # A(oo) - A(-z) U (Ai - yi(oo))-1 U (Ai - A{-z))~l

# . (11.18)

Since the eigenvalues of yi(oo) lie in the interior of T, it follows that tf(Ai-.A(oo)r1t
+ |A|)- 1

forA^IntT.

(11.19)

It is also readily seen that 1

SVl.

(11.20)

Furthermore since doAo = d, observing that the entries of the matrix A(—z) are bounded and that \j(—z), j — 2,... ,m, are bounded away from 0 for z G V(d) U S^j and using the formula for the entries of the inverse matrix (see, e.g., Horn & Johnson [85]), we obtain, with Aji(X;—z) the cofactors of the respective entries of Ai — A(—z), for all A ^ IntT, A G T>(XQ), and

(Ai - A(-z))-1

!/(< C max \An(\; -z)\ |det(Ai -

1-1 A(-z))\

l<jl<m

\X\r~1 J ] |A - Xji-zT1 < C\X - \!(-z)\-1. 3=1

(11.21)

11.2. AUXILIARY RESULTS RESULTS

235

Since X\(—z) G T ' for z G S ^ , it holds |A - Ai(-z)| A i ( - z ) | > c(|A| + \\i{-z)\)

for A g IntT and z G

and it follows from (11.16) that that

- ^ >B for

|A -

Using these observations, observations, instead instead of (11.21) we get, for all A ^ I n t T , A G X>(Ao), and z G TA,

| (Ai - y i ( - z ) ) " 1 # < C(|A|

(11.22)

Combining (11.18), (11.19), (11.19), (11.20), (11.20), (11.22) (11.22) and and taking taking (11.15) (11.15) into into account account therefore yields, for all A A ^ I n t T , A G £>(Ao), and z G T\, < <

(11.23)

C(|A[ +

Also, similarly to (5.7), we have, for k G {0,K], A ^ Int (T U V(6k)), and \\{zI-kA(t))-l\\
(11.24) (11.24)

Using now (11.19), (11.23), (11.23), and and (11.24) (11.24) for for estimation estimation on onthe the basis basisofof (11.17), (11.17), we find, for all Jfe G (0, K] K] and and A AG GInt Int (T (T U UV(Sk)) V(Sk)) n £>(A0),

_ A(-kA(t)))-% m)

< C If (AI - A(-


1

.

(11.25)

Finally, note that the estimate estimate % (Ai - A{-z))-'

- (Ai - .A(oo))- 1 1 < C(\z\ + IAD"1

obviously holds for A £ Int (T U V(X 0)) and z G V{d) U S V l as well. Using basis of (11.17), with r \ 0 in this fact and (11.19) for for an estimation on the basis place of TA, gives, for A £ Int (T U £>(A 0)),

Combined with (11.25), (11.25), this this inequality inequality shows shows the the claim. claim.

236

CHAPTER CHAPTER 11. 11. LINEAR LINEAR MULTISTEP MULTISTEP

METHODS METHODS

Remark 11.1. The assertion assertion of of Theorem Theorem 11.1 11.1 yields yields that, that, under under the the asassumed conditions, the operator satisfies satisfies operator 2l(t 2l(tn) = k~1A(—kA(tn)) G B(X^) the Af- condition, in particular, particular, the the A\-condition A\-condition ifif aa >> 0,0, in in the the space space 3£( 3£(m\ Theorem 11.2. Suppose Suppose that that A(t) A(t) G G S((p,a), S((p,a), Cl(Dom Cl(Dom A(t)) A(t)) == X, X, and and the the linear multistep method method is is of of type type LMi(ip), LMi(ip), for for some some tp tpEE (0,TT/2) and a G R. Let further seminorms seminorms \\ \t\t and and \\ |* |*;t form respectively (7|> 00 and and let let aa rational rational function ui(z) be ((p)-regular ((p)-regular and and subject subject to to deg[cj] deg[cj] << — [7J — 1. Then, with K > 0 sufficiently small, small, the the seminorms seminorms

i,.. .,um)Tln

= max l<j<m

\u(-kA(t n))uj\t nn

and KXl,

,Xm) T | . ; n

are well defined for k G G (0, (0, K], K], ttn G fijj. and form respectively Aj(7)Aj(7)- and and (A^j)- and A\(*; /y)-concordant if a > 0) Ai(*;j)-concordant 0) pairs pairs with with In the case where 77 is is an an integer, integer, under under the the extra extra 2l(i n ) = k~lA{—kA{tn)). requirement that the pairs (| |t;.A(£)) and (|(| \*-t\A(t)) \*-t\A(t)) are are respectively respectively (TMV 7)'7)

~

an

d (*;7l7l (Picr) ~ concordant,

i / 7 > 1, and \u
|xl*;t
for for all all tt ee [0,T], [0,T], uu ee X, X, and and

X

e X*

ifj = 0, it is sufficient to to suppose suppose that that deg[u] deg[u] << —[7] —[7] instead instead ofof << — [7J — 1Furthermore, for any fixed fixed ££ G G (0,1] (0,1] and and n\ n\ >> 00 sufficiently sufficiently large, large, with with some K > 0 sufficiently small, small, the the seminorms seminorms

I(«i,...,u m ) r I n =A: c muc ||(^(t n) + ml) (a ol +

kboA^yWl

and m

(A*(tn) + /iiJ) (05/ +

kblA*{t kblA*{tn))-lXjh

3=1

are well defined for k G and form respectively G (0, (0, K], K], ttn G ^ respectively Aj(l Aj(l — ^)and Aj(*; 1 —C) -concordant -concordant (Ai(l (Ai(l — Q- and Ai(*; 1 —C)- concordant if a > 0) pairs with ^i(tn). Apart from everything everything else, else, in in the the case case aa >> 00 the the above above holds with K = 00.

11.2. AUXILIARYRESULTS RESULTS 11.2. AUXILIARY

237

237

Proof. Thisresult result isestablished established asin inTheorem Theorem11.1, 11.1, applying aswell wellthe the Proof. This is as applying as argument employed inthe theproof proof ofTheorem Theorem 6.1. As concerns the seminorms argument employed in of 6.1. As concerns the seminorms I ||||nn and andI I |* |*;;nn,, to toshow showthe theclaim claimwewe letK, K,8,8, Ao,T, T,and and be the same let Ao, T\T\ be the same 15 15 as in inthe the proofof ofTheorem Theorem 11.1 and,taking taking inplace placeof ofT\ T\if if |A| > >Ao, Ao, as proof 11.1 and, F^ooF^ in |A| we use usethe the operatorcalculus calculus formula, forall all G (0,K], K],tt G G[0,T], [0,T],and and operator formula, for kk G (0, we

70 70 70 70 (AI Jfc2l(i))®u(-kA(t)) u(-kA(t)) ==(XI (XI-- A;2l(oo))A;2l(oo))® uj(-kA(t)) uj(-kA(t)) (AI -- Jfc2l(i))® ®

+ 7T- II w(-z) w(-z) + 7T2-KIJ JTx 2-KI Tx

10 1 10 1 U\\-A(-z))-™-(\\-A{oo))-' )®(zI-kA(t))dz, U\\-A(-z))-™-(\\-A{oo))-' )®(zI-kA(t))dz,

with 70==L'yL yJ1. + 1. Applying thesame sameformula formula 70==1 1 and with with 70 J'+ Applying the withwith 70 and with m l m (ao— —boz)~ boz)~ placeof ofui(z) ui(z)allows allows usto toprove provethe the desired estimates forthe the (ao place us desired estimates for other two seminorms involved. other two seminorms involved. A(t) S((p,a), Cl(Dom Cl(Dom =XXfor for any Theorem11.3. 11.3. Suppose that GG S((p,a), A(t))A(t)) = any Theorem Suppose that A(t) t G G[0,T], [0,T],and andthe the linearmultistep multistep method isof oftype typeLMj((p), LMj((p), forsome some linear method is for ipGG(0,7r/2) (0,7r/2) aG GR. R.Let Letfurther further bea atesting testingfunctional functional forA(t), A(t), ip andand a A(t;A(t; ,,)) be for let seminorms t, m = 0, *0, form respectively (-y\ip,a)and (*;7*|y, o~)let seminorms || | m; t, m = 0, *0, form respectively (-y\ip,a)and (*;7*|y, o~)m; concordant pairs A(t),and and satisfyhol(i?;.4(£; hol(i?;.4(£; |m;t,tn= = concordant pairs with with A(t), let let A(t)A(t) satisfy ,, ;;|| |m;t,tn 0, *0),for forsome some7,7* 7,7* G[0,1) [0,1) and -dG G[0,1]. [0,1]. Then Thenthe theoperator operator21 21(t (tnn) = = 0, *0), G and -d k~lA(—kA(t A(—kA(t )) satisfies hypothesis H O L ^ ; m j|j = | 0, *0) in the space k~ )) satisfies hypothesis H O L ^ ; j| j | ; , m = 0, *0) in the space m nn nn m X( mm)), where where X

(11.26) (11.26)

Proof. Aneasy easycalculation calculation shows Proof. An shows that that 00

0 0

00

0 0

/i(t,s) /i(t,s)

where Vj(t,s)==aamm-j(—kA(t)) where Vj(t,s) -j(—kA(t))

-j(—kA(s)), a — a— m-j(—kA(s)), m

(t,s) VmmV (t,s)

At j j = =ll, ,. . .. ,.m, m . . At t ht eh e

same time wehave, have,denoting denoting forshort shortCjCj = aam _j6o — —bbmm-jao -jao and and g(t) same time we for = g(t) = = m_j6o

15 15

Therefore Therefore 5 5==00 and andK K==00 00 if ifaa>>0.0.

238 238

CHAPTER LINEAR MULTISTEP METHODS CHAPTER 11. 11. LINEAR MULTISTEP METHODS

With the the aid aidofofthis this representation, representation,we we find, find, for for jj == 11, .,....,.m , mand and With forfor all all ueX andxx GG X*, ueX and X*,

=

{{A*(s)Q*(s)x,Q(t)u) -kkCj Cj{{A*(s)Q*(s)x,Q(t)u)

=

(g*(s)X,A(t)g(t)u)) ,A(t)g(t)u)) (g*(s)

-kcjAA(t,s]Q*(s)x,Q(t)u), -kcjAA(t,s]Q*(s)x,Q(t)u),

u as above, above, AA(t,s;x, AA(t,s;x, —A(t\Xi A(t\Xiu) ~~ -4( -4(s ;X> ;X>u )- Using Using the the last last where, as where, ) — equality and andrecalling recallingthat thatA(t) A(t) satisfies satisfieshol(i9, hol(i9, ,m = = 0, 0,*0), *0), equality ,, ;;| | ||mm;;tt,m it follows followsthat thatfor for jj == 11,,......,,m m and andfor forall allt, t, [0,T], T], t ii ee ll, , and andxx GGX*, X*, s sG G [0,

k-'Kx,Vj(t,s)
11.3 11.3

Stability Stability

We are are now nowinina aposition position to to state state our our results results on on stability. stability. As As above, above,we we We distinguishbetween betweenthe the cases cases with withaa constant constant and andaavariable variable operator. operator. distinguish Theorem 11.4. 11.4.Suppose Suppose that thatA(t) A(t) ==AAisisindependent independent oft, oft, AA GG S(ip,a), S(ip,a), Theorem is of of type type LMj(ip), LMj(ip), and and aaseminorm seminorm ][-][t ][-][t forms forms the linear linear multistep multistepmethod method is the (£| >0.0. a (£| > 00sufficiently sufficiently small, small,for for all allkkGG(0, (0, and satisfies K]K] and G flk, flk, tn G

M-kA) YYn][tn< C C M-kA) o~ > > 00 in in addition, addition, the the above aboveholds holdsfor for TT ==KK——oo. oo. If o~ Proof. It is ispointed pointed out outin inSection Section 11.1 11.1that that if if YYn is is the theC-extended C-extendedsolution solution Proof. It (11.1),then thenthe thevector vector Y Y n specified specifiedby by (11.5) (11.5) fits fits(11.10) (11.10)with with y°, y°,FFn , and and of (11.1), 2ln = = 21 21given givenby by (11.6), (11.6), (11.7), (11.7),and and (11.9), (11.9),respectively. respectively.Also, Also, the the seminorm seminorm ]['][«—}[ ]['][«—}[u(~kA)-}[t (~kA)-}[tn forms forms a Aj(£)-concordant Aj(£)-concordantpair pair with with 21, by by 21, Theorem Theorem11.2. 11.2.

11.3. STABILITY 11.3. STABILITY

239

239

thusremains remains toapply applyTheorem Theorem 2.1to toproblem problem(11.10), (11.10),taking takinginto intoaccount account It thus to 2.1 that that m m

| | Fn || | |( m C ^ | | /n ++mm_ j || || ||F
(11.28) (11.28)

j=0 j=0

and and ||Y Y0 | | ( mm))< C C

max ||^||. ||^||. max

(11.29) (11.29)

0<j<m—l 0<j<m—l

Concerning the case case with withaa variable variableoperator, operator, wepresent present the the following. following. Concerning the we 11.5.Let LetTT< < and operatorA(t) A(t) and andthe the linear multimultiTheorem11.5. Theorem oooo and let let the the operator linear step method methodbe beunder under the theconditions conditionsof of Theorem Theorem11.3 11.3 for some some(p(pG G (0,TT/2), (0,TT/2), step for aG GR, R, 7,7* 7,7*GG [0,1), andti andtiG G(0,1] (0,1]such such that that7+7* 7+7*< < 11 + + tf. tf.Let Letfurther further Ait) Ait) [0,1), satisfy hypothesis hol^J^ for ] some some $2 $2 G G(0,1] (0,1]subject subjectto to 77 < [O,min{£i,i92}]>where where £1 == 11— — 7* 7*++ i9. i9. Then Then the the VVA(t) for £1 extended solutionY Yn of of (11.1) (11.1) is is defined definedand andsatisfies satisfiesthe the stability stability estimate estimate extended solution A(tn) substituted substitutedfor for AA and andwith with {I-\-kA{t {I-\-kA{t ))~l substituted substitutedfor for (11.27)with withA(t (11.27) n))~ ui{~kA), wherethe thefactor factor t~ t~ isisreplaced replacedby byt~ t~ ++log(i,-+i/A;) log(i,-+i/A;)if if f; f;= = $2 $2<<£1 £1 ui{~kA), where and by bytj* tj* ++tptp log(t log(tj+1 /k) ifif££==£1 £1<
240

CHAPTER CHAPTER 11. 11. LINEAR LINEAR MULTISTEP MULTISTEP METHODS METHODS

with 16

- k~lam-j{-kA{tn)),

j = 0,..., m - 1. 1.

It follows from Theorem Theorem 11.3 that the operator 2l(i n ) satisfies hypothesis as in the statement of the HOL(i9; I |m;n,tn = 0, *0) with the same present theorem where where the seminorms ||| \\m-n, tn — 0, *0, are given by (11.26). By Theorem 11.2 these seminorms form respectively respectively A-j^)A-j^)- and Aj(*;7*)-concordant pairs pairs with with 2l(t 2l(t n ). Further, applying the following identity, with with Cj denned the same as in the proof of Theorem 11.3, for j — 0 , . . . , m — 1, WnJ = bm-jiaol + kboA{tn+m))~1(A(tn+j) %

^tn) ^tn)

-

A(tn+m))

-- A(tn+m))

(a0I +

kb0A{tn))'1,

t — tn+j,tn, extend where the factors (aol+kboA(t n+m))~1 (A(t)—A(t n+m)), to linear bounded operators operators on X, and using the fact that A{t) satisfies > 0, for all x £ X * and u G X, hypothesis hol? feJi?2] we have, with some that

\{x,WnJu)\ < C^ 2 ||(^*(i n ) + Mi/)(aS/ + A;6^*(in))-1xll*ll«ll- (H-30) Noting now t h a t by (11.4) it holds, for any ( x i , . . . , X m ) T 6 X^*

and

. Xm)T, ® n (ni, . . . , Um)T)(m) = (Xm, with the aid of (11.30) we obtain | ((Xl.

, Xm) T , ®n(wi,

Umm))TT)( )(mm)) \\ U

< Ck^\\(A*(t n) + /xi/) (a*0I + fc&5^(t n))-1 < C||(Xl,

,Xm)r|||*;tf2;n||(ui,

,um)T\\{m)

Umm))TT\\\\{m) ,, U {m),,

where the seminorm

*n) + ^ W 3=1

We remark that actually Wn,j 6 B(X), which follows from the above comment.

11.3. 11.3. STABILITY STABILITY

241

241

forms a A^(*; 1 1— pair with ), because of Theorem 11.2. 11.2. — i^-concordant i^-concordant pair5t(t with 5t(t nn ), because of Theorem forms a A^(*; Now weweuseuse the the aboveabove shown properties of the operators 2l(£ operators 2l(£nn ) and Now shown properties of the and 1*B n and thatthat the seminorm ]|[-][n ][(^ ++^^(*n))~ ][t nn forms *B andnote note the seminorm ]|[-][n== ][(^ ^^(*n))~1][t formsa Aj(£)a Aj(£)17 ), ininvirtue of Theorem 11.2, 11.2,17 while and and concordant pairpair with with 2l(i 2l(i nn ), concordant virtue of Theorem while(11.28) (11.28) (11.29) areare validvalid in theinconsidered case. Thuscase. the claim follows Theorem (11.29) the considered Thus thebyclaim follows by Theorem 2.17 which is applied to problem (11.10) with 2l(i with 2l(i nn ) ininplace of 2lof 2l nn , with place withthethe 2.17 which is applied to problem (11.10) statement of the of present with 71 =and 7, with 71 = 7, same 7,7*,$ same 7,7*,$ as asininthethe statement the theorem presentand theorem $1 = $0= =$00, = 7*0 0,— 17*0 - $2$1 ==$,$,70 70 — 1 - $2-

O

O

As aapossible application, we consider family ofthe m-step BDFofmethAs possible application, we the consider family m-step BDF methods, ods, mm== 1,2,..., 1,2,..., for forwhich which

and and m mm m

E E ;=o ;=o

n, n

m ——0 0

\

\ In,

;=o ;=o

In, 1

m

\ 1 13n, \

i=o

O 13n,

O <)<)

99

m m

f

f 11 11 ^9^ ^9^

i=o

Note that (11.31) and the equality (11.32) together imply together condition imply condition Note that (11.31) andfirstthe first inequality in (11.32) (11.11). by by (11.31) it is immediate that equation (11.13) has only(11.13) has only (11.11). Further, Further, (11.31) it is immediate that equation one root = 0 (of m). Also, itm). easilyAlso, followsit from the follows order from the order one rootA(oo) A(oo) = 0multiplicity (of multiplicity easily conditions (11.32) that all the roots of (11.12), the single simplethe rootsingle simple root conditions (11.32) that all the roots except of (11.12), except Ai(0) belong to thetoopen Intl?(l; It is known Hairer (see, e.g., Hairer Ai(0)= =0, 0, belong thedisk open disk1).Intl?(l; 1).(see, It e.g., is known & [81])[81]) that that for thefor m-step methods withmethods 1 < m < 6,with condition & Wanner Wanner the BDF m-step BDF 1 < m < 6, condition

mm],], with withipipmm €€ (0,TT/2) (0,TT/2) depending depending (ii) 11.5 holds (ii) ininDefinition Definition 11.5 for holds for

=d\—(1 = +—(1 20), + oi2 20),

== oi2 == 9,9, bobo == 1, 1,b\ = b\ b = b22 ==== 0,0,

where where 9 9is aisfixed a fixed parameter. parameter. (For the sake (Forofthe simplicity sake we of assume simplicity 0 to we assume 0 to be real.)AnAn elementary (but cumbersome) shows that for this be real.) elementary (but cumbersome) computation computation shows that for this method method condition condition (ii) in(ii) Definition in Definition 11.5 is fulfilled 11.5forisany fulfilled fixed (pfor £ (0,any fixed (p £ (0, TT/2) TT/2) ifif 99££ [-1/2,1/2] [-1/2,1/2] and andforfor

>1/2,1/2, where where

tpe ——arctan -— -— tpe arctan

(1 (1 ++464611 )33 /22 - I)I ) 33 / 22 '

In In tthhee case case ££==0 0wewe also also can c a n take take|[-][n= |[-][n= |||| II,II,byby Theorem Theorem11.1.11.1.

242 242

CHAPTER CHAPTER 11. LINEAR 11. LINEAR MULTISTEP MULTISTEP METHODS METHODS

equation (11.13) has Further, Further,it it is iseasily easilychecked checkedthat that equation (11.13) hasaaroot root A(oo) A(oo)= = 11 ofof multiplicity 2 and thethe other multiplicity 2 while whileone oneof ofthe theroots roots of of (11.12) (11.12)is is Ai(0) Ai(0)==0 0 and other lies in the lies in the theopen open disk diskInt IntD(l; D(l; 1) 1)ifif66>>—1/2. —1/2. Consequently, Consequently, theconsidered considered method is isof oftype type LMi(ip) LMi(ip)for forany anyfixed fixed ipipe (0,TT/2) (0,TT/2)if if 00G G (-1/2,1/2] (-1/2,1/2]and and method for any for anyfixed fixed yy €€(0, (0,tp tpg} if 00>>1/2. 1/2.

11.4 Comments andand bibliographical remarks 11.4 Comments bibliographical remarks Apparently, multistep approximation in has Apparently, multistep approximation in aaBanach Banachspace spacesetting setting hasbeen been examined for in opexamined forthe thefirst first time timein in Crouzeix Crouzeix[60], [60], in the thecase case with withconstant constant operator but erator butfor forgeneral generalCo Cosemigroups. semigroups.At Atthe thesame sametime timean anearlier earlierpaper paper of of 18 Le Roux Le Roux[102], [102],18 although althoughwritten written in in aaHilbert Hilbertspace spacesetting, setting, demonstrates demonstrates techniques techniquesthat thatare areextended extendedto tothe theBanach Banachcase, case,dealing dealing withwith sectorial sectorial opoperators. Note erators. Notethat that both both mentioned mentioned paperspapers aim aimat atshowing showingconvergence convergence onlyonly donot notallow allowus usto tosettle settlethe thequestion questionof ofstability. stability.The Theconsideraconsiderabut they but theydo tion tion of oflinear linearmultistep multistep methods methods in inthe thepresent presentchapter chapter covers covers the thecases casesboth both with aa time-independent with time-independent and andaatime-dependent time-dependent operator; operator; in infact, fact, our ouranalyanaly19 Later [129], sis sis here hereis isclose closeto tothe thelines linesfound foundin in [28]. [28].19 LaterPalencia Palencia [129],considering considering the autonomous case only, shows the approxthe autonomous case only, shows thestability stabilityof ofgeneral generalmultistep multistep approximations. For Foraaspecial specialclass classof ofsuch such approximations, approximations, some some strong strong stability stability imations. and and nonsmooth nonsmootherror error estimates estimates are arepointed pointedout outby byHansbo Hansbo[83]. [83].The Thecase case has recently in with aa variable with variableoperator operator hasalso also been beenconsidered considered recently in Gonzalez Gonzalez& & Palencia [74], if Palencia [74],but buttheir their results resultsare areapplicable applicableonly only if the theoperator operatorA(t) A(t)isis Lipschitz-continuous while Lipschitz-continuous while the thesituation situationin in which whichA(t) A(t)isisproperly properlyHolderHoldercontinuous continuousis is not notcovered coveredin in[74]. [74].

18 18

it it also also treats treatslinear linear multistep multistep ones.ones. Apart Apart from fromsingle single stepstep methods methods arenow nowbased based on onaadifferent different form formof ofHolder-continuity Holder-continuity for forA(t). A(t). We We are

19 19

Part IV

INTEGRO-DIFFERENTIAL EQUATIONS UNDER DISCRETIZATION

This Page is Intentionally Left Blank

245

In this part of the book book we we present present some some results results concerning concerning stability stability of discretizations of abstract abstract linear linear integro-differential integro-differential equations. equations. As As the the starting object, we consider consider the the initial initial value value problem problem rt

yt + A{t)y=

Jo

P(t,s)y{s)ds P(t,s)y{s)ds + +f(t) f(t)

forte(0,T], forte(0,T],

y(0) y(0) == y°, y°, (11.33) (11.33)

where P(t, s) is a linear (generally (generally unbounded) unbounded) operator operator in inXX for for 00<< ss < > 00 sufficiently sufficiently large large and and for for any any t,s 6 [0,T] such that t > > s, s, the the operator operator (A(t) (A(t) +/iiJ)~ +/iiJ)~ 1 P(t,s) is extended to a linear bounded operator operator defined defined on on the the whole whole X. X. Further Further requirements requirements on A(t) and P(t, s) are imposed imposed in in the the subsequent subsequent statements. statements.20 Problem (11.33) is more complicated complicated in in comparison comparison with with (5.1) (5.1) since since an an extra term is involved in in (11.33), (11.33), which which isis of of nonlocal nonlocal nature nature and and brings brings about additional difficulties. difficulties. As As concerns concerns discretizations discretizations ofofproblem problem (11.33), (11.33), they are accordingly harder harder for for analysis analysis than than those those studied studied above abovefor forproblem problem (5.1). In our opinion, this topic topic deserves deserves to to be be the the main main subject subject for for aaseparate separate book; nevertheless, in what what follows, follows, just just to to reveal reveal further further the the possibilities possibilitiesofof our approach, we discuss certain certain selected selected discretization discretization schemes schemesfor for problem problem (11.33). In Section 12.1 we show the the stability stability of of these these discretizations discretizations on on finite finite time intervals while Section Section 12.2 12.2 isis devoted devoted to to discussing discussing suitable suitable sparse sparse quadrature rules for an approximation approximation of of the the memory memory term, term, which which are are ofof considerable interest in practical practical applications. applications.

20

Here we derive no error error estimates estimates so so that that no no conditions conditions that that provide provide the the unique unique solvability and high regularity regularity properties properties of of (11.33) (11.33) are are employed. employed. The Thebelow belowrequirements requirements on A(t) and P(t,s) are merely merely used used to to show show the the stability stability ofof the the discretizations discretizations that that are are considered.

This Page is Intentionally Left Blank

Chapter 12

Integro-Differential Integro-Differential Equations under Discretization Discretization Stability of discretizations discretizations 12.1 Stability chapter we examine two concrete discretizations discretizations schemes schemes of In the present chapter Euler method method and the second the Crankwhich the first is the backward Euler methods, when when applied applied to purely differential differential Nicolson one. 1 Both of these methods, with the aid of certain assertions assertions equations of the form (5.1), are analysed with established in Part II. grid in the segment [0,T], [0,T], with with stepsize stepsize Let, as above, Q/. be a uniform grid let &k &k retain the same meaning meaning as before. Since Since our techniques k > 0, and let throughout are now based on making use of the results of Chapters 4 and 6, throughout that T < oo. The The memory term term in (11.33) is this chapter chapter it is assumed that approximated by applying the quadrature formula, approximated formula, with with some some weights weights KKUti, rtn rtn ™zl A"W>) = Y* nn,rtl « / il>{s) ds, with i} = tl>{ti),

(12.1) (12.1)

function. Our main where tp(t) can be thought of as a scalar or (X)-valued function. requirement below below on the quadrature formulas formulas used used is that with some some p > 0, requirement A n ( l ) = t n, KUtl > 0, and 1=0 lr

The latter method method is considered in the case A{t) = A only.

247

248

CHAPTER 12. INTEGRO-DIFFERENTIAL INTEGRO-DIFFERENTIAL CHAPTER

EQUATIONS EQUATIONS

discretization for all tn, tj G Qfc such that 0 < tj
subsequently stated stated the operator I + kA(tn+\) Since under the conditions subsequently method in the following has a bounded inverse, we take this discretization method + kA(t))-lP(t, s), form, denoting i ^ = (/ + kA^i))'1 and P(t, s) = (I + Yn+l

= n

Y^

kilnf(tn+l) for t n G kilnf(t

1=0

Yo = y°,

(12.3) (12.3)

which can be equivalently rewritten in the form (4.2) with Xn = A(t n)(I + kA(t n))-\

(12.4) (12.4)

S n = 2t n + 1 - 2l n ,

(12.5) (12.5)

Fn=iinf(tn+1),

(12.6) (12.6)

and y nj

= Kn+i,iP(t n+i,tl).

(12.7) (12.7)

theorems we show the stability of the backward In the following three theorems Euler method (12.3) in different situations. T h e o r e m 1 2 . 1 . Let the operator A(t) and a seminorm | |t satisfy the conditions of Theorem 6.12 for some ip E (0,TT/2), a G R, ~d G (0,1], 7i G [0,1), and let seminorms ][-][t and | |*;t form respectively (£\
\\P*(t, s)X\U < C\((I +

fcA(t))- 1)*xU;t-

(12-8)

(12.3) satisfies satisfies the stability Then the solution Yn of the discrete problem (12.3) estimate

kA{tn))-lYn\tn
(12.9)

for all k G (0,K) and t n G fifc. Furthermore, in the case ^ — 0 the last inequality holds with }[Yn][tn substituted for ][(/ + kA(t n))~1Yn][tn.

12.1. STABILITY STABILITY OF OFDISCRETIZATIONS DISCRETIZATIONS 12.1.

249249

Proof. Werecall recall that thatproblem problem (12.3) is equivalent equivalentto to (4.2), (4.2), (12.4) (12.4) (12.7). (12.7). Proof. We (12.3) is by Lemma Lemma 5.3, 5.3,the theoperator operatoriln iln isisuniformly uniformlybounded, bounded, for FFn given given Since, Since, by for by (12.6)we wehave have by (12.6)

\\Fn\\
foralltnGftfc. foralltnGftfc.

(12.10) (12.10)

Now, observing observingthat that the backward backwardEuler Eulermethod method is of oftype type Vi(ip) Vi(ip)for forany any the is Now, (p (0,TT/2),by by Theorem Theorem6.1 6.1ititfollows followsthat thatthe theseminorm seminorm (p £ (0,TT/2),

))^.}[tntn n))^.}[

ifif ££ >> 00, , if €€==0,0,

forms aa A^(£)-concordant A^(£)-concordantpair pair 2ln given given by by (12.4), (12.4),and, and,assuming assumingfurfurforms withwith 2l ther that that all all the theconditions conditionsof of Theorem Theorem4.1 4.1which which do do not notinvolve involvethe thesemisemither aresatisfied satisfiedfor forsome somesuitable suitable$, $, 71, 71,70, 70,73, 73,7*3, 7*3,the theclaim claim directly directly norm ]|[-]|[ ]|[-]|[ norm n are in view view of of the theabove above reasoning. reasoning. follows from fromthat thattheorem theorem follows in Belowwe wewill will show showthat thatthe theneeded neededconditions conditions ofTheorem Theorem4.1 4.1are areverified verified of Below for 70 70 == max{l max{l ——#,71}, #,71}, 73 73 == 0,0,and andforfor same ,, 71, 71,7*3 7*3asasinin for thethe same thethe statementof of the thepresent present theorem, theorem, which is sufficient sufficient to to conclude concludethe theproof. proof. statement which is In the the matter matter of of the theconditions conditionson on the theoperators operators2l 2ln and and Q5 Q5 givenby by In n given (12.4) and and (12.5), (12.5), respectively, respectively, their verification canbe bedone done by by using usingthe the (12.4) their verification can resoningemployed employed in the theproof proof of ofTheorem Theorem6.12, 6.12,choosing choosingsuitably suitably and resoning in |||||| || n and Il;n. III Il;n. Note further furtherthat that in virtue virtueof of (12.8) (12.8)the theoperator operatortyty specifiedby by (12.7) (12.7) Note in H:i specified H: fulfils condition condition(4.14) (4.14) with 7rM) = CK CKn+ | * 3; nn = = |il* |il* \*-t \*-tn+1 and 7r |||||| |*3 fulfils with M)/ = n+1, and n+iti, \h;n— —IIII II-II-Although Althoughby by Theorem Theorem6.1* 6.1*the the seminorm seminorm|||||| §*3 §*3;n;n forms forms aa III ' \h;n Aj (*;7*3)-concordant 7*3)-concordantpair pair with +i> 2l not notwith with 2l 2ln , it it is iseasy easy to tosee, see,using usingthe the Aj (*; with 2ln+i> above techniques, techniques, that Theorem 2.11remains remainsin in force force even evenif if the theseminorm seminorm above that Theorem 2.11 in its itsstatement statementis is assumed assumedto to form form aa A^(*;£*)-concordant A^(*;£*)-concordant ]|]|[-]|[* [-]|[*;n in pairpair withwith 4.1, whose proof uses Theorem 2.11, the operator operator2l2l +iHence Theorem the +iHence Theorem 4.1, whose proof uses Theorem 2.11, n still applicable applicablein in the thecase case where wherethe the pair pair (||| (||| |||*3 |||*3;n;2ln-i-i) n;2ln-i-i) is is Ai(*;j^)Ai(*;j^)is still concordant.It It is isalso also obvious obviousthat that thepair pair ((|| | 33; nn; 2l 2ln ) is is A^(0)-concordant. A^(0)-concordant. concordant. the of the theabove aboveobservations observations it remains remainsto to establish establishthat that(4.15) (4.15) and In viewof it and In view (4.16) hold holdwith with73 73 == 00and andwith with «;„+!,/ «;„+!,/substituted substituted for nnnj. To Tothat that end, end, (4.16) for since 7*3<< 1,1,by bya a simplecalculation, calculation, changing theorder order of of summation summationand and since 7*3 simple changing the

250

CHAPTER CHAPTER 12. 12. INTEGRO-DIFFERENTIAL INTEGRO-DIFFERENTIAL

EQUATIONS EQUATIONS

using (12.2) we have, for for all all titi < < ttm < ts < tn, that s-l n-1

nn--11

j=m q=j

q=m q=m n-1

min{<j,s-l} min{<j,s-l}

£ (« j=m j=m min{g,s-l}

*93

s-l s-l

£

and s—1

s—1

m—1 m—1

s—1 s—1

m—1 m—1

j=m

j=0

j=0 j=0

j=0 j=0

j=0 j=0

Combining both these estimates, estimates, itit follows follows that that condition condition (4.15) (4.15) isis satisfied satisfied established for 73 = 0, with K n+ i^ substituted for 7rnij. The same is similarly established in respect of (4.16). So the proof is complete. complete. Theorem 12.2. Let the the operator operator A(t) A(t) and and seminorms seminorms | | |® |®;< and \ | m;t , m = 0,1, *0, satisfy the the conditions conditions of of Theorem Theorem 6.13 6.13 for for some some ifif ££ (0,TT/2) ; [0,1) and and € (0,1] such that-y + ^ = a e E, 7 e ,7,7i,7*,^.,i9*. € [0,1) 1> 7i + 7* < 1, i9 > i9. i9. ++ i9*«. i9*«. iieeii ako ako seminorms seminorms << and and | | |*|*;t form respectively (£|y, c)- and and (*; (*; 7*3|y, 7*3|y, o^)-concordant o^)-concordant pairs pairs with with A(t) A(t) for for some some £ € [0,1 - 7* + min{t?,t?i}) min{t?,t?i}) and and 7 + 3 € [0,1 + min{tf - 7 , ^^ -- 77ii..--££}}))-Suppose further that the the conditions conditions (12.8) (12.8) and and (12.2) (12.2) hold hold for for the the operator operator P(t, s) and for the quadrature quadrature formula formula (12.1), (12.1), respectively. respectively. Then Then the the solution solution Yn of problem (12.3) satisfies satisfies the the stability stability estimate estimate (12.9) (12.9) in in which, which, if£ if£ == 0,0, the expression on the left-hand left-hand side side can can be be replaced replaced by by }[Yn][tn

The claim is shown by by checking checking the the conditions conditions of of Theorem Theorem 4.2 4.2 with with the aid of the argument argument employed employed in in the the proof proof of of Theorem Theorem 6.13 6.13 since since the the backward Euler method method is is both both of of type type Vi((p) Vi((p) and and of of type type Vj{G(0,7r/2). Theorem 12.3. Let A(t) A(t) = = AA G G S(ip, S(ip, a) a) for for some some

0 sufficiently sufficiently large large and and some some ££ GG (0,1], the operator P(t,s) = (A + H\I)~ lP(t, s) satisfies the condition, condition, for for allt,ti,s allt,ti,s G G [0,T] [0,T] such that s < min{£,£i}, min{£,£i}, )\\ + \t-t1\-<\\(P(t,s)-P(t1,s))\\
(12.11) (12.11)

12.1. STABILITY OF DISCRETIZATIONS DISCRETIZATIONS

251 251

Then, for Yn the solution of problem (12.3) (12.3) we we have, have,for for all allkk €€ (0,K\ (0,K\ and and tn £ f2/-, that

Proof. So far as the backward Euler Euler method method isis -A/()-stable -A/()-stable for for any any ip ip££ (0,7r/2), by Theorem 6.2 it follows follows that that the the operator operator 21 21 == A(I A(I ++ kA)~ kA)~l satisfies the A^-condition. A^-condition. Therefore, Therefore, using using the the estimate estimate (12.10) (12.10) which which isis obviously valid in the considered considered case, case, the the claim claim follows follows by by Theorem Theorem 4.3 4.3ifif we assume that condition condition (4.22) (4.22) holds. holds. It thus remains to verify (4.22). (4.22). Note Note that that in in view view of of the the identity identity

(21 + /zo/)~V + kA)- 1 = ((1 + it suffices to show (4.22) with with K Kn+itiP(tn+i,ti) substituted for ^p ^pn,i- By the obvious inequality \\Kq+l,lP{tq+i,ti) - K n>lP(tn,tl)\\

<

\K q+hi-Kntl\\\P(tq+i,tl)\\ +Kn,l||£(*9+l,*i)--P

taking (12.11) into account account now now gives, gives, for for all all titi
Finally, using actually the the same same reasoning reasoning as as in in the the proof proof ofof Theorem Theorem 12.1 12.1 , we when proving (4.15) with with 73 73 == 00 and and with with KKn+\j substituted for conclude that the expression expression on on the the right-hand right-hand side side of of the the last last inequality inequality isis p bounded by Ck log(T/k) + Ci s _ m . This completes the proof. Now we also discuss, for for the the sake sake of of simplicity simplicity in in the the case case A(t) A(t) == AA only, the Crank-Nicolson discretization discretization method method whose whose formal formal application application toto problem (11.33) then yields, yields, with with tn+\/2 — tn + k/2,

Y n) = A n+1(P(tn+1/2,-)Y) Y0 = y°.

+ f(

252

CHAPTER CHAPTER 12. INTEGRO-DIFFERENTIAL INTEGRO-DIFFERENTIAL

EQUATIONS EQUATIONS

Since under the below restrictions the operator I+^A has a bounded inverse, we take this method in the following form, with £2 = (J + I-A)" 1 , 2t = AQ,

Fn = Qf(tn+l/2),

and P{t,s) = QP(t,s),

n

1=0

Yo

= y°.

(12.12) (12.12)

Clearly, (12.12) can be equivalently rewritten in the form (4.1) with 2l n = 2t, and with the same Fn as here. tynl = nn+ijP(tn+i/2,ti), We present only one stability result for problem (12.12). T h e o r e m 12.4. Let A(t) = A ee S(ip, a) for some

,o~)that seminorms concordant pairs with A for some £, 7*3 G [0,1/2] subject to £ + 7*3 < 1 and that the newly defined operator P(t,s) P(t,s) satisfies satisfies condition condition (12.8). (12.8). Then Then the solution Yn of problem (12.12) is under the stability estimate (12.9), (12.9), with with /(fj-1/2) substituted for f (tj). Furthermore, the expression on the left-hand side of this estimate can be replaced by \\Yn\\ in the case where £ = 0 and

I-I— 11-11Proof. The result follows by applying Theorem 4.2 whose conditions, in the case 2l n = 21 now considered, are checked with the aid of Theorems 2.1 and 2.2. At the same time the fact that the last two theorems are applicable can be established by using Theorems 6.5 and 6.5* since the Crank-Nicolson method method is Am(<£>)-stable for any ip G (0, TT/2) and fulfils the condition deg[a(-) — a(oo)] — —1. D D

12.2

Quadrature rules rules

Here we give examples of quadrature rules that satisfy satisfy our above assumptions. The most obvious choice is the rectangle rule for which KUII — k for */ < tn-i- A drawback of this method is that all already computed values values of the solution are involved in further calculations so that all of them have to be stored for future use. Following the philosophy of Sloan Sz Thomee [147] and Zhang [168], we now turn to describing certain sparse sparse quadrature quadrature rules rules for which the storage requirement is really reduced.

12.2. QUADRATURE RULES RULES

253 253

We begin with a quadrature quadrature formula formula based based on on the the trapezoidal trapezoidal rule ruleon oninintervals of length O (fc 1/2), with a slight modification modification near near ttn. More precisely, with v — \k~~xl2\ and k\ = vk, we put fy == lk\, lk\, and and let let lln be the largest integer with t; n < tn. For the interval [0,t n] we then apply the composite composite trapezoidal rule with stepsize trapezoidal stepsize k\ k\ on on [0,t;n], then the one-interval trapezoidal rule on [fyn,£n-i]i a n d finally the left side rectangle rectangle rule rule on on [tn-\,tn}. We therefore come to the quadrature quadrature formula formula

+ \ (tn-l - */„) (^(*n-l) ++ (*/J) (*/J) Since the rule is second order order in in k\ k\ over over [0,ij [0,ijn] and [fyn,£n_i], and first order on the the whole. whole. Observe Observe that that KKn>i < fc1/2 on [tn-i,tn], it is first order in k on for ti < t n-i, and it is easy to see see that that (12.2) (12.2) holds holds for for pp == 1/2. 1/2. The The number of time levels that that enter enter the the computation computation isis OO (&"" (&""1/2) for this rule, rectangle rule. rule. as compared with O (k" 1) for the standard rectangle Developing the idea of reducing reducing the the storage storage requirement, requirement, we we set set vv— — following. We We first first use use [£T1/4J and k 2 = vzk = O (k 1^) and do the following. Simpson's rule on as many many intervals intervals of of length length 2&2 2&2 that that can can be be fitted fitted into into [0,t n _i], and then, on the remaining remaining interval, interval, which which isis of of length length atat most most O(A;1//2), the composite trapezoidal trapezoidal rule rule on on as as many many intervals intervals of of length length trapek\ = u2k = O (&;1/2) as fit in, thus reaching titin, then the one-interval trapezoidal rule on the interval interval [i; n ,t n _i], and finally the left rectangle rectangle rule rule on on [tn-\, tn\. Similarly to above, this this quadrature quadrature rule rule isis first firstorder order on onthe thewhole, whole, while (12.2) holds for p — — 1/4. 1/4. Note Note that that the the number number ofof time time levels levels that that need to be stored per unit unit time time isis now now O O (k^ (k^1) + O {k^k^) + 1 — O (A;" 1/4). Also, it is not hard to construct construct quadrature quadrature rules rules which which further further reduce reduce the the storage requirement. We remark that our considerations considerations in in Section Section 12.1 12.1admitting admitting quadrature quadrature rules that reduce the storage storage requirement requirement are are related related to to Bakaev, Bakaev, Larsson, Larsson, and Thomee [40], where condition condition (12.2) (12.2) appears appears for for the the first first time. time. In conclusion, note that certain certain quadrature quadrature rules rules with with OO (fc~ (fc~1//2) time levels that are needed for for the the computation computation are are presented presented in in [147] [147]and and [168], but the techniques used in in these these works works do do not not allow allow one oneto to deal dealwith with further further reducing the storage requirement. requirement. At At the the same same time, time, for for instance, instance, for for the the Crank-Nicolson method the the results results of of [147] [147] and and [168] [168] are are applicable applicable under under weaker conditions on the kernel kernel P(t, P(t, s) s) than than in in the the preceding preceding section. section.

This Page is Intentionally Left Blank

Appendix A

Functions of Linear Operators Here we collect some facts facts of operator theory which are in use in the course of our main considerations. considerations. As As therein, therein, we we let X be a complex Banach space and accept all the attendant notation introduced introduced in the book.

A.I

Analytic continuation continuation of the resolvent

In this section, given a linear operator £, we denote by p(£) the resolvent set of £. It is well known that if £ is a linear operator and if A G p(£), then p(£) contains as well some open disk centered centered at A (cf. Hille & Phillips [84, TheoTheorem 5.8.2]), which can be shown by analytic continuation. As a consequence of this fact, we obtain that that p(£) p(£) is, in fact, an open set. For our purposes in the main parts of the book, we need, however, to have at our disposal some quantative estimates accompanying accompanying the procedure of analytic continuation.

Theorem A.I. Let £ be a linear operator and let fi G p(£), with Then, for any A G C such that |A — //| = r < M, there exists an operator 6 (jit, A) G B(X) which is permutable with the resolvent of £, obeys the equality

and satisfies the estimate

255

256

APPENDIX APPENDIX A. FUNCTIONS OF LINEAR

OPERATORS OPERATORS

As a consequence, it holds

\\(\i - nr'w < 1 M -rM Proof. The claim follows when inserting inserting

into the Taylor expansion of the resolvent

with the aid of a standard argument (cf. Hille & Phillips [84, Section 5.8]).

Now the last result is used for analytic continuation of the resolvent of a sectorial operator. ^ A . 2 . Let £ e <S(M; tp, 0) for some

Theorem A.2. and let
( ^ ^cos(ip -

v

and satisfies the estimate <

1 1 — M sin(tp —

As a consequence, it holds, for all X <

- Msm(ip —

Proof. Let A € C be arbitrary with argA = ip\. Then, for fj, = cos/ _i we have |/x - A| = |//| sin(

A.I.

ANALYTIC

CONTINUATION CONTINUATION

OF THE RESOLVENT

257

since £ G S(M\ ip, 0) and arg /J, — (p, by Theorem A.I we therefore find, with A with arg A =
where

Now, in view of the inequality |/z| > |A|, the claim follows at least for all A It is however with arg A = ip\ and, by symmetry, for all A with arg A = clear, using the above reasons, that the same assertion remains remains valid valid for all A £ Int E m as well. applications of the above results. Now we consider some applications L e m m a A . I . Suppose that £ G S( 0 suc/i i/iai /or all A ^ ^ ( S ^ j U

(A.I) In the case a > 0, (A.I) holds with R = 0. Proof. By Theorem A.2 there exists a ipi G (0, ip) such that -l

if A—a ^ Int Y, 0 sufficiently large to be confident that (A.I) holds for A ^ Int ( E ^ U V(R)). The more precise wording wording in the case a > 0 is obvious. L e m m a A.2. Let £ G S( 0, and m G N N suc/i i/iai £ < m. T/ien i/iere exist a ip\ G (0, 0 suc/i i/iai for all A £ I n t ( 2 ^ U V{R)) and u G X,

In the case a > 0 f/ie above holds with R — 0. Proof. The claim is established in a manner similar to that used in the proof of Lemma A.I. In fact, it suffices to mention that it follows from Theorem A.2 that with some tp2 tp2 G (0, ip), for arg(A — a) = (p%, arg(/x — a) — ip, and \X — a\ = \/J, — cr\ cos(tp —

(XI - £ ) ~ m - (fil - £)- m ©(A) m ,

258

APPENDIXA.A.FUNCTIONS FUNCTIONS LINEAR OPERATORS APPENDIX OFOF LINEAR OPERATORS

where where ||<5(A)||
A.2

Functionsof bounded bounded and and sectorial sectorial operators operators Functions

We now now discuss discusssome somemeans meanstotodefine define functions of linear operators; functions of linear operators; usingusing such functions functions isisananessential essentialingredient ingredient techniques in the present such of of ourour techniques in the present book. For For our our purposes purposesweweemploy employa simplified a simplified notion of contour which book. notion of contour which differs slightly slightlyfrom fromwhat whatisisusually usually accepted in the theory of functions differs accepted in the theory of functions of a of a complex variable. variable. More Moreprecisely, precisely, a contour mean a simple piecewise complex byby a contour we we mean a simple piecewise smooth curve curve inin the thecomplex complexplane, plane,which which either is finite of finite length smooth either is of length and and closed or or passes passesthrough throughoo. oo.It Itis isaccepted accepted book a contour closed in in thethe book thatthat a contour is is always oriented oriented counter-clockwise counter-clockwiseor ordownwards downwards is finite or infinite, always if itif isit finite or infinite, respectively, unless unlessititisisspecified specifieddifferently. differently.Instead Instead of contour sometimes respectively, of contour sometimes say on on equal equalterms termsintegration integrationpath. path. we say first ££ ££ B(X) B(X) and andletletE Ebebea finite a finite contour such Let first contour such thatthat the the openopen set set of which which 55 isisthe theboundary, boundary,contains contains spectrum of Let £. Let further A, of thethe spectrum of £. further /(A) /(A) 1 A. Then Then complexfunction functionthat thatis isholomorphic holomorphic a neighbourhood of A. be aa complex in in a neighbourhood operator / /((££) ) given givenbyby the operator

^L

££))d lA

~

(A2) (A

-

denned and and bounded boundedononX X(see, (see,e.g., e.g., Hille & Phillips Chapter is denned Hille & Phillips [84,[84, Chapter V]). V]). Furthermore, the thefollowing followingimplications implications hold, ai,a: C, Furthermore, hold, for for ai,a: 2 £ C, /(A) == 11=* =*/(£) / ( £ ) ==I,I, /(A)

(A.3) (A.3)

/(A) A=» =»/(£) / ( £ ) ==£,£, /(A) == A

(A.4) (A.4)

a2 / 2 (A) =* =*
+ aa2 / 2 (£), (£), +

g(£) (£). g(£) = = /i(£)/ /i(£)/2 (£).

(A.5) (A.5) (A.6) (A.6)

Apart from from these theseproperties, properties,there there exist some other premises allowing Apart exist some other premises allowing us tous to think of of the the operator operator/(£) / ( £ )given givenbyby(A.2) (A.2) a function of £. above think as as a function of £. TheThe above representation thethe Dunford operator calculus formula. representation (A.2) (A.2)isisoften oftencalled called Dunford operator calculus formula. Next defining functions of closed unbounded Next we we consider considerpossibilities possibilitiesofof defining functions of closed unbounded operators. it it suffices to deal withwith sectorial oper-operoperators. Actually, Actually,for forour ourpurposes purposes suffices to deal sectorial ators. ators. x

neighbourhoodof aa set set Q QCCCCwe wemean mean any any open open set containing containing C\Q. C\Q. By a neighbourhood

A.2. BOUNDED AND SECTORIAL SECTORIAL OPERATORS OPERATORS

259 259

Let now £ G <$(, c) for some some < € (O,tp), with some Ci/> > 0 depending depending possibly possibly on ontp, tp, /(A) is holomorphic in a neighbourhood neighbourhood of of £
(A.7) (A.7) (A-8)

|/(A) - /(oo)| < C(l + |A|)-^

(A.9) (A.9)

for all A e E ^ + a.

By Lemma A.I there exist exist aa ipi ipi G G(0, (0,> 00such such that that (A.I) (A.I) holds. holds. Then, letting, for any 6 > > 0, 0, E$ E$be be the the contour contour given given by by3$ 3$ == 9(E 9(E Vl owing to (A.I) and to the the accepted accepted conditions conditions on on /(A) /(A) the the expression expression

/(£) = /(oo)/ + ^ r j [

(/(A) (/(A) -- /(oo)) /(oo)) (A/ (A/ -- £)£)-xdA

(A.10)

makes sense and defines aa linear linear bounded bounded operator operator on on XX for for any anyfixed fixedee>>00 subject to i? + e > 0. It is is easy easy to to show, show, using using Cauchy's Cauchy's Theorem Theorem (see, (see,e.g., e.g., Hille & Phillips [84, Theorem Theorem 3.11.1]), 3.11.1]), that that the the definition definition ofof /(£) /(£) does does not not depend on e. Moreover, by by Cauchy's Cauchy's Theorem Theorem there there exist exist other other deformations deformations of the integration path in in (A. (A.10), 10), which which are are tolerable tolerable as as far far as asitit isisregulated regulated by the accepted conditions conditions on on ££ and and /(A). /(A). Further, Further, by by standard standard means means (cf. Taylor [160] or Hille & & Phillips Phillips [84, [84, Chapter Chapter V]) V]) itit isis established established that that the properties (A.3), (A.5), (A.5), and and (A.6) (A.6) are are still still in in force force while while (A.4) (A.4) isis now now absurd for the reason that that formula formula (A. (A.10) 10) involves involves bounded bounded functions functions only. only. In principle, following Taylor Taylor [160] [160] one one could could show show that that formula formula (A.10) (A.10)pospossesses all the features that that are are intrinsic intrinsic in in an an operator operator calculus, calculus, dealing dealingwith with the class of functions /(A) /(A) with with the the above above stated stated properties properties (A.7) (A.7)—(A.9). —(A.9). Fortunately, there is no necessity necessity in in showing showing this this fact fact because, because, for for the the aims aims of this book, it suffices to to treat treat the the algebra algebra of of functions functions which which isis constituted constituted by e~x and all suitable rational functions. functions. At At the the same same time, time, ifif /(A) /(A) isisaararational function, using Cauchy's Cauchy's Theorem Theorem yields yields that that formula formula (A.10) (A.10) defines defines /(£) just the same as Taylor's Taylor's operator operator calculus calculus formula formula (see (see [84, [84, Formula Formula (5.11.2)]), while in the case the operaoperacase /(A) /(A) == e~ e~x, applying (A.10) defines the £ tor e~ in conformity with semigroup semigroup theory theory (see, (see, e.g., e.g., Reed Reed &: &:Simon Simon[133, [133, Section X.8]). Nevertheless, Nevertheless, we we remark, remark, following following [160], that if /(A) satisfies (A.7) —(A.9) and if with with some some m mG GN, N, there exists a finite limit g(oo) g(oo) == lim lim (/(A) (/(A) A Am),

(A.11)

|X|-.oo,

for any fixed ip G (0, ip), it then holds, with g(X) = = /(A)A /(A)Am, for u G Dom£ m , = /(£)£ m,

(A.12) (A.12)

260

APPENDIX A.A. FUNCTIONS FUNCTIONSOF OF LINEAR OPERATORS OPERATORS APPENDIX LINEAR

where and 1, to to ourour assumptions, the the constant constantMMvaries variesconsiderably considerablyafterwards, afterwards, cannot assumptions, wewe cannot nownow Lemma A.I A.I ininorder ordertotoobtain obtain(A. (A. 10).AtAtthe the same time, if we assume apply Lemma 10). same time, if we assume addition to to the the conditions conditions (A.7) (A.7) (A.9) (A.9)that that£ £G G S(M;ipi,a) with with in addition S(M;ipi,a) with the the same same >max{—a, max{—a, even if the then formula 0}0} even if the sizesize of of varies considerably; considerably; however, however,the theinfluence influenceof ofMMclearly clearly appears M varies appears in in anyany estimates involving involvingthe thenorms normsofoffunctions functionsofofthe theoperator operator Note estimates £. £.Note alsoalso thatthat the new new assumption assumptionon on££the theadditional additionalcondition condition(A.(A. implies, under the 11)11) implies, as as above, (A. (A.12). 12). above, the next next section sectionwe weconsider consideralso alsoother otherfunctions functionsof of sectorial operators. In the sectorial operators.

A.3 Fractional Fractional powers powersof operators operators Fractional powers are areusually usuallydefined definedforforsectorial sectorialoperators, operators, however, if the Fractional powers however, if the operator in in question questionisisunbounded, unbounded,one onethen thenmeets meets certain technical difficuloperator certain technical difficul(see, e.g., e.g., Komatsu Komatsu [93]). [93]).AtAtthe thesame sametime, time,forforthethe aims book, ties (see, aims of of thisthis book, suffices to to treat treat the thepositive positivefractional fractionalpowers powers bounded sectorial operait suffices of of bounded sectorial operaand certain certain negative negativefractional fractionalpowers powersofofsectorial sectorial operators possessing tors and operators possessing a bounded bounded inverse, inverse, which whichfacilitates facilitatesour ourconsideration. consideration. (0,TT/2). In principle, principle, there there first ££ G GS(tp,0) S(tp,0) flflB(X) B(X) for forsome someipipGG(0,TT/2). Let first several ways ways equivalent equivalenttotoeach eachother otherthat thatdefine definethethe fractional powers are several fractional powers 0. For For the the sake sakeofofbeing beingdefinite, definite,wewe take definition suggested £^, £ >> 0. take thethe definition suggested by by Balakrisnan [45] [45] (cf. (cf. also alsoKomatsu Komatsu[93]), [93]),avoiding avoidingthethedifficulties difficulties that Balakrisnan that areare proper for for unbounded unbounded operators. operators. More Moreprecisely, precisely,given given £ G — l,m)for for proper £G (m(m — l,m) N, the the operator operator £^ £^isisintroduced introducedbyby m G N,

=

J_ / A^ m £ m (AI -

£)-x dA, £)-

(A.13) (A.13)

where H H == dY.^, dY.^, while while£^ £^isisdefined definedininthe thestandard standardway way is an integer. where if £if is£ an integer. 2 then follows follows from from [45, [45,93] 93]that that It then £*£" == ££€+r)

0. for f,f, r/r/ >> 0.

(A. 14) (A. 14)

significant that that since since££isisbounded, bounded,bybyCauchy's Cauchy's Theorem conIt is significant Theorem thethe contour E E in in (A.13) (A.13) can canbe bereplaced replacedbybya asuitable suitableclosed closed (that finite) contour, (that is, is, finite) contour, 2

B(X). We take take into into account accountthat that£ e B(X).

A.3.

FRACTIONAL

POWERS POWERS OF OF OPERATORS OPERATORS

261 261

sufficiently for instance, E can be replaced replaced by by EL EL == <9(£ <9(£vn£>(L)) with L > 0 sufficiently large. Now let £ e S((f,0) for for some some

/ \ 2m JE

where the contour S is is defined defined the the same same as as above. above. Also, Also, itit isis proved proved inin aa 3 standard way (see, e.g., e.g., Krein Krein [96, [96, Chapter Chapter I,I, Theorem Theorem 5.1] 5.1] ) that

£-«£-*? = £-K+»7)

for for

£, 1£, T) > 0 such that £ + r? < 1-

(A.16) (A.16)

In principle, £~^ is defined defined for for any any ££ >> 00 and and the the property property (A.16) (A.16) isis then then valid for any £, r\ > 0, but but we we do do not not deal deal with with such such an an extension. extension. It is clear that the integral integral representations representations (A. (A.13) 13) and and (A.15) (A.15) are are related related to the operator calculus calculus formulas formulas (A.2) (A.2) and and (A.10). (A.10).4 Moreover, it appears that fractional powers are are included included in in the the algebra algebra of of functions functions ofof aa bounded bounded sectorial operator, as stated stated in in the the following following result. result. T h e o r e m A . 3 . Let £ G G 5(^,0) 5(^,0) nn B(X) B(X) for for

0 be sufficiently large so that that the the spectrum spectrum oo//££ belongs belongsto to the the set set Ajr, Ajr, == Int Int ((SS^^nn T>(L)) U {0}, and let /(A) /(A) be be holomorphic holomorphic in in aa neighbourhood neighbourhood ofof A^. A^. Then Then it holds, with Hj, = dAi, dAi, for for any any fixed fixed ££ >> 0, 0,5

In fact, this is established established with with the the aid aid of of the the representations representations (A.2) (A.2) and and (A.13) by using a standard standard argument argument (see, (see, e.g., e.g., Krein Krein [96, [96, Chapter Chapter I,I, Section Section 5]). We state also the so-called so-called convexity convexity inequality. inequality. 3 It is assumed in [96] that the the operator operator in in question question has has dense densedomain. domain. This This assumption assumption is not really needed for showing showing (A.16). (A.16). 4 The difference is that the the integrand integrand in in (A.13) (A.13) and and (A.15) (A.15) isis unbounded unbounded (at (at 0), 0), asas a function of A. It is easily easily seen seen that that the the corresponding corresponding integrals integrals nevertheless nevertheless converge converge under the accepted conditions. conditions. 5 In principle, HL may be replaced replaced by by another another closed closed contour, contour, among among those those which which are are obtained by a deformation deformation of of EL EL within within the the bounds bounds of ofpossibility, possibility,as asregulated regulatedby byCauchy's Cauchy's Theorem.

262

APPENDIX APPENDIX A. A. FUNCTIONS FUNCTIONS OF OF LINEAR LINEAR

OPERATORS OPERATORS

Theorem A.4. Let £ <<==S(
for all u EX.

(A.18) (A.18)

If instead of £ € S(3£) i£ i£ is is assumed assumed that that £~ £~ x € 23(3£), £/ie iosf estimate estimate holds holds rf 7 with £""£ and £~ in p/ace o/£^ and fi *, respectively, where £ + 77 < 1. For a proof, see, e.g., Krein [96, Chapter Chapter I,I, Theorem Theorem 5.2]. 5.2].

A.4

Functions of ordered ordered non-commutative non-commutative operoperators

To meet our needs it suffices suffices to to discuss discuss the the subject subject for for bounded bounded operators operators only. Let therefore £ i ,,££22 €€ B(X) B(X) and and let let Si Si and and HH2 be closed contours surrounding the spectrum spectrum of of £1 £1 and and that that of of £2, £2, respectively. respectively. Let Let further further 2 /(A2, Ai) be a holomorphic function function on on aa connected connected set set in in CC containing £2 x Hi in its interior. By By the the Dunford Dunford operator operator calculus calculus formula formula (A.2), (A.2), one one can define the operator operator )= - ^ /

2TU J

/(A 2,A)(A 2/-£2)-1dA2.

It is readily ascertained ascertained that that /(£2,A) /(£2,A) isis aa (#(£))-valued (#(£))-valued holomorphic holomorphic funcfunction of A on a connected connected set set containing containing Hi Hi in in its its interior. interior. This This fact fact further further yields that the expression expression

£i)]] -

J

/(£ 2,A)(A/-£1)-1dA

(A.19)

makes sense and defines defines aa linear linear bounded bounded operator operator on on X. X. 2

1

Definition A . I . The operator operator [[/(£2,£i)]] [[/(£2,£i)]] given given by by (A.19) (A.19) isis called calledaafuncfunction of ordered operators. operators. In fact, the above notation notation arises arises from from Maslov Maslov [111]: the indices over operators show the order order in in which which they they act act while while autonomous autonomous brackets brackets ]] are employed to restrict restrict the the region region where where this this symbolism symbolism isis valid. valid. It is easily checked that that if, if, with with Vj Vj CC C, C, jj — —1,2, 1,2, some some open open disks disks cencentered at 0 and containing containing Hj, Hj, jj = = 1,2, 1,2, respectively, respectively, /(A2, /(A2,Ai) Ai) isis holomorphic holomorphic in the bidisk T>i x T>\ so that /(A2)A!)=

A.4. FUNCTIONS FUNCTIONS OF OF ORDERED ORDERED OPERATORS OPERATORS

263 263

then the expression expression oo

JlJ2=0

defines a linear linear bounded bounded operator operator on on XX which whichcoincides coincideswith withthat thatgiven givenbyby (A.19).

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Appendix B

Linear Cauchy Problems Problems in in Banach Space Here we give a brief report of introductive facts from from the theory of linear abstract differential equations. equations. Let there be given a complex Banach space X, an interval [0, T], a linear (generally, unbounded) operator operator A(t) : D o m i ( t ) C X —> X depending on t e [0,T], a function /(£) : [0,T] —> X, and a vector y° G 36. We consider the following Cauchy problem problem yt + A(t)y = f(t)

forO
y(0) y(0) == y°. y°.

(B.I) (B.I)

By a classical solution of problem (B.I) we mean any function y(t) : [0, T] —> X that is strongly continuously differentiable differentiable in (0, T] and strongly continuand fits the ous at 0, possesses the property y(t) € Dom A(t) for t G (0,T] and differential equation and the initial condition in (B.I). For brevity, we say everywhere in the book solution instead of classical solution. In the parabolic case, which is under consideration in this book, the unique solvability of problem (B.I) is extensively examined in the literature. For the earliest results, we refer to Sobolevskii [148] and Tanabe [159], where the situation when A(t) has dense, constant 1 domain is studied. In the case of variable or nondense domains, more more or less recent results are contained in Amann [9] or Lunardi [108]. However, for showing convergence, higher regularity results results are needed, about which not much is known, particTherefore, in the general case with ularly for variable or nondense domains. Therefore, a variable operator, in order to avoid quoting isolated results results of higher reglr

That is, independent of t 265

266 266

APPENDIX B. CAUCHY CAUCHYPROBLEMS PROBLEMS INBANACH BANACH SPACE SPACE APPENDIX B. IN

toorestrictive restrictiveconditions, conditions, weassume assumethe theunique unique ularity,posed posed often under too we ularity, often under solvability ofproblem problem(B.I) (B.I) and muchregularity regularity asneeded. needed. solvability of and asas much as Howeverwe weact actdifferently differentlyin inthe thecase casewhere where theoperator operatorA(t) A(t)isisindeindeHowever the oft22t since sincein in Chapter Chapter77 we wedeal deal with witherror error estimates interms termsof of pendent estimates in pendent of the data.It It isisknown knownthat that if AAG GS((p,a) S((p,a) for forsome some

for t 0, > 0, for t>

(B.2)(B.2)

2-rn7 7H 2-rn H tA makessense sense anddefines defineson onXXaalinear linearbounded bounded operator e~tA whichpossesses possesses makes and operator e~ which all the theproperties propertiesof ofholomorphic holomorphic semigroups, continuity at00 all semigroups, exceptexcept strong strong continuity at unless A has has densedomain domain (see,e.g., e.g.,Sinestrari Sinestrari[146]). [146]). Inthe theautonomous autonomous unless A a adense (see, In weemploy employthe theconcept conceptof ofextended extendedsolution solution andits its development introcase we case and development introduced further. duced further.

Definition B. .I I. .Under Underthe theassumptions assumptionsthatthat AG GS(ip,a) S(ip,a) for forsome some ip ip GG Definition B A (0,TT/2) andaaGG and that f(t) f(t) isisstrongly stronglycontinuous continuous fort tGG [0,T] T ]; the the (0,TT/2) and NN and that for [0, function function ;

((

B B

.

. 3

3)

)

3 is calledthe theextended extendedsolution solution ofproblem problem (B.I). (B.I). is called of

Definition B..22. .Suppose Supposethat that AG GS((p,a), S((p,a), f(t) f(t) isis times strongly strongly continDefinition B A pp times continuously differentiate differentiate uously for fort tGG [0,T], [0,T], and andy?.s y?.sG GDomTl, DomTl,jj = =0 0,,... . ,p ,p . — —1, 1,for for some tpG G(0,7r/2), (0,7r/2),a aG GN, N,and andp pG GN, N,where where some tp y°0) = y° y°and and y°{j) - f^f^l\0) y° y° \0) 0) = {j) -

Ayf^, j =j 11= ,p . -- 1.1. - -Ayf^, ,,... . ,p

(B.4) (B.4)

Then Then the thefunction functiony(t) y(t)given givenby by(B.3) (B.3) isis called calledthe thep-th p-th order orderextended extended solution solution (B.I). of of problem problem (B.I).

It isnot nothard hard to tosee, see,using usingthe theproperties propertiesof ofthe theintegral integralin in (B.2) (B.2)(see, (see, It is e.g., Sinestrari [146]), thep-th p-th orderextended extended solution of(B.I) (B.I)obeys obeys e.g., Sinestrari [146]), that that the order solution of = //^^""^^( O (O Ay^_ the equality, equality, with the with y°{p) = ) )-- Ay^_ {p)y° x), x) s)A {v tA yW(t) =e' e'tA y°{p) + (( ee-{t{t-s)A \s) yW(t) = y° f{v\s) {p) + Jo o 2

ds forfor > 0. (B.5)(B.5) ds t >t 0.

wewrite writetherefore therefore Ainstead insteadof of A(t). A(t). In In this thiscase case we A If y° y°ee Cl(Dom Cl(DomA), A),the the concept conceptof ofextended extendedsolution solution coincides coincides with that withof of that mild mildsolution solution If e.g.,Sinestrari Sinestrari[146]). [146]). however the that extendedsolution solution isdefined definedunder under the (cf., (cf., e.g., NoteNote however that the extended is the acceptedconditions conditions ify° y°££Cl(Dom Cl(Dom A). accepted eveneven if A). 3

267 Observe that it is then not not necessary necessary that that y9 y9 -. G Dom A. Also, it is worth worth noting that the conditions conditions imposed imposed on on f(t) f(t) and and y9.,, y9.,, jj == 00,,.. . . ,p,p — 1, in Definition B.2 express, in in fact, fact, aa certain certain kind kind of of time time regularity regularity for for y(t). y(t). In In particular, the extended extended solution solution of of order order >> 11 isis simultaneously simultaneously aa solution solution of (B.I), as it follows from from Sinestrari Sinestrari [146, [146, Theorem Theorem 4.4]. 4.4].

This Page is Intentionally Left Blank

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Index I, xii LK{TUT2-X),

£*, xiii Hn, 4

147

CIV, 5 IntV, 5 Aj-condition, 6, 19 Aj(M)-condition, 25 Aj-condition, 5, 19 ftfc, 3 T,v, xiii

2n nJ -, 72 £*, xii , xii

L-J,29 , 262 Uit, 179 , 103 5 ^ (V is a set), 5 dZn (Zn is a discrete function), 33 dZn (in Chapter 8), 179 Ran£, xiii p(£) (in Appendix A.I), 255 255 p(z), 96 ^-method, 211 On, 63 a i;n , 63 Un (in Chapter 2), 35 iOn (in Chapter 3), 63 Onj (in Chapter 2), 35 iXy (in Chapter 3), 63 fflinj, 72 $n,;, 76

II-II. xii II ' II*, x i i a{z), 96 fife, 3 U fc , 179

B{X), xii V{r), 5 P(2;r), 5 ;V.,<7), 124 ) , xiii , 213

2ln, 67 %;n, 67 iin, 67 deg[w], 89 Dom£, xiii Dom | |, xiii Yn, 72 D(t), 104

r(z), 95 a, 95 i, 95 b, 95 c, 95 e, 95

T>n, 104

a (in Part II), 103 »(*), 104 284

INDEX

285

v(z), 95 w{z), 95 231 Amplification matrix, 231 Backward Euler method, 100, 100, 248 248 BDF method, 241 Commutator, 54 Contour, 258 Convexity inequality, 261 261 Crank-Nicolson method, 100, 100, 251 251 Discrete evolution operator, operator, 44 Discrete semigroup, 4, 19 19 Equation integro-differential, 245 with memory, 71 Evolution equation in discrete time, 4, 19 Family of grids, exponentially balanced, 185 185 locally balanced, 186 quasi-quasiuniform grids, grids, 186 186 quasi-uniform grids, 187 Fractional power, 260 Function of linear operator, 258 ordered operators, 262 Functional Qn(-), 148 QniO, 148

Gauss-Legendre method, 99 99 Hypothesis ...), 33 ( . . . ) , 33

H

, 33

hol(...), 86 hol[...], 84 hoi®[...], 84 ( ^ ( . . . ) , 85 >(...), 85 hol^[...], 232 holf f c ) [...], 85 lip, 86 lip(...), 86 COM1[...], 54 COM2[...], 54 coml[...], 220 com2[...], 220 Integration path, 258 228 Linear multistep method, 228 of class Mi, 231 of type LM{ip), 231 of type LMj (
Non-uniform grid (partition), (partition), 179 179 Operator (M; ip, cr)-sectorial, 124 (
286

INDEX INDEX

(.. .)-concordant, 6, 19 strictly (.. .)-concordant, 86 86 strictly Aj(.. .)-concordant, .)-concordant, 77 strictly A^(.. .)-concordant, .)-concordant, 77 Parabolic case, xiii differential equations, xiii xiii Parameters of set of Aj-configuration, Aj-configuration, 55 Radau method, 100 RK method, 95 Tl-stable, 96 i4(v?)-stable, 96 .Aj-stable, 97 Aj((p)-stab\e, 97 of class Sj, 97 of typeC(yj), 96 of type Cj(¥>), 97 of type V(
7M, 97 Sectorial estimate, xiii Set Aj-proper, 6, 25 Aj(...)-proper, 7 Aj-proper, 6 A5(...)-Proper, 7 of Aj-configuration, 5 Simplifying assumptions, 98 98 Singular points of operator, 6 set of Aj-configuration, 55 Solution classical, 265 extended, of Cauchy problem, 266 of order p, 266

of ofRunge-Kutta Runge-Kutta procedure, procedure,104, 104, 148, 148, 180 180 of of Cauchy Cauchy problem, problem, 265 265 of of multistep multistep procedure, procedure, C-extended, C-extended, 229 229 ^-extended, ^-extended, 229 229 Sparse Sparse quadrature quadrature rule, rule, 252 252 Stability Stability function, function, 95 95 Testing functional, 85, 232 232 Triplet Triplet M M -- O-concordant, O-concordant, 77 A$(.. A$(.. .)-concordant, .)-concordant, 77