Spectral Methods and Their Applications

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Spectral Methods and Their Applications

Guo Ben-Yu Shanghai University P R China

,.p World Scientific I

Singapore· New Jersey· London· Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd.

USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

POBox 128, Farrer Road, Singapore 912805

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Library of Congress Cataloging-in-Publication Data

Spectral methods and their applications I Guo Ben-yo.

Guo, Ben-yo, 1942p.

cm.

ISBN 9810233337 (alk. paper)

Includes bibliographical references. 1. Spectral theory (Mathematics) QA320.K84

I. Title.

1998

515'.7222--dc21

97-44642

CIP

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Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.

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PREFACE In the past two decades, spectral methods have developed rapidly. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to numerical simulations in various fields. The purpose of this book is to present the basic algorithms, the main theoretical results and some applications of spectral methods. Particular attention is paid to the applications to nonlinear problems. The outline of this book is as follows. Chapter 1 is a colloquial introduction to spectral methods. In Chapter 2, we discuss various orthogonal approximations in Sobolev spaces, used in spectral methods. We also discuss the filterings and recov ering the spectral accuracy. Chapter 3 is a survey of the theory of stability and convergence. Chapter 4 consists of two parts. In t "e first part, we present some basi c spectral methods with their applications t o nonlinear problems. In the second part, we consider the spectral penalty methods, the spectral viscosity methods and the spectral approximations of isolated solutions. Chapter 5 is devoted to the spectral approximations of multi- dimensional and high order problems. T he spectral domain decomposition methods and the spectral multigrid methods are also introduced. We consider the mixed spectral methods for semi-periodic problems in Chapter 6, and some combined spectral methods with their. applications in Chapter 7. T he final chapter focuses on the spectral methods on the spherical surface. I am gratef ul to Prof. Wong, R. S . C. and Dr. Dai, H. H . . T hey organized the Workshop at City University of Hong Kong, which motivated me to write this book. I thank Dr. Caldwell, J. for his help in polishing up the manuscript. I also thank Misters Chan,oH. C. and Cha, K. H. who typed the manuscript. Guo Ben-yu Shanghai University City University of Hong Kong December 1996

v

CONTENTS

PREFACE CHAPTER CHAPTER

2.1 2.2 2.3 2.4 2.5 2.6 2.7

1.

2.

v

1

INTRODUCTION ORTHOGONAL APP . ROX�MATIONS IN SOBOLEV SPACES

Preliminaries . . ; . . . Fourier Approxima tions Orthogonal Systems of Polynomials in a Finit e Interva l Legendre Approximations .' . . . . . . . . . . . . . . . Chebyshev Approximations . . . . . . . . . . . . . . . Some Orthogonal Approximations in Infinite Intervals . Filterings and Recovering the Spectral Accuracy . . -

CHAPTER 3 . STABILITY AND CONVERGENCE

3.1 Stability and Convergence for Linear Prob lems . 3.2 Generalized Stabili ty for Nonlinear Problems . 3.3 Initial-Value Problems . . . . . . . . . . . . . CHAPTER

4.1 4.2 4.3 4.4 4.5 4.6

4.

10

10 '16 24 31 43 56 66

82

82 86 94

SPECTRAL METHODS ANR) PSEUDOSPECTRAL METHODS 100

Fourier Spectral Methods and Fourier Pseudospectral Methods Legendre Spectral Methods and Legendre Pseudospectral Methods . Chebyshev Spectral Methods and Chebyshev Pseudospectral Methods Spectral Penalty Methods . . . . . . . . . . . Spectral Vanishing Viscosity Methods . . . . . Spectral Approximations of Isolated Solutions Vll

100 120 143 152 164 181

Contents

viii

CHAPTER' 5. SPECTRAL METHODS FOR MULTI-DIMENSIONAL AND HIGH 'ORDER PROBLEMS 188

5 . 1 Orthogonal Approximations i n Several Dimensions .

188

5.2 Spectral Methods for Multi-Dimensional Nonlinear Systems .

196

5.3 Spectral Methods for Nonlinear High Order Equations

210

5.4 Spectral Domain Decomposition Methods .

231

5.5 Spectral Mu ltigrid Methods . . . . . . . .

248

CHAPTER 6. MIXED SPECTRAL METHODS

257

6 . 1 Mixed Fourier-Legendre Approximations .

257

6.2 Mixed Fourier-Chebyshev Approximations

265

6.3 Applications . . . . .. . . . . . . . . . . .

273

CHAPTER 7.1

7.

COMBINED SPECTRAL METHODS

280

Some Basic Results in Finite Element Methods . .

280

7.2 Combined Fourier-Finite Element Approximations .

284

7.3 Combined Legendre-Finite Element Approximations .

289

7.4 Combined Chebyshev-F inite Element Approximations .

295

7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . .

300

CHAPTER

8.

SPECTRAL METHODS ON THE SPHERICAL SURFACE 313

8.1 Spectral Approximation on the Spherical Surface

313

8.2 Pseudospectral Approximation on the Sph erical Surface .

318

8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . .

322

References

331

Spectral Methods and Their Appli�ations

CHAPTER 1

INTRODUCTION Spectral methods have developed rapidly in the past two decades. They have been applied successfully to numerical simulations in many fields, such as heat conduction, fl uid dy namics, quantum mechanics and so on. Nowaday s, they are some of t he powerful tools as well as finite difference methods and finite element methods, for numerical solutions of partial differential equations. Th e main f eat ure of th e spectral met hods is t o take various orthogonal sy st ems of infinit ely differentiable global functions as trial functions. Different trial functions lead to different spectral approximations. For instance, trigonometric polynomials for periodic problems, Legendre polynomials and Chebyshev polynomials for non periodic problems, Laguerre polynomials for problems on the hal f line, and Hermite polynomials for problems on the whole line. The fascinating merit of spectral methods is the high accuracy, the so-called con vergence of " infinite order" . For example, we consider numerical solution of the Laplace equation on a cube, and divide the domain into N3 uniform subcubes. If we use linear finite element methods, then th e error in the L 2 -norm between the geiLUine solution and the numerical one is of order N-:- 2 , no mat ter how smooth t he genuine solution is. If we use the standard " nine points" difference scheme with the uniform mesh size if, then the error is of the same order. However, the smoother the genuine solut ion, the higher the convergence rate of the numerical solution by spectral meth ods. I n particular, when the genuine solution is infinitely differentiable, the numerical one converges faster than N-a , a being any positive constant. The basic idea of spectral methods stems from Fourier analysis. In 1 820, Navier 1

Spectral Methods am; Their-"Applications

2

considered a thin plate problem governed by the equation 84ij' 84 U 84 U 2 2 2 + f, + 0 < x, y 8 x8 y 8y4 = 8x4 with the boundary conditions 82 U x = O, 7r , U 8x 2 = 0, 82 U y = O, 7r. U 8y 2 = 0, Suppose that f( x, y) can be expanded as 00

f(x, y) =

<

7r,

(1.1)

( 1 .2) (1.3)

00

L fl,m sin1x sinmy. L 1=1 m= 1

By the orthogonality of trigonometric functions, fl m , =

4 1"1" f(x, y) sin1x sinmy dxdy .

2'

7r

a

a

We look for the solution as 00

00

U(x, y) = L L UIm (1 .4) , sin1x sinmy. 1= 1 m=1 Obviously it satisfies the boundary conditions ( 1 .2) and (1.3). By substituting ( 1.4) into (1. 1 ) and using the orthogonality again, we deduce that fl,m UI m , = 14 + 21 2m 2 +m4 and so , sin1x sinmy. (1 .5) U(x, y) = L L 4 [j,2 /m2 m 2 1 + +m4 /= 1m=1 Finally we truncate this series to obtain the approximate solution N N UN (X, y) = L I.: U/ m , sin1x sinmy 1= 1 m=1 which is a solution of the Fourier spectral method for problem ( 1 . 1 ) - (1.3). Spectral methods can also be applied to evolutionary problems. Consider the heat conduction problem 8U o < x < 7r, t > 0, 8t 8x 2 ' ( 1 .6) U(O, t) = U(7r, t) = 0, t > 0, U(x, O) = Uo(x), O:S;x:S;7r 00

{

00

Chapter

1.

Introduction

3

where Uo(x) is a given continuous function and vanishes at

x =

0, 71'. Let

U(x,t)=L U,(t) sin Ix 00

{

1=1

dU, = z 2 [/;I, dt U,(O) = Uo,,, 1 = 1 , 2, . . . ,

( 1 .7)

and substitute this expansion into (1 .6), we find that -

( 1 8) .

where UO,I is the Fourier coefficie nt of the initial state Uo(x),

2

UO, I = 71'

( 1 .9 )

0

1" Uo(x)sinlxdx.

By ( 1 .8) ,

We now truncate the series ( 1 . 7 ) and denote the corresponding one by UN ( X , t), N UN ( X , t) = L U,(t) sin Ix. 1=1

This is an exceedingly good approximation to ( 1 . 7 ) for any t > O. In fact, IU(x,t) - UN ( X , t)1 =

�

I I=f:N+1 Uo,le-I't sin lx l

2 max lUo(x) I iNroo e-tll'dy; 0$'"$"

Thus the error goes to zero very rapidly as N -+ 00 . There are several kinds of spectral method s, called spectral methods, pseudospec tral methods and Tau methods, respectively. All of them can be derived from the method of weighted residuals. Let n be a spat ial domain with the boundary an, and n = n u an. We consider here an initial-boundary value problem as follows

{

xE n, t > 0, LU=J, xE an, t > 0, BU=O, U(x,O)=Uo(x), xEn

( 1 . 10)

where L is a differential operator and B is a linear boundary operator, J(x,t) and Uo(x) are given functions. Assume that some conditions are fulfilled for ensuring the

Spectral Methods and Th�ir..¥\.pplications

4

existence, un iqueness and regularity of � olution of ( 1 . 10). The method of w eighted res iduals is to find an approxi� ate solution of the' form N (1.1 1 ) UN(X,t) = UB(X,t) + 2:>N,I(t)
where the trial functions
{. in

in (UN(X,0) - Uo(x)) tPj(x)dx = 0,

(LUN(.X,t) - f(x,t)) tPj(x)dx = 0,

o:5j:5N, t > O:5j:5N

9,

( 1 . 12)

where the weight functions tPj(x),0:5j:5 N, are linearly independent. We first derive the spectral methods. In this case, each
(v,w) = Accordingly, ( 1 . 12) stands for

{

in v(x)w(x)dx,

(LUN(t),
Il vll

=

(v,v)�.

o :5 1:5N,t > 0, 0:51:5N.

( 1 . 13)

It is more convenient to describe scheme (1.12) by a projection operator PN. To do this, we set a finite dimensional space VN as VN = span {
{

PNLuN(X,t) = PNf(x,t), UN(X,O) = PNUO(X).

t > 0,

(1.14)

Chapter

1.

Introduction

5

' We next deduce the pseudos pect ral method s. In this case, ¢>/(x) are the same as in spectral methods and UB(X, t) == o. We choose suitable collocation points x(i) , 0 � j � N , such that :f O.

( 1 . 15)

The weight functions are

tPj(x)=S(x - xU»), 0 � j � N where Sex) is the Dirac delta function. Then ( 1 . 12) stands for

{

LUN(XU), t)=f(xU) , t), UN(XU) , O)=Uo(xU»),

O � j � N, t > O, 0 � j � N.

( 1 . 16)

It is also more convenient to express scheme ( 1 . 16) by an interpolation operator IN. For any v E C(fi), the interpolant INv E VN, satisfies

By ( 1 . 15), the interpolant INv is determined uniquely. Thus ( 1 . 16) is equivalent to

{

INLuN(x, t)=INf(x, t), t > 0, UN(X, O)= INUo(x).

( 1 . 17)

There is another way to express pseudospectral methods. F or instance, if xU), are distributed uniformly and if we define the discrete inner product as 1 (V, W)N = __ t v(xU»)w(x(j»), N + � j=O

then ( 1 . 16) and ( 1 . 17) are equivalent to

{

(LUN(t), (p,)N = (J(t), (p,)N' (UN(O), ¢>dN=(Uo, ¢>,)N'

o � I � N, t > 0, o � I � N.

( 1 . 18)

We now turn to the Tau methods. Suppose that ¢>/(x), 0 � I � N are orthogonal in L2(fl), but do not fulfill the boundary conditions. The component UB ( X, t) i n. ( 1 . l l ) is of the form N+k UB(X, t)= L UN,/(t)¢>/(x) /=N+l

Spectral Methods and Their ?ipplications

6 where

k

rPj (x), O

is t he number of independent. bound� y conditions. We take :::; I :::; N. Then i� rea. d as

( 1.12) ), rPl ) = (f(t ), rPl ) , 0 :::; I :::; N,t > 0, { (LUN(t (UN(O), rP') = (Uo, rP, ) , 0 :::; I :::; N

k

'ljJj (x) = ( 1.19)

with additional eq uations given by the boundary conditions. The Tau methods can also be descri bed by a projection operator, namely

{ UN(PNLuN(X, t) = PNf(x,t), X, O) = PNUo(x)

k

t > 0,

( 1.20)

with additional equations on an. In practice, mostly used methods for problems with non-homogeneous boundary conditions are some combinations of pseudospectral methods with Tau methods. It means that partial differential equations are approximated by collocation approach in which 1:::; N, do not satisfy the boundary conditions, and the boundary conditions , are treated as in Tau methods. When we solve initial-boundary value problems nUlIl erically, we need to discretize the derivatives of unknown functions with respect to time Denote by [Tl the integer part of any fixed positive constant Let be the mesh size for the variable and

rPl (X), O :::;

R.,.(T)

T.

r

= {t = kr I k = 1, 2, . . . , [�] } ,

t.

f4(T)

= Rr(T) U { o} .

t ( 1.21)

v ( x, t) = 2"1 v (x, t + r) + 2"1 v (x, t - r), D.,. v (x, t) = �(v (x,t + r) - v (x, t» ) , D.,.v ( x, t) = �(v ( x, t) - v(x, t - r ) ) , D.,. v(x, t) 21r (v(x, t + r ) - v(x, t - r)) , DTTV (X, t) = :2 (v(x, t + ) - 2v(x, t) + v(x, t - r)) . D.,. v and D.,. v approximate �; at time t with the trunc ation error of order O(r), provided that I �:� I is bounded. D.,. v approximates �; with the error O(r2 ) , when We use the following notations

r

Chapter 1. Introduction

I �:� I 04V as that I at4 l is bounded.

7

DT�V approximates 0ot22 V with the error O( r 2), as long . ov at t + T WIt' the truncat'Ion But DTv approxImates ot

is bound�d. Similarly,

error O(

r 2). It can be verified that 2 ( DTv (t), v(t ) ) 2 ( DTv(t), v (t)) 2 ( DTTV(t), D Tv(t))

{

11

'2

DT l v (t) 1 I 2 - r I D'Tv(t ) 1 I 2 , D'Tll v(t ) 1 2 , D'T I D'Tv (t) 11 2 .

( 1.22) ( 1 .23) (1.24)

As an example, we consider the problem at

au = a2u + I, O<x<7r, ON,I (t) sin Ix.

!

1=1

A spectral scheme of Crank-Nicolson type for ( 1 .25) is

( DTUN(t), (h) ( UN(O), ¢>I)

=

(21 02UN (t + r ) + 21 02UN (t) ox 2 ox2 0� I � t E Ji, (T), +I(x,t + � ) ' ¢>I) ' 0� I� ( Uo, ¢>I) , N,

=

N.

( 1 .26) Further, by the orthogonality of trigonometric functions, we obtain from ( 1 .26) that

{ UN,D'TUN,I(O)I(t)= Uo,= -�/ , UN,, (t +. r ) - �UN,,(t) + II (t + t) ,

where

� 1� N, t E R'T(T), � 1� N

o o

UO,I is the same as in ( 1.9) , and II (t + �). = � f I (x, t + �) sin Ix dx.

i1'

Spectral Methods and Their Applications

8

Finally

Therefore we can evaluate the coefficients UN,I(t), 0 :5 1 :5 N, t E .R,.(T) explicitly. This is indeed another advantage of spectral methods. Conversely, if a finite difference scheme of Crank-Nicolson type is used for (1.25), then we have to solve a linear algebraic system for the values of unknown function at each time step. It is also true for any finite element scheme of Crank-Nicolson type. On the other hand, the matrices occurring in finite element methods may weaken the stability of computation. Sometimes, the "lumping" technique is used to overcome this trouble, but it usually lowers the accuracy. Even though the spectral methods have many advantages, they were not used widely for a long time. The main reason is the expensive cost of computational time. However, the discovery of the Fast Fourier Transformation (FFT), see Coo ley and Tukey (1965), removed this obstacle. Let N be the number of terms in one-dimensional Fourier expansion. Then FFT reduces the total number of opera tions from O ( N 2 ) to O ( N log 2 N ) . Especially, FFT saves a lot of work for multi dimensional problems, since trial functions in several dimensions are the products of several trial functions in one dimension. For example, we consider the summation

N N N

� �o NE =

} {. (Nl+P 1 + + 1 + Nnr) +1 . (N . (N ))

a',m,n

� � �e +1 1=0

exp 2?r�

'"' 2NmQ' � e +1 m=O

""'

mq N

�a" m,neN+1 n=O

�

.

If we use the usual method to evaluate the left-hand side of the above formula, then the total number of operations is O ( N6 ) . In contrast, it is reduced to O ( 3N2 1og 2 N ) , if the right-hand side is calculated by F FT. Since Chebyshev polynomials can be changed into trigonometric polynomials by a transformation of independent variables, FFT is also available for Chebyshev spectral approximations. Furthermore, the coming of the Fast Legendre Transformation (FLT) , see Alpert and Rokhlin (1991), and the , possible parallelization of FFT and FLT make the spectral methods more efficient.

Chapter

1.

Introduction

9

The first serious applic ation 'of the spectral methods to partial differential equ a tions was due to' Silberman (1954) . However, they have become practical only af ter Orszag (1969, 1 970) , and Eliasen, Machenhauer and Rasmussen ( 1970) devel oped transform technique for evaluating convolution sums arising from quadra tic non-linearities. The colloc ation approach was first used by Slater ( 1934) and Kan torovich (1934) in specific applic ations, and the foundation of orthogonal colloc a tion was laid by Lanc zos ( 1938). But the earliest applic ation of the pseudospec tral methods to partial differential equations was made by Kreiss and Oliger (1972) , and Orszag (1972) . Lanczos (1938) also developed the Tau methods. Gottlieb and Orszag ( 1977) summarized the state of art in the theory and applic ation of spectra l methods. They provided numeric al analysis for linear problems. H ald (1981 ) , Ma day and Quarteroni ( 1981 , 1982a, 1 982b), and Bernardi, Canuto and Maday ( 19 86) considered nonlinear problems. While Guo (1981a, 1 985) and Kuo (1983) devel oped spectral methods for nonlinear problems with numeric al analysis independently. There have been numerous introductory and review articles, e.g. , Mercier (1981 ) , H ussaini, Salas and Zang ( 1985), Gottlieb (1985) , and Quarteroni (199 1 ) . Some new developments in this field are contained in the books by Canuto, Hussaini, Qu arteroni and Zang (1988) , by Bernardi and Maday (1992a) , and by Guo (1996). On the other hand, applic ations of spectral methods in meteorology have been covered in the review by Jarraud and Baede (1985), and in the book by H altiner and Williams ( 1980) .

CHAPTER 2 ORTHOGONAL APPROXIMATIONS IN SOBOLEV SPACES

The remarkable convergences of spectral methods owe much to the rapid conver gences of expansions in series of orthogonal systems of smooth functions. We present a summary of the relevant theory in this chapter. Fourier approximation, Legendre approximation, Chebyshev approximation, Laguerre approximation and Hermite ap proximation are considered. The error estimations of different projections in Sobolev spac es are discussed. We also introduce the filterings which are used to improve the stability of pseudospectral methods for nonlinear partial differential equations. In order to recover the spec tral acc uracy of orthogonal expansions for discontinuous functions, some essentially nonoscillatory approximations and reconstruction methods are used.

2.1.

Preliminaries

Let IRn be the n-dimensional Euclidean space and x = (Xl,X 2 , , xn ) E IRn. Denote by n a bounded open domain in IRn with the boundary an. If for any x E on, . there exists a system of orthogonal coordinates Y = (yI , Y2,... ,Yn), a hypercube U'" = n;=l [-aq,aq] and a Lipschitz continuous mapping �'" from n;;} [-aq, aq] into [-!an, !an ] such that . • •

nnu", annu",

{y E U"" Yn > �'" (Yl, ... ,Yn-l)} , {y E U"" Yn = �x (Yl,' .. ,Yn-l)} , 10

Chapter 2. Orthogonal Approximations in Sobolev Spaces

11

th en we say that an i s Li pscni tz continuous. Further, let m b e a non-negative integer. A relati� ely open part r E a n i s sai d to be of class cm,1, if for any x E r, we can choose such a mapping �'" that its deri vatives up to the order m are Lipschitz continuous. In thi s book, we assume that an i s at least Li pschi tz continuous, and n = O U ao. Let V(O) be the space of infini tely differenti able functions wi th compact supports in 0, and Coo (n) be th e space of i nfinitely differenti able functions on n. The dual space V'(O) of V(O) i s the space of di stributions on O. By the theory of di stributions (see Schwarz (1966)) , for any v E V'(O), the associ ated parti al derivatives of any order a , and \l = (a1, a2, . ., an). Let kq, 1 � q � n exist. The notati on aq stands for aXq be non-negati ve integers. For any n-tuple k = (kl,k2,· . ., kn), Ikl = k1 + k2 + ... + k n etc . . The symbol a; = a�la;2 ... a�n. When n = 1 , a", = a1. In addition, at = We now introduce the Sobolev spaces. For any real number p, 1 .� p � 00, let .

£P(n) = {v I

with the norm

•

:t '

v

is measurable and IivIiLP(o)

(10

Iv(x}IPdx Iivliv(o) = Iivliv(o) = ess sup lv(x) l,

1

y,

<

oo}

1 � p < 00, p = 00.

"'EO

LP(O) i s a Banach space. In parti cular, L2 ( 0 ) is a Hilbert space equipped wi th the

i nner product (V,W)L2(O) =

10 v(x)w(x)dx,

where w is the conjugate of w, if w is a complex-valued function. Since L2 ( n) contains V(O) and Coo (n) as dense subspaces, the duality pai ring between 1J'(n) and V(O) i s an ext ension of the inner product in L2 ( 0 ) . Next, let m : b e any non-negati ve integer, and set It is provided with the following semi-norm and norm, (2. 1 )

.-

Spectral Methods and Their Applications

12

( 2 .2)

Wm,P(U) i s a Banach space. In particular, wm,2(U) = Hm(u). It is a Hilbert space for the associated inner product (V,W)Hm(o)=

L:

Ikl$m

Vv,wEHm(u).

(8!v,8!w)L2(O)'

In order to define the space Wr,P(U) for any non-negative number r, we need the theory of space interpolation. Firstly, let B be a Banach space with the norm II· liB. For any real numbers a and b, define the Banach space P(a, bj B) = {f I 1: (a, b)

with the norm

-+ B is strongly measurable,

IIIIILP(a,b;B) =

(l

)

lll(t)ll� dt

IIII1LP(a,b;B) = ess sup 111(t)IIB, tE(a,b)

,

P,

P=

III01lB ELP(a, b)}

1 :::; P < 00, 00

where the integral on the interval (a, b) is the Bochner integral. Now let Bl and B2 be two Banach spaces. Bl is continuously imbedded and dense in B2• For any real numbers v and P � 1 , we denote by W(v, Pi B1, B2) the space containing all functions 1 such that tV1 EP(O, OOj B1), This is also a Banach space with the norm IIII1W(v,p;BloB2) = max ( 1It"IIILP(o,oo;Btl, 11t"8dIlLP(o,oo;B2») .

Assume () = � + v < 1 . Denote by (v, Pi B1, B2) the set of initial val ues of functions in W(v,pj Bb B2). It is also a Banach space with the norm

T

infB,.B) 111I1w(v,p;B"B2). IIvIlT(v,p;BloB2) = JeW(v.p; 2 !(O)=_

If 0 < () < 1 , then for any v ET(v,Pi Bl, B2),

v ll1i;;(o,oo;B,)11t"8dll�(o,oo;B2)· , IIvIlT(v,p;BloB2) = J E W(inf 2 IIt ",J', B1. B) !(o)=_

Chapter 2. Orthogonal Approximations in Sobolev Spaces

13

If in additio n v E Bl, then v E T(II,Pi Bl, B2) and Hereafter Cl denotes some positive constant independent of any function v. On the other hand, let Bl be continuously imbedded and dense in B2 as well. If L is a linear operator from Bl into Bl, and for any v E Bl, L,

then there exists a unique continuous extension, denoted by

v E B2,

If in addition 0 < 8 T(v,PiBl,B2) and

<

such that for any

1, then for any v E T(v,Pi Bl, B2), we have that Lv E

IILvIlT(v,p;B"B2) � c�-B c�lIvIIT(v,p;Bl,B.)'

and 0 < 8 < 1 , then T (v,p;Lq(O), LP(O) is 8 +�. continuously imbedded in 1"(O),.!. = P q 'Y Now let 1 < P < 00 . For any integer r . = m � 0, Wr,P(O) = wm,p(o). If r = m +O' and 0 < 0' < then define the Sobolev space Wr,P(O) as As an example, if 1 � P � q

Wr,P(O) =

1,

{v I

v

< 00

E wm,p ( O ) ,a v E

; T (1 -

0'

It is also a Banach space with the norm

(

IIvllwr.P(fl) = IIvlI�m,p(n) +

-�

)

,p, Wl,P(O), LP(O) , Ik/

t: Ila;vll�(l-O'-t'P'WI'P(fl),LP(fl)))

Ik m

This definition can be generalized to the case P = 1 or p = A n equivalent norm i s as follows,

Ilvllwr.p(fl) =

(Ilvll�m,p(n),

(

+

2:

Ikl=m

)

1

00

)P x dy ,

ess sup IIVIlWr,oo(fl) = max I/vl/wm,OO(fl), Imax kl=m z,yEfl .

) -

I x _ Y /n+pO'

%:;49

la;v(x) - a;v(Y) I IX

YI

0'

}

m .

I

P

( see Adams

l a;v(x)-a;v(y) IP d ll fl fl

=

(2.3)

( 1975)).

I

.

1 �p<

00 ,

_

� ' 0, let W;'P(f!) be the closure of V(f!) in wr,p(f!). Then for < p < 00, define the Sobolev space wr,p(f!) by 1 1

For

1

..

Spectral Methods and T� eir Applications

14 r

-

p

+q

-

=l

r <

0 and

.

In particular, wr,2(f!) = Hr(f!). For r > 0, it is a Hilbert space for the related inner product. For r < 0, (v,g) ( 2.4 ) IIvllw(o) = sup .eHor(O) ."' .

It is more convenient to use another notation for the interpolation between two ' separable Hilbert spaces Xl and X2, Xl being continuously imbedded and dense in X2• For 0 < () < 1 , let

In this case, we have Moreover if 0 ::; A

< 8::;

In particular, for 0

( 2.5 ) 1 , then

::; p ::; r,

( 2.6 ) When af! is of class em-l,l for certain positive integer m, a general interpolation exists for three spaces. It states that if A - ! is not an integer, A � 0, 0 < () < 1 and o ::; p ::; r ::; m, then This property also holds when f! is a hypercube without limitation on r and p. Now let f! = ( 0, 2 1r ) n and Vp(f!) be the space of infinitely differentiable func tions with the period 2 1r for all variables. Similarly we can define L;(f!), H;(f!) and w;,q(f!). Moreover let A(v) =

10

vex) dx ,

Chap ter 2. Orthogonal Approximations in Sobolev Sp aces

15

and Then for any v E

HJ(O) or v E H�(O) n L�(O), there exi sts the Poincare inequality

(2.7) Hereafter c denotes a generi c positive constant depending only on the geometry of O. Thus for any non-negative integer m, the semi-norm I . IHm(o) is a norm of H/{'(O) or H;:'(O) n L�(O), equi valent to the norm II . IIHm(o). We give a simple version of the Sobolev imbedding theorem. It states that m � - m > 0, i n C (0) for W ,P(O) i s continuously imbedded in Lq(O) for � � P

=

q

p

0, and in Lq(O) f or m � and any q, 1::; q < 00. p We now turn to the trace theory. Let ( VI," " n ) be the uni t outward normal n If 00 i s of class cm - l,l , then for a relatively open vector to 00, and v

-

part

=

m

<

n

f

v=

O = qL:vqOq. =l

n

v

of 00, the mappi ng T'!;: v --+

m -!. If

T'!; v = (

vir, ovvlr, ... , o;'-llr)

(0), can be extended by a density argument to any space Hr(o), r >

00 i s a union of finite number of i"i, and

defined on Coo

class C m -l, l , then for any r

H�(O) =

<

fj

are relatively open parts of m + ! such that r -! i s not an integer, we have

{ I

v vE

W(O), Tlr�!lV ° for all j

=

}.

On the other hand, let Hr-!(r) denote the im age of Hr(o) by the trace mapping Tf with r > !. It i s shown that 11 .4> IIH r-.l2" (r)

=

veHr(o) vll'=';

inf IIvlw(o).

The details of the above results can be found in Lions and Magenes

(1975). Finally we give the following L ax-Milgram Lemma (1954).

(2 .8 ) (1972) and Adams

Spectral Methods and -r:h�ir Applications

16

Lemma 2 . 1 . Let B be a reflexive Banach space and a( . , . ) be a continuous bilinear

elliptic form on B

X

Vw E B .

B, i. e., there exists a positive constant

a ( w , w) ;::: all wl 1,

Then for any 9 E B', there exists

v

a

such that

(2.9)

E B uniquely such that

a ( v, w ) =< g, w > ,

(2.10)

Vw E B.

For simpli ci ty, the inner products (',')£2(0), (',')W(O) and the norms 11,11£2(0), II· Ilw(o), II'IILP(O), 11'llwm,p(o) will be denoted by (' , .), (', .).,11·11, 11·11., II·IILP, 11'llwm,p respectively. Similarly the spaces C (a, bi Hr(o) ) , C (a, bi Wm ,P(O ) ) , H'Y (a, bi Hr(o) ) and H'Y (a, bi Wm ,P(O)) will be denoted by C(a, biHr) , C(a, biwm ,p) , H'Y(a , biHr) and H'Y (a, bi Wm,P), etc . .

2.2.

Fourier Approximations

We consi der the Fourier approximations in thi s secti on. Let n =1 and A = (0, 27!'). The set of functions eilx , 1=0,±1, . . . , is an orthogonal system in L2(A). The Fouri er transformation of a function v E L2(A) i s

Sv=

00

1=-00

L

vleilx

(2.11)

where VI i s the Fourier coefficient, VI = -

11 -ilx d 1=0,±1,.... 27!' A V ( )e According to ll iesz Theorem, Sv converges to v in L2(A), and the Parseval equality •

x,

x

holds, namely IIvl1

2=27!'

00

L IVt!2. 1=-00

00

(2. 12)

latl2 < there exi sts a uni que function in L2( A) such that its Fourier coefficient VI = al for alI I . Thus we 1=-00

Conversely for any complex sequence { al } such that L

can write

00 " ilx v= L vie 1=-00

00 ,

Chapter 2. Qr,t!! ogonaJ Approximations in Sobolev Sp aces

- -

17

A general result i s the Young-Hausdroff inequality, see Hardy, Li ttlewood and P6lya (19 52) . It states t hat for 2 $ p < 00 , 1 + 1 = 1 , p

q

( 2� L Iv(x)IPdX) $ (J�:oo IVt!q) , (too IVt!p) $ ( 2� L Iv(x)lqdX) ; . 1

"

�

(2. 13)

1

P

(2. 14)

The convergence in L2(A) does not imply the pointwi se convergence of Sv to v at all poi nts of A. However, Carleson (1966) proved that Sv converges to v for all x outsi de a set of zero measures in A. Some other results are as follows,

(i ) if v i s continuous, periodi c and of bounded vari ation on A, then Sv i s uniformly convergent to v on Ai

(ii ) if v i s of bounded vari ation on A, then !v(x-) for any x E Ai (iii ) if v i s continuous and periodic,

x E A.

Sv

Sv is convergent-pointwi se to !v(x+) +

does not necessarily converge at every point

-

For di scussion of differenti ation, let m be a positive integer and fI';(A) be the subspace of Hm(A) whi ch consi sts of all functions who�; first m 1 derivatives are periodic. In thi s space, it i s permissible to differenti ate termwi se the Fourier series m times, in the sense of square mean. In deed, o:r:ei/:r: = ileil:1 and so . U:r:V 1= (!lu:r:v,eil:r:) 2 7r (!'}-)

Therefore, for any v E

iI�(A),

=

-(

o:r:v(x)

!l eil:r:) V,U:r: 00

=

I:

1=-00

= Z

·Z (v,eil:r:)

ilVlei1:r:.

By repeating the above procedure, we obtain 00 o:;v(x)= I: (ilrvleil:r:. 1=-00

=

' 2 7rZ'ZVI.

Spectral Methods and -r:itir Applications

18 Consequently, the norm

For r = m + u and 0 <

and by (2.5) , for v E

IIvllm

u

in fr;'(A) is equivalent to

< 1 , by the space interpolation (2.6) ,

H;(A), (2. 15)

For r < 0, let norm

H;(A) = (H;r(A))'. Therefore, for any r,H;(A) has the equivalent (2.16)

In particular for r

:::::

0, the space

H;(n) has the equivalent semi-norm

Now let N be any positive integer and VN be the set of all trigonometric polyno mials of degree at most N, i.e. ,

(2. 17)

VN

VN denotes the subset of consisting of all real-valued functions. In numerical analysis of spectral methods, we need some kind of inverse inequalities. The first one is due to Nikolskii (1951), stated below. Theorem

2. 1 .

If 1 ::; P::; q ::; 00, then for any 4J E VN,

where Po is the least even number greater than or equal to p.

Ch ap ter 2. Or,hogonai Approximations in Sob olev Spaces

Proof. For simplicity, let 1> E VN and DN(t) Ntl DN(t) sin2 SIn2. 2 t =

Then

1>( X)

2 t

.!. f 7r JA

=

19

be the Dirichlet kernel, =

-21 + I:/=1N cos It.

(2.18)

1>(t)DN(X - t ) dt

and thus by the Cauchy inequality,

Furthermore,

1> 2 E V Ell.

� 2

1 1>

1 IIL � �

and so

C�: 1 ) :; 1 1>11 · 1

(Np;7r+ 1 ) ' 1 1>�1 I � (Np;7r+ 1 ) ' 11 1>11 �111>lllp, .

I

1

111>�IIL� �

Hence

p

Since �

q,

.

1

we get

•

Another kind of inverse inequalities is called the Bernstein inequalities. Some of them can be found in Butzer and Nessel

(1971).

Theorem 2 . 2 .

1> E VN,

Let m be a non-negative integer and 1 � p

Also for any r ;:::: 0,

11 1>11 , � cN'II1>II·

�

00.

Then for any

20

Spectral Methods and Their Applications

Proof.

Let DN(t) be the Dirichlet kernel. By differentiating Ox
JA

(2.18),

(-f.l sin It) dt. 1=1

N-1 Since
2N f
where FN_1(t) is the Fejer kernel, i.e. , N-1 N-1 FN-1(t)= DI(t) = 2 +

,

1� N

1 � (1-

I

N

) cos It.

On use of the Holder inequality,

Then an iteration leads to the first conclusion. By p = we obtain that for r = m + 17 , 0 < 17 <

2,

v

1,

The L 2 -orthogonal projection PN : L2(A) E L 2 (A),

Indeed,

M oreover

(2.12 ) implies that

(2.15) and the previous result for

•

-+

VN is such a mapping that for any

PNV= L v[eilx. IIISN

Chapter 2. Orthogonal Approximations in Sobolev Spaces

If

21

v is s uitably smooth, then

I v - PN v l Loo IL>N lfid· Thus thE) error between v and PN v depends on how fast the Fourier coefficients decay to zero. This in turn depends on the regularity and the periodicity of v. I n fact , for any v E C1 (A) and [ #- 0, �

and so iii = 0 (t). Because the last integral is the Fourier coefficient of ax we conclude that if E C2 (A) and (211'-) = (0 + ) , then = 0 (�). The repetition of the above procedure tells us that if E cm (A) an d are periodic for all k � m-2, then = 0 (l-m). In the early work of Kreiss and Oliger (1972), Guo (1981, 1985) an d Kuo (1983), the error estimates between solutisms or differential equ ations an d their spectral approximations were carried out based on the above facts . But recent work is associated with global approximations , started by Pasciak (1980).

v

v

v

VI

Theorem 2.3. ff r � 0 and

For

Proof.

f.1

f.1

<

C

and

� r, then for any

v E H;(A),

L (1 + [2 " I 1 2 � cN2/1--2r L [2r-2/1- (1 + [2 " liid2

liI>N r

< cN2/1--2 II

v,

VI a;v

� 0, we have from (2.16) that

Ilv - PNvl � For

v

I I>

< 0, we hav e that for any

so

II v

- PNV II/1-

) VI I I>N r 2 c 2 N ILN (1+1 r I d � 2/1--2 lvl;·

=

9

fi

)

E H;;/1-(A),

9EH:;�(A) 9,,0

sup

(v - PNv,g) _< cN/1--r lvIr' Ilgll -" �

•

22

Spectral Methods and" Their Applications

We have seen that PN8",v = 8",PNv. Hence PN is also the best approximat ion of v in VN for the norm II . 11 m,m being any non-negative integer. However it is not so for the LP-norm, 1:5 p:5 OO,p i- 2. But we still have some results. For instance,

In particul ar,

c =

in!

.peVN

Ilv - 4>IILP:5 cN-mIl8;;'vIlLP,

� for

1:5 p:5 00.

p = 00. Also, we have

with u(p) = 0 for 1 < p < 00 and u(p) = 1 for p = 1, 00. Anyhow PNV approximates v in the D'-norms with the same order as the best approximation, and IIv-PNvIILP-+ o as N -+ 00. In many applications, the numerical algorithms based on Fourier transformation cannot be implemented precisely. So we prefer to use discrete Fourier transformation. Let AN be the set of interpolation points,

AN = {xU) I xU) = � 2N +l'

J'

-

- 0"

1 ... , 2N} .

(2.19)

The discrete Fourier transformation of function vEe (X) is

INv(x) = L: Vleilx III$N

such that

INv (xU)) = V (x(j)) , 0:5 j :5 2N.

if p = (2N +l)q, q is an integer, otherwise,

Due to the orthogonality, i.e. ,

{ 2N0, +1,

�eiPX(j) =

we have

(2.20)

j=O

� v (x j) ) e-ilx(j) . VI = -2N 1+1 L...J ;=0

_

The coefficient VI can also be expressed in terms of the coefficient

1 � VI - _ � 2N +1 ;=0 _

_

_

(� vpe ) L...J p=-oo

A

i(P-I)XU»

VI. In fact,

Chapter 2. Or't�ogona1 Approximations in Sobolev Spaces =

1_ t _ 2N + 1 p=-oo 00

23

(fj=O vpei(p-I)"'W)

( 2.21 )

1: VI+p( N+l ' p=-oo 2 )

The interpolation IN c an be regarded as an orthogonal projection upon VN with respect to the discrete inner product. To do this, we define the discrete inner product and the norm as

( 2 . 22 ) Then and by

(2 . 20),

VvE C (X) ,'E VN. ( 2 . 23 ) This shows that INv is the orthogonal projection of v for the inner product ( ·)N. /

-,

As in the c ase of Fourier transformation, the following results hold, (i) if v is continuous, periodic and of bounded variation on uniformly on Xj

X, INv

tends to

v

(ii) if v is of bounded variation on X, INv is uniformly bounded on X and converges to v at every continuity point for Vj

(iii) for any integer I ::f and any po sitive N such that N > I I I , let VI = vf ) be the N I-th Fourier «oefficient of INv. If v E c;:" (X), then vI ) decays faster than cl-m , m being any positive integer, uniformly in N. If v satisfies the hypotheses N for which VI = " (l- m ), then Vf ) possesses the same asymptotic behavior.

0,

I I

We now deal with the global error between v and INv. Theorem

2.4.

Ifr > ! and 0:51l:5 r, then for any vE H;(A),

N

24

Spectral Methods and Their Applications -

I ENVIl� c IL�N ( 12Y' I VI iid2 cN2P. I LI �N II:VI+P(2N+l,12 :::; cN2P. II:�N (L I I p(2N 1) 1-2r) (L p(2N 1) 12r 1VI+P(2N +ll I 2) :::; cN2P. L (N-2rIPI-2r IqI:N Iql2r IVqI2) :::; cN2p.-2r L Ipl-2rlvl; :::; cN2P.-2rlvl; which together with Theorem 2.3 imply the desired result. :::;

1+

#0

<

#0

p#O

+

+

#0

11 +

+

l>

#0

•

The following result provides the estimate in Loo-norm, Kenneth ( 1978) studied the estimates in em -norms. =f. Finally unlike But for any E

oxINv INoxv

2.3.

PN.

v

H;(A) with r > �,

Orthogonal Systems of Polynomials in a Finite Interval

w(x)

Let be a non-negative, continuous and integrable real-valued function in the interval A = ( - 1 , 1 ) . The associated weighted space of real-valued functions is L� equipped with the norm

<

=

I v I L1.:(Al (L I v (xWw(x)dxY, =

If p =

(A) { v I l vIl L1.:(Al } 1

00

1 :::; P <

2, then L� (A) is a Hilbert space for the weighted inner product «(V,W)L�( Al L v (x)w(x)w(x)dx, Vv,w L� (A) . =

E

00 .

Chapter

2.

25

Orthogonal Approximations in Sobolev Spaces

For any non-negative integer m, the Sobolev space

W;"P(A)

W;"P (A)

I

=

{v a;v E L� (A), k �

is defined as

m} .

For any. real r � 0, W�'P(A) is defined by the space interpolation wi th the corre sponding norm . W;:� (A) is the closure of D(A) in W�,p(n). For r < 0, W�,p(n) = (wo�,, q(n))' ,�p + �q = 1. In particular, W�,2(A) = H:(A). It is a Hilbert space for the related inner product when r 2: 0. While for r < 0, 9€Hor(A)

IlvIIH':;(A) = sup

(v,g)

g;'�

For simplicity, we denote ( , ')L�(A)' II'IIL�(A)' II'IIL�(A)' II·IIH':;(A) andll·lIw.:;,p(A) by ( , ')w, 11·llw, II'IIL�' 11·llr,w and ll·llw.:; p , respectively. Also, C (a, b; W�'P (A)) and H'Y ( a, b; W�'P(A) ) are denoted by C(a, b; W�'P) and H'Y(a, b; vV�'P) with the norms II· IIC(a,b;W':;'P) and 11'IIH�(a,b;W':;'P)' respectively, etc . . Let (/>I (x), I = 0, 1 , . . , b e an orthogonal system of algebraic polynomials with respect to the weighted inner product ( , ')w' By Weierstrass Theorem, such a system is complete in L�(A). The formal series of a function v E L�(A) in terms of {1>I } is '

'

'

.

'

00

1=0

Sv =I:VI1>I (x) with

VI =

1 1 1> II �

�

(2.24)

L v(x)1>I(x)w(x)dx.

By the completeness of the system {1>I } ,Sv is convergent to v in L�(A). The rate of convergence depends on the choice of the weight function w(x). The different choices correspond to different spectral methods. Denote by the set of all algebraic polynomials of degree at most N in A. In this set, several inverse inequalities are valid. We first consider the relation between 111>IIL� and 1 11> I IL� ' 1 � P � q � 00. Let L be a linear operator defined on L is said to be of (p, q) type, if there exists a positive constant d depending only on p,q and N such that II L1> IIL� � dll1>IILE for any 1> E According to Riesz Thorin Theorem, we know that if L is of both (PI, ql) type and (P2, q2) type for

IPN

IPN.

IPN.

Spectral Methods and THeir Applications

26

Pl ;P2 <00 , 1 $ qI, q2 $00 , then for qIq2 PIP2 0 < 9 < 1, q = (1 - 9)q P = (1 - 9)P + 9PI' 2 + 9qI ' _ _ 2 the operator L is also of (p, q) type. If in addition II L4>I L� $ d "I 4>II L�" 1$

(2 . 25) 1= 1 , 2, then

(2.26) where

c (Pl ! P2 )

Theorem 2 . 5 .

number 6,

then for any 4>

PI and P2 . If 4>1 E y"(A) for alii, and for certain positive constant and real

is a positive constant depending only on

Co

E

(2.27) IPN

and aliI $ P $ q $00,

where u(N ) = N2 6+1 for 6 >

6 < -!.

-!,

u(N) = In N for 6 = -!, and u(N) = 1 for

Proof. Let us consider, for example, the case 6 > - ! . By (2.24 ) and (2.27), (2.28)

d CN2 6+I .

Clearly , ) type with the constant = Thus the identity operator is of ( 1 00 for any r = � , 1 $ is of ( r, r ) type with = 1 . For 1 :5 < 00 , we set = 1 , = 00 , = = r and = � i n (2.25 ) . Then (2.26) implies the desired result. On the other hand, we know from (2.28) that is also of ( 1 , 1 ) type and 00 ( ,00 ) type. The proof is complete. •

PI

qI

L p:5 q:5 oo, L P2 q2 9

d

P

L

We now turn to another kind of inverse inequalities. The first result in the fol lowing theorem is due to M arkov (1898), e.g. , see Timan (1963). Theorem 2.6.

For any 4> E IPN,

Chapter

2.

27

Orthogonal Approximations in Sobolev Spaces

If, in addition, then

for

Proof.

all 2 � p �

(2.29) 00 ,

Set x = cos y ,

(2.30)

vex) = v(cos y).

From the proof of Theorem 2.2 , I Ox¢>(cos y) sin yl::; N O:O;v$ max" 1¢>(cos y) 1 2 and so (2.31 )

Next let ¢>N be an algebraic polynomial of degree N, and distinct points in A. Set

Then we have the Lagrange interpolation as N- l Ox
x (j) = cos 7r(22jN+ 1 ) '

xU), 0

::; j :::;

(xU») qN - l,j (X). 0 :::; j

:::;

N - 1,

we obtain that 1 cos(N arccos x), 2N - 1 ( - l )j N N 2 - 1 sin ,,(22jN+ l) •

C onsequently, 0'"

x'l'N( X ) =

cos(N arccos x ) L.. N J=O

� _ J (j» ) ,)1- (X (J» ) 2 ( 1 ) Ox
N - 1 be

Spectral Methods and Their Applica,tions .

28

Hence by virtue of (2. 3 1 ) ,

If x (O) < I x l :::; 1 , then all the differences x - x U ) have the same sign and so I O,, ¢N (X ) 1

< <

(:�:(��S X ) 1 N sin�os x) 2 I I ¢N I I Loo I 1 - x2 l :::; N 11 ¢N I I Loo .

I I ¢N II L OO

I};

c OS

If x (O) 2:: x , then the same conclusion follows immediately from (2 . 3 1 ) . If (2.29) holds , then the second conclusion comes from the space interpolation and the previous result 'yl.s shown in Quarteroni ( 1984) .

v

•

The L�-orthogonal projection PN : L� -+ lPN is such a mapping that for any E L� (A ) , (2.32)

Indeed,

N L v/ ¢ / (x ) . / =0 There are some close relations between orthogonal systems and Gauss-type intePN V(X)

=

(i) Gauss integration. Let x(O) , . . . , x( N) be the roots of the (N + 1 )-th orthogonal polynomial ¢N+1 (X) , and w(O) , . . . , w( N ) be the solution of the linear system

grations. We have the following results.

Then w(j )

>0

N L ( x(il r w u ) j=O

=

1 x m w(x) dx , A

0 :::; m :::; N.

(2.33)

for 0 :::; j :::; N, and for any ¢ E lP2N+1 , N L ¢ ( xU» ) wU) j=O

=

1 ¢( x ) w (x ) dx . A

(2.34)

(ii) Gauss-Radau integration. Let x { O) , . . . , x(N ) be the N + 1 roots of the poly nomial .,p (x) = ¢ N+l (X) - ¢N+l ( - I ) ¢N1 ( - 1 ) ¢N(X) , and w(O) , . . . , w( N ) b e the solution of (2.33) . Then (2.34) holds for any ¢ E lP2N .

Chapter 2. Or�hogonaJ Approximations in Sobolev Spaces

29

..

(iii)

x(O), . . . , x( N) be the N + 1 roots of the polynomial (1 - x 2) "" ( x) ",, (x) =
Gauss-Lobatto integration.

valid for any

x U),

. • .

E

x ( N) =

0 � j � N are called Gauss-type interpolation points. Clearly The points 1 = - 1 in Gauss-Radau integration and Gauss-Lobatto integration, and 0 � j � N are called Gauss-type in Gauss-Lobatto integration. The numbers weights. In pseudospectral methods, the fundamental representations of a smooth function are in terms of its values . at Gauss-type )ntepolation points. The interpolant is given by

x(O)

w(j) ,

v

INv(x ) = N

l: VI
such that

(2.35)

We introduce the discrete inner product and the discrete norm as

I v l N,w = (v, v)1,w' 1

The Gauss-type integrations imply that

(2.36) (2.37)

=

where A 1 , 0, - 1 for the Gauss interpolation, the Gauss-Radau interpolation and the Gauss-Lobatto integration respectively. Obviously by (2.35) ,

(INv ,W)N,w = (V , W)N,w V V , W C (A) . Thus INv is the orthogonal projection of v upon IP N with respect to the inner product given in Furthermore, the orthogonality of {
E

(2.37)

(2.36).

(
=

=

l:

j= O

� I, m � N

Spectral Methods and Their Applicat�ons

30

where b',m is the Kronecker function. Therefore by virtue of (2.38),

(v, ¢>,) N,,,, = (INV, ¢>,) N,,,, pL:N=o vp ( ¢>p , ¢>,) N,,,, "'fIv/, =

The coefficient

VI

0 51 5 N.

can be expressed in terms of the coefficients {VI } , i.e. ,

I . I ", I . l i N,,,,

( , · ) N ,,,, . .

=

(2.39)

( · )w and

and There are some relations between and between . , In fact, for the Gauss integration and the Gauss-Radau integration,

Moreover,

v, w E e (X), (v,W)", - (V,W)N,w (v,w)w - (INv,w)N ,,,,

(2.40)

(2 .37) and (2.38) imply that for any =

and so for the Gauss integration and the Gauss-Radau integration, =I For the Gauss-Lobatto integration, usually. :aut for mostly used orthogonal systems in A, they are equivalent, namely, for certain positive constants Cl and C 2 ,

I ¢> I N,,,, I ¢> I ",

¢> E IPN, I (v, ¢» '" - (v, ¢» N,w l 5 I (v, ¢» '" - (PN -1 V, ¢» ",I I (PN -lv, ¢» ", - (INv, ¢» N ,,,, I c ( l I v - PN-lvll ", + I PN-IV - INv I N,w) I ¢> I ",

In this case, we derive from

(2.37) that for any

+

<

(2.42)

This formula plays an important role in numerical analysis of pseudospectral methods.

Chapter

2.

Orfhogonal Approximations in Sobolev Spaces

2. 4 .

Legendre Approximations

Let A

=

is

( - 1 , 1 ) and w(x)

31

1 in this section. The Legendre polynomial of degree 1

==

L,(x) =

( - 1 )1 , ' ""'2il! 8", ( 1 - x 2)' .

(2.43)

It is the I-th eigenfunction of the singular Sturm-Liouville problem

(2.44)

>. , = 1(1 +. 1 ) . Clearly Lo (x) = 1, L 1 (x) = x , and they

relat ed to the I-th eigenvalue

satisfy the recurrence relations

1 21 +. 1 xL, (x) L _ (X) , 1 � 1 , --z:tl 1 +. 1 ' l ( 21 +. 1 )L,(x) = 8",L'H ( X ) - 8",L, _ 1 ( X ) , 1 � 1 .

L 'H ( X) =

It can be checked that L, ( 1 ) ( _ 1) ' + l i l(1 +. 1 ) . Moreover,

ILI (x) 1

=

:::;

1 , L, ( - 1 )

=

(2.45) (2.46)

( _ 1 ) ' , 8", L, ( 1 ) = i l(l +. 1 ) and 8",L, ( - 1 ) =

1,

(2.47)

The set of Legendre polynomials is the L2-orthogonal system in A , i.e. ,

(2 .48) By integrating by parts , we deduce that

(2.49) The Legendre expansion of a function v E L2(A) is v(x) with

v, = (I +.

00

= E v,L, (x) 1=0

�)

L V ( X )L,( X ) dx .

(2.50)

Spectral Methods and Thelc Applications

32

We now consider the differentiation. Let vf 1 } be the coefficients of Legendre expansion of oxv, 1 = 0, 1 , . . . . By (2.46) ,

oxv(x)

=

On the other hand,

00

oxv(x) = L VloxLI(X)' 1= 1

Hence

v[t}

00 = (2 1 + 1 ) L

vp.

p= l + l p+1 o d d

This formula is valid for any v E H 1 (A). We turn to the inverse inequalities in the space IPN. Theorem 2 . 7 . For any rfJ E IPN and 1 � p � q � 1

00,

1

IIrfJII L4 � ((p + I )N2) p-o llrfJIILP. Proof. Assume that

IrfJl reaches the maximum value at X* E A . Thus

and by Theorem 2.6,

IrfJ (x*) I - lrfJ(x) 1 � N 2 1 x-x* l lrfJ (x*) I · We suppose that x* � 0 for definiteness. Then

i .e. ,

( 2 .51 )

Chapter 2. Orthogonal Approximations in Sobolev Spaces

33

Further, we have

which leads to the conclusion.

•

We can derive the above theorem from Theorem 2.5 apart from the constant. Indeed by ( 2.47 ) and ( 2.48 ) , we have 6 = � in (2.27) . . Theo rem 2. 8. Let m be a non-negative integer and 2 ::; p ::; 00. Then for any v E IP N, Also for any r 2:: 0,

Proof. Let �l and � P) be the coefficients of Legendre expansions of

and ox
By (2.51 ) ,

Let a N = 1

+

2�

::; � . Then

from which, Theorem 2.6 and a repetition, the first conclusion follows. We now prove the second one. If r = m + (J and m is a non-negative integer, 0 < (J < 1 , then by (2.15),

v

The L 2 -orthogonal projection PN : L 2 ( A ) E L2 (A) ,

• ->

IPN is such a mapping that for any

34

Spectral Methods and Their Applications

or equivalently,

N PNV( X) = L v,L,(x). '=0 We now estimate the difference between v and PNv. L'emma 2 . 2 . For any v E Hr(A) and r � OJ Proof. Assume at first that r = 2m, m being an integer. We have I v - PNv l 1 2 = '=LN+l vr ll Ld l 2 • Let Av = -all' ((1 - x 2 ) a.,v). B y (2.44) , V, I L� 1I 2 i v (x)L, (x) dx = 1 ( 1 1\ I L1I1 2 i v( x)AL,( x) dx Av(x )L, (x) dx. l(l 1\ I L 1I1 2 i 00

+

+

Iterating this procedure yields

and thus

which is the desired result for space interpolation as before.

r = 2m. For any r � 0, we deduce the result from the •

a.,PNV( X) '" Piva., v(x ). But we have the following result . Lemma 2 .3 . For any v E Hr(A) and 0 ::; ::; r - I J Usually

j.l

Chap ter 2. OIlthogonal Approximations in Sobolev Spaces

35

Proof. According to the space interp olation, it suffices to consider the case r � 1 , I-' = O. Assume first that O",V is continuous and so (2.51) is valid; Also let

Then by

N N-1 o",PNv(x) = L w/L/ (x), w/ = (21 + 1 ) L vp. /=0 p=l+l p+1 odd

(2.51 ),

if 1 + if

Accordingly,

PNO", V

( X ) - u",PNv ( x ) .

where

£l

_

{

N

is odd ,

1 + N is even.

1 . ( 1) ", I ( ), v V. (1) ",0 ( ) 1 N 'f'N X 2N i: 3 N+ 1 'f'N X I V. ( 1 ) ",0 ( ) v. ( 1 ) ",1 ( ) , 2N + 3 N + 1 'f'N X 2N + 1 N 'f'N X 2N

I

+ +

+

¢>�(x) = L (2p + l)Lp(x) , ¢>}.(x) O
=

if N is even, if N is odd

(2.52)

L (2p + l)Lp(x).

l
E H1 (A) , it can be approximated by a sequence of infinitely differentiable func tions for which (2.52) holds. Then we can pass to the limit . Next by Lemma 2.2,

If

v

A similar estimate exists for

Iv�LI. On the other hand, (2.48) leads to

1I ¢>� 1I 2

=

and a similar estimate is valid for

implies

which completes the proof.

L 2(2p 1) $ cN2

O
II ¢>}. II 2 .

+

Finally the orthogonality of

¢>�

and

¢>}.

•

36 Theorem

2.9.

For any

v

Spectral Methods and Their Applica.tions

E HT (A) , r ::::: 0 and

J-L

:::; r ,

for J-L ::::: 1 , for 0 :::; J-L :::; 1 , for fJ, < o .

where

Proof. Lemma 2.2 implies the desired result for

J-L =

(2 ..53)

O. Now let

J-L

> O.

The space

interpolation allows us to consider positive integer fJ, only. We now use the induction. Assume that the result is true for

fJ,

-

1 . Then

+

+

from which and Lemma 2.3, it follows that

l l'-l I PNoxv -oxPNvl l l'-l I l v - PNvl l I v - PNvl l 1' :::; Iloxv( -PNoxvl l l , , ( :::; cNO' I'- r- ) l I v I i T cNO' I' r) l I v l r . > - 1) +

0- ( fJ" r ) , the conclusion for J-L argument leads to the conclusion for fJ, < O. Since o-(J-L

-

1,r

:::;

0 follows . Finally a duality •

The result of Theorem 2.9 is provided by Canuto and Quarteroni ( 1 982a) , except the case

J-L

<

I l v - PNvl l v'" cN! -m v (o;' v ) , I l v - PNv I L� :::; cNi -Tl l v l r �, V( v)

O. Other estimates are as follows, <

where m is any positive integer and r :::::

best approximation error in LP-norm , 2 < p � L2-norm, i.e. ,

inf

denotes the total variation of

00

I l v - rfJ l LP cN-r l v l wr,p . �

The convergence rate in Theorem 2.9 is not optimal for

v m (v,w)m 2: (o;v, o;w) ,

J-L

> O.

=

k=O

The

v

Let m be non-

negative integer. Since Hm (A) is a Hilbert space, the best approximation to be the orthogonal projection of

v.

decays as the truncation error in

upon IP N in the following inner product

should

Chapter 2. Qrfhogonal Approximations in Sobolev Spaces

37 ( 2 . .'54)

The corresponding orthogonal projection satisfies

(v - PiV v, ¢)m I ; - PiV v il m

and thu s

= 0, 4> E IP N

= inf

I v - ¢I I m .

(2.55)

wi �h homogeneous boundary conditions, other kinds of projections are needed for Furthermore, when the spectral methods are used for partial differential equations

obtaining the optimal error estimates . Let

and

lP� lP�l =

(2.56)

for simplicity. Set

am (v, w) W:v, a;:w), v, w E a(v,w) al(v,w). lP�,o P;:/'o V

=

and

The commonly used inner product in Hl{' (A) is

=

The Hl{'-orthogonal projection

vE

Hl{' (A)

:

Hl{' (A)

--+

am (v,w).

is such a mapping that for any

Hl{' ( A ) ,

We now estimate the difference between

v PiV,ov . and

Theorem

2. 1 0 .

v PJilv, any v E and

and the difference between

Let m be a positive integer. Then for

m s r,

(2.57)

Hr (A)

and 0 S

ft

S

Proof We first assume that ft = m and let

wE 1 a; v I · 1 m. 1 I · lI w I · 1 m v*(x) = q(x) + i: PiV:l ayv(y) dy.

Because for any

W and k S m -

,

vanishes at certain point , so the Poincar�

inequality (2.7) is valid. Hence W is a Hilbert space for the inner product and the norm

=

is equivalent to the norm

For each

vE

ame- , · ) ,

Hm ( A ) , let

Spectral Methods and Their Applications

38

The polynomial q ( x ) of degree m - 1 , is chosen in such a way that ( == 0 for 0 $ k $ m 1 . We now use the induction. By Theorem 2 . 9, the conclusion is true for m = O. Now suppose that it is valid for m 1. Then

I v - v*lI w I v - v*lm l o",v - PN�io",v lm-1 =

=

A

o;v* - o;v)

$ cNm - r ll v ll r

and thus by (2.55) ,

This completes the induction and implies the conclusion for p, = m. Next we consider the case p, = O. Let 9 E and consider the auxiliary problem

L2(A)

(W,Z)m (g,z), We know from Lemma 2.1 that possesses a unique solution. Moreover, the we get regularity tells us that I w I 2m cllgll . P utting z v - PNv in (v - PNv ,g) (v - PNv,w)m (v - PNv,w -PNw)m ' (2.58)

=

(2.58)

=

$

(2.58) ,

=

=

Consequently,

Then by a duality argument,

I v - PNv ll

gEL'(A) g�O

= sup

(v - Ifftv,g) 9

$ cN -m l iv - PN v i l m $ cN - r ll v ll r .

The result for 0 $ p, $ m follows from the space interpolation. Theorem 2 . 1 1 . Let m be a positive integer. Then for any v E H r

0 $ p, $ m $ r,

Proof We first assert that for any x

=

n

and

v E H[{' (A),

;:;, v ( ) 1'" P;:;::i ,Ooyv (y)dt ,

p o

(A ) H[{'(A)

•

-1

m � 1.

(2.59)

Ch apter 2. Orthogonal Approximations in Sobolev Spaces

39

x

Indeed the function on the right-hand side of (2.59) is in IPN, vanishes at = - 1 , and its derivatives u p t o the order of 1 vanish at = - 1 , 1 . S o i t suffices to verify that it also vanishes at = 1. Let = (1 By the definition of

m x m 1 x e - x2) - . PN'o , L a;- l (PN::i ,oa", v (x)) a;-l e(x)dx L a;- l (a", v (x)) a;-l e(x)dx. Integrating m - 1 times by parts, we deduce that (2(m - I ) ) ! l a", vdx -

=

(2( m - 1 )) ! (v(l) - v( - 1 ) )

which ends the proof of (2.59) . Furthermore,

=

0

I v - pN,ovlm la",v - PN::mi ,oa",vlm_1 . . . mlI a;v - PN-m (a;v ) I ::; c(N - m) - rl a;vl r_m ::; cN -rl v l r =

whence the conclusion is true for

I' = m.

For

=

I' =

==

0, we consider the problem

m ( , z) (g , z) , z H;;'(A). It has a unique solution and I wl b m ::; cl g l . Following the same line as in the proof a

w

=

v

E

of Theorem 2.10, we complete the derivation for consequence of the space interpolation.

I' =

O. The general result is a •

The proof of Theorem 2 . 1 1 is given by Bernardi and Maday (1994) . We now turn to the discrete Legendre approximation. There are three kinds of Gauss-type interpolations. (i)

Legendre- Gauss interp olation.

LN + I (X), and

(ii)

Le g endre-Gauss-Radau interpolation.

of L N(X) + LN+ I (X), and

W (0) - (N +2 1) 2 ' _

x(j) are the N + roots of O ::; j ::; N. In this case, xCi ) are the N + 1 roots

In this case,

1

Spectral Methods and Their Applications

40

( iii ) Le g endre- Gauss-Lobatto interpolation. In this case, 1 , xU), 1 � j � N 1 are the roots of oxLN(X), and 1 wU) N(N + 1 ) (LN (XU) ) 2 ' -

= - 1 , x(N ) =

x(O)

2

_

v E e (K) is N INv(x) I: v/L/(x) 1=0

The Legendre interpolant of a function

=

where

=

�

t v (x (j) ) LI (x U) ) w U) , 11 1=0 ( N + �rl for the Legendre-Gauss interpolation II = ( l + �rl for 1 < N" N for the Legendre-Gauss and the Legendre-Gauss-Radau interpolation, and IN VI

=

=

Lobatto interpolation. Consequently in the third case,

!

Usually oxINv # INo",v. In order to estimate the error between need the following lemma. Lemma 2.4. Let 0 < s <

for any

v E W (A)

1 . There exists a projection lIN : HI' (A)

and 0 � J.L � s � r,

(2.60)

v

--+

and

lPN

INv, we

such that

where lIN is the orthogonal projection with respect to a suitable inner product, whose associated norm is equivalent to the standard norm II . II

Proof. B y an affine mapping, we transform A into A * = (0, 1t-). For any • .

v E H B (A*),

denote by Mv the function obtained by the even reflection with respect to the origin and so Mv E H; (-7r, 7r) . This space is equipped with the inner product (V, W)H'(-1f,1f) =

t

1=-00

(1 + 12) ' VIJ,1

Chapter 2. Orthogonal Approximations in Sobolev Spaces

41

where VI and WI are the coefficients in Fourier expansions of v and w . So we define the following inner product in H' (A* ) ,

and denote b y lI N the orthogonal projection upon IP N with respect t o this inner 1 product. It is easily checked that ( v, V) 1 '( A*) is a norm equivalent to I I . I I , . In order to prove this lemma, we first consider the case f.l = s . We have Il v - IIj.,r vl l , � cll vl l s >

Further let II}y : H l (0, 1I") v E Hr (A*) and r 2

1,

--+

"Iv E H' (k) .

(2. 61)

IPN be the H l -orthogonal projection. Then for any

(2. 62)

When s � r � we use interpolation taking into account and We next consider the case f.l = 0, arguing by a standard duality argument. Denote by J, : (H' (A*)) ' --+ H' (A*) the Riesz isomorphism. It means that for any w E H' (A*),

1,

(2. 6 1) (2. 62).

We assert that Jo maps L 2 (A* ) into H2 o (A*) with continuity. To prove it, let WI , 91 and 1, be the Fourier coefficients of Mw, Mg and M J.g respectively. Then the above expression is equivalent to + 1 2 ) ' l,w, t (1 t 91 Wl, 1=-00 1=-00 =

Notice that

WI = w -I, 91 = 9-1

whence M J,g E H 2 s (

-

11" , 11"

and i, = i-I. Taking

) with

(1 + I 2) ' j,.

=

g,

Vv E H' (A * ) . w=

cos

lx, we get

Spectral Methods and Their Applications

42

L2 (A*) H2• (A*)

into It means that J. maps with the continuity. By this fact, (2.62)" and the property of Riesz isomorphism described in the above, we obtain that IIv - IIivvll

9EL�(A·) IlglI='

9€L2(A-) IIgll='

sup ( v - IIivv, g) = sup ( v - IIivv, J.g)H ' (A " )

=

(v ::; cNs- r UvU r · N-s g eL2(A ' ) IIgll='

- IIivv, J.g - IIiv J.g)H '(A ' )

sup

We deduce the result with 0 Theorem 2 . 1 2 . For any

< f! <

v Hr(A), E

Proof. We first consider the case

f! =

g eL 2 (A') IIgll='

sup / I J.g lI 2 . ::;

cN- rl vll r.

by the space interpolation.

s

r>

�

and 0 ::;

O. For any

By (2. 60 ,

c

f!

•

::; r,

> 0,

) I IN (v - IIt+< ) I IN ( IIt+· v) I : Il - t+ Il : ::; c Il v IIt+< v ICo . Let us denote by B;,q(A) the Besov space of order and index see Bergh and Lofstrom (1976) . We know that Bi, l (A) LOO(A) with continuous injection, and for any O,Bt l (Ht +« A),Ht -« A)), . Thus Il v � IIt+< 1 := c Il IIt+< l : c Il IIt+< I If IIt+< I v 11 2

c

>

I

v

::;

1

=

::;

=

v-

v

II

eV

-

p, q ,

s

�

2, 1

v

v

t., (A) ::;

v�

v

H !+' (A)

v

v-

Using Lemma 2.4 and letting c --+ 0, we get the desired result . For -

J.t

>

0, we have

H ! -' (A) .

Ilv - INv l 1' ::; I v - PNvl 1' + cN21' I PNv -INvl cN"(I',, rJl l vl r cN21'( l v - PNvl I v - INvl ) cN"(I' r) I vl r + cN21'- r+ ! Ilvl r. <

+

+

<

•

Chapter 2. Orthogonal Approximations in Sobolev Spaces

43

Canuto and Quarteroni ( 1982a) first proved Theorem 2.12. It could be improved in some special cases. Bernardi and Maday ( 1992a,1992b) showed that for the Legendre G auss interpolation and any v E Hr (A), r > � ,

-<

and for the Legendre-Gauss-Lobatto interpolation and any v E Hr (A) , 0 2r - 1 ,

:::;

J1.

:::;

1 , J1. < (2.63)

They also showed that for the Legendre-Gauss-Lobatto interpolation and any positive integer m ,

I v - INV I I

:::;

cNl- m ll v ll m .

We also refer to Maday (1990) . In the end of this section, i t i s noted that Theorem 2 . 1 2 and (2.42) imply that for 1 any v E Hr (A) , r > 2' and ¢i E lPN, (2.64)

where A = � in general. But A = 0 for the Legendre-Gauss interpolation and the Legendre-Gauss-Lobatto interpolation.

2.5.

C hebyshev Approximations

Let A = ( - 1 , 1 ) and w ( = ( 1 of the first kind of degree is

x)

- x2f in this section. The Chebyshev polynomial [ T,( x) cos arccos x) . (2.65) It is the l-th eigenfunction of the singular Sturm-Liouville problem (2.66) ax ( ( 1 - X2) t ax v ) + A ( 1 - x2r t v 0, x A, related to the l-th eigenvalue A l [2 . Obviously To ( x) 1,T1 ( x) x, and they

satisfy the recurrence relations

1 2

=

=

(1

=

=

E

=

(2.67) (2.68)

Spectral Methods arid Their Applications

44 It can be verified that T/(1) (- 1 Moreover

) /+1 [2 .

(2.69) The set of Chebyshev polynomials is the L�-orthogonal system in A, i .e. , (2.70) with Co

=

2 and

c/

=

1 for

1�

IS

v

1. The Chebyshev expansion of a function E L�(A)

v(x) L: /=0 v,T, (x) v/ � 'lr c/ JAf v( x)T/ (x)w(x) dx . 00

=

with

(2.71)

=

We consider the differentiation. Let sion of 1 = 0, 1 , . . . . By (2.68) ,

oxv,

and so

VA/(l ) v

=

if) be the coefficients of Chebyshev expan

� � q

�

pVP ' A

p= '+ l p+l odd

(2.72)

This formula is true for any E H� (A). We now present the inverse inequalities. Theorem 2 . 13. For any rp .

where

Po

E lPN and 1 � P � q � 00 ,

is the same as in Theorem 2. 1 .

Proof. Let A *

=

(0, 2'1r) and make use of the transformation

x

=

cos y,

v ( y ) v (cos y ) . *

=

(2.73)

Chapter 2. Orthogonal Approximations in Sobolev Sp aces .

Smce

dy dx

=

4.'5

-w(x), we have

I I I/> II L:';

=

G)

�

1 :::; p :::;

IW IILP(A') '

(2.74 )

00.

Then the desired result comes from Theorem 2. 1 .

•

Theorem 2 . 14. Let m be a non-negative integer and 2 :::; p :::; I/> E IP N ,

00.

Then for any

Also for any r � 0, Proof. Let

�/

and

�il)

respectively. By (2.72 ) ,

(C/�P») 2

:::;

be the coefficients of Chebyshev expansions of I/> and ox rP

( P�' p2) ( P�' ��)

4

p+r odd

Accordingly,

Il ax l/>II�

=

N-l

2 /=0 ( �P » )

?!:. L 2

q

:::; N ( N + 1 ) (2 N + 1 )

�

p+l odd

:::;

N-l N ?!:. N (N + 1 ) (2N + 1 ) L -1 L 3

/=0 c/ /=0 ��

:::;

��r 4N4 1 1 rP ll�

from which, Theorem 2. 6 and a repetition, the first conclusion follows. ,We now prove the second one. If r

=

m+

(J'

and m is a non-negative integer, 0

an inequality parallel to (2. 15 ) ,

< (J' <

1 , then by

•

v

E

The L�-orthogonal projection PN : L� (A) L� ( A ) ,

o r equivalently, PN V(X)

=

-+

IP N is such a mapping that for any

N

L vm(x) .

/=0

We now deal with the error between v and PN v.

Spectral Methods and Their Applications

46 Lemma 2 . 5 .

For any v E H�(A) and r ;::: 0,

�

[ sin y [ :5 1 , we get

Proof. By the space interpolation, it is enough to rove it for non-negative integer r.

So by transformation (2.73 ) , v* E H; (A* ) . Since

I �:I

=

Let PN be the symmetric truncation of the Fourier transformation up to degree N as in Section 2.2. Then for any v E L!(A), ( 2 . 75)

Therefore by Theorem 2.3 and (2.74) ,

•

Generally 8",PN V(X) -=I PN 8",v(x). But we have the following result. Lemma 2.6.

For any v E H�(A) and 0 :5 p. :5 r - 1 ,

Proof. It suffices to prove it for r ;::: 1 and p. = O. As in the proof of Lemma 2.3, we can assume that v is a smooth function. Let N-l N 8",PN v(x) = .E wai (x), c/w/ = 2 .E / =0 p=1+ 1

pUp.

p+1

By virtue of ( 2. 72 ) ,

odd

if I + N is odd, if 1 + N is even.

Chapter 2. Orthogonal Approximations in Sobolev Spaces

So

PNO v(x) "

_

a

(1 (1 PN v(x) - { Vv..NN(1 ))",0'l'NrP0N( X( )X +) +v. NV. +1(1) ) 'l'rP",NN11 (( XX )) ,, N +1 _

:&

where

47

if if

N is even, N is odd

By Lemma 2.5, we have

I V��l l .

II rP� lI w

and a similar result for On the other hand, we know from (2.70) that and Thus, noting that and are orthogonal, we obtain �

Ni

Il rPkr l w N L

rP�

rPkr

�

•

Theorem

2. 1 5.

For any

v E H�(A) , r � 0

and Jl-

:$ r,

where q(p" r) is given by (2. 53} .

I l v - PNv lll',w

:$ <

Jl-

0

is true by Lemma 2.5. Now let Jl- > O. The space interpolation allows us to consider positive integer Jl- only. We shall use induction. Assume that the result is valid for Jl- 1 . Then by Lemma 2.6, Proof. The conclusion for

=

Il o"v - PNo"v l l'-l,w + Il PNo"v - o"PNvlll'- l ,w + Il v - PNv ll w cNO' (I'- 1 ,r - 1 ) l I vl l r,w + cNO' (I',r) l I v l l r,w :$ cNO' (I',r) lIvll r,w '

Finally a duality argument completes the proof for Jl- < O.

•

The proof of the above theorem is due to Canuto and Quarteroni (1982a) , except the case Jl- < O. Another estimation is that for any E W P ( A ) and 1 :$ p :$ 00 ,

v

r

,

48

Spectral Methods and Their Applications

()

where O"N p = 1 for 1 < p < 00 and O"N (I ) = O"N (oo) = 1 + In N. In order to define the best approximation in we introduce the inner product =� ' V E (2.7 6 )

H;:: ( A), (v, w)m,w m (8!v, 8!wL v, w H;:: (A). The H;:: -orthogonal projection P;:; : H;:: ( A) is such a mapping that for any v E H;:: (A), k=O

-+

IPN

Consequently,

(2.77)

In the numerical analysis of Chebyshev spectral methods applied to partial differential equations with homogeneous boundary conditions, we need other kinds of projections for the derivations of optimal error estimates. To do this, let

lZm,w(v,w) (2 . 7 8 ) am,w (v ,w) (8;'v,8;'(ww)) . (2.79) and aw(v,w) a l ,w (v,W). We first analyze some In particular, lZw (v,w) properties of am,w (v, w). For the sake of simplicity, we consider the case 1 only. For any v E HJ,w (A), I vw21 w :$ I v iI ,w. Proof. Let g (t) .!.t Jorf(s)ds. From Theorem 4.1 in Chapter of Lions (1967) , = lZ l ,w(V , W)

I

=

m

=

Lemma 2 . 7.

=

•

x

3

Set t = + 1 . Then

1 00 Ig (x + lW (x + 1)-� dx :$ 196 100 I f (x + lW(x + 1 )-� dx.

Now take f(x 8"v(x) for - 1 :$ x :$ 0 and f(x + 1 ) 0 for x > O. Then g(x + "!lV(X) and so ll lv (X)12w5(X)dx :$ 196 v'2 i: 1 8",v(xWw(x)dx. �

1)

=

�

+ 1) =

=

Chapter 2. Orthogonal Approximations in Sobolev Spaces

We can get a similar result on the interval [

v 2 ( x )w3 ( x)

which implies that integrating by parts,

49

0,1 ) . The combination of them reads

tends to zero as

x

goes to ± l . Furthermore, by

l 8"v8,,(vw) dx l l 8"vl 2wdx - � 1 (2x2 + 1 ) v 2w5 dx, 1 8"v8,, (vw) dx l I 8,, (vwWw - 1 dx + � 1 V 2W5 dx. Subtracting ( 2.81 ) from ( 2.80) yields l I8"vI 2w dx - l (x 2 + 1 ) V2W5 dx = l I8,,(vwWw- 1 dx "2, 0

( 2.80) ( 2 . 81 )

and the desired result follows.

•

any v , w E HJ,j A), aw( v , v) "2, 4'1 l I v l k2 w , l aw(v, w) 1 ::; 2 I v h,w l w h, w. Proof. We get from ( 2.80) and ( 2 .81) that aw(v, v ) = �4 11 8",v ll � + -41 JAf V2 (X)w3 (x) dx + �4 JAf 18,,(v(x)w(x)Ww - 1 (x) dx which leads to the first conclusion. Next by Lemma 2.7, for any z E L� (A) and w E HJ,j A), I (z, 8",(ww)) I < l ( z , 8xw)wl + l (z , xwW2 ti < Ilzll w l w h ,w + I zl l w ll ww 2 11 w ::; 21 I zll w l w l l , w . Putting z 8xv, we complete the proof. The HJ,w -orthogonal projection Pj/ HJ,j A) IP� is such a mapping that for any v E HJ,jA), Lemma 2 . 8 . For

•

=

:

-t

.

Spectral Methods and Their Applications ·

50

The other HJ,w -orthogonal projection pfl : HJ,w(A) for any v E HJ,w (A),

-+

1PRr is such a mapping that (2.82)

We now estimate the difference between v , PJ"v , Pilv and pilv. Theorem 2 . 16. For any

Proof. We first set I-'

=

v E H:(A) and 0 :$ I-' :$ 1 :$ r,

1 . Let

L w(x )w(x ) dx,

{w E H�(A) I Aw (w )

=

}

o .

Ww is a Hilbert space for the inner product aw( · , . ) . In fact, the Poincare inequal1 ity (2.7) is valid for any Ww and the norm 1 I · l I w., = a� (. , . ) is a norm equivalent to . For each v H�(A) , put

E

I l h ow .

.

wE .

The constant q is chosen in such a way that Aw (v - v*)

=

o. Thus by Theorem 2. 15,

from which and (2. 77) , we complete the proof for I-' = l . We next deal with the case I-' = O. Let 9 E L!(A) and consider the problem •

=

(w, zh,w (g, z )w,

T/

z E H� (A)

.

(2.83)

Since the left-hand side is the inner product of H�(A), the existence and uniqueness of is ensured by Riesz representation form. Taking z = w in (2.83) , we get that I I l kw :$ cl l l l w . Now, let vary in V{A) and then in. the sense of distributions,

w w

g

w - 8., (8.,w(x)w(x)) (g(x) - w(x )) w(x ). =

(2.84)

51

Chapter 2. Orthogonal Approximations in Sobolev Spaces

Since

1 8"w(x)w(x) - 8"w(x*)w(x*) 1 = 1 1: (g(y) -w (y ))w(y )dy l I g -wl l w arccos x - arccos x* l " , 8"w(x)w(x) is meaningful at x ±1. Multiplying (2.84) by any z E H� (A) and integrating by parts, we· obtain that 8"w(1)z(1)w(1) - 8"w(-1)z(-1)w(-1) = L 8"w( x)8"z(x)w(x) - L (g (x) - w( x)) z(x)w(x) dx, z E H! (A) . Thus by (2.83) , 8., w (1)w(1) = 8., w ( - l )w( - 1) = Furthermore by (2.84) , - 8� w (x) = g (x) -w(x) + 8.,w (x) 8.,w (x)w-1(x). Therefore it remains to prove that 8"w8",ww-1 E L!(A) . Because of 8",w (x)w - 1(x) = x w2(x), we deduce from (2.84) that L: (8",w (x)8.,w (X )) 2W-1( X )dx II (8"W( X ) ) 2W5(X )dx . L: [w2(x) i: (w(y ) -g (y ))w(y ) dyf w-1(x) dx " (w(y) -g(y»w (y) dy] 2 dx . j c jO [ � l X + -1 Using Hardy inequality see Hardy, Littlewood and P6lya (1952)) , we obtain L: (8"W(X )) 2w5(x) dx elI (w(x) -g(X))2W (X ) dx. A similar result holds on the interval ( 0, 1 ). So 1

l

�

=

"I

O.

�

v'XTI

�

(

�

-1

�

z = v - Pi.v in (2.83), it follows that I (v - P"t- v , g )w l I (w,v- - p"t-h,w l = I (w - p"t-w , v-- p"t-vh ,w l eN 1 1 I w I l 2,w l v - p"t-v l t .w eN 1 1 l g l w l v - p"t-v l h ,w . Then by the previous result, I v - PNI V I 01 sup (v - IPi.I wv , g).., eN-1 1 v - PN1 V I 1 ,01 eN-r l v I r,w' We can derive the result for 0 by the space interpolation and complete the By taking

�

�

�

-"' .

9

geLi!,(A)

<

proof.

p.

�

<

�

1

•

Spectral Methods and Their Applications

52 Theorem 2 . 17.

For any v E H� ( A ) HJ,w ( A ) . and r � 1, n

Proof. Let v * (x) Mv(x) Then

i: PN- 1ayV(Y) dy, i: (PN- 1ayV(y) dy - � V*(l) ) dx .

Mv E lPR, and l I axv - a,,(Mv) ll w :::; l I axv - PN-1axv ll w + � (L w(x) dX) � I v * (I) I ·

On the other hand,

I v *(l) 1 = Iv (l) - v * (I) 1 and so

I L (axv(x) - PN_1axV(x)) dx l :::; (L w-1 (x) dX) ' l I axv - PN-1 axvll w 1

pJ/v is the best approximation to v in the norm

Finally t� conclusion comes, since associated to the inner product (2.78). Theorem 2 . 18.

n

For any v E H� ( A ) HJ,w ( A ) and 0 :::;

11

:::; 1 :::;

•

r,

Pr:oof. By Lemma 2.8, for any ¢ E lPR" Il v - ph'°v l : ,w :::; cilw (v - Ph'°v, v - Ph'°v) = caw (v - Ph'°v, v - ¢) :::; cllv - Ph'°V lll ,w ll v - ¢ I h.w ' Therefore

Chapter 2. Orthogonal Approximations in Sobolev Spaces

53

and the conclusion for IJ. == 1 follows from Theorem 2.17. For the case IJ. let us consider the auxiliary problem ==

0,

(2.85)

aw (w, z )

satisfies the condition (2.9), this problem has a Since the transpose form solution. Further, we can verify that unique By virtue of (2.82) , (2 : 85) , Lemma 2.8 and the result for IJ. 1 , we know that

Il w ll 2 .w :5 c ll g ll w .

==

I (v - P�Ov,g) J

l aw (v - p�O v , w) 1 l aw (v - p�ov , w - p�O w) 1 o o :5 cl l v - p� vllt.w ll w - p� w llt .w :5 cN-r l w Il 2 .w l v ll r.w :5 cN-r Jl g ll w ll v ll r.w ==

whence

0

The result for <

IJ.

< 1 follows from the space interpolation.

•

The first proof of the above three theorems can be found in Maday and Quar teroni (1981) . . For the discrete Chebyshev approximations, there are also three kinds of Gauss type interpolations. (i)

Cheby shev- Gauss interpolation.

N�l for

(ii)

0 :5 j :5 N.

Chebyshev- Gauss-Radau interp olation.

0 :5 j :5 N, and w(O) 2;+1 ,W (j) ==

(iii)

In this case, xU)

. J

==

< o-

< -

N

N�l for 1 :5 j

2N and wU)

== ..!!...

The Chebyshev interpolant of a function IN

v

==

N :L v,1/ (x) 1=0

:5 N.

==

In this case, xU) . L for 1 < J < l. N - -

v E C(A) is ==

cos "'J�t:) and wU)

In this case, xU )

Cheby shev- Gauss-Lobatto interpolation.

, w(O) == weN)

==

N-

cos

2-:Jl

==

for

cos ti N for '

Spectral Methods ana Their Applica,tions

54 where

VI = ..!:.. t v (x (i) ) T, (x U» ) w(i) II ; =0

= �CI for 1 < Nj IN = � for the Chebyshev-Gauss interpolation and the Chebyshev Gauss-Radau interpolation, and IN = 'If' for the Chebyshev-Gauss-Lobatto interpola tion. As a consequence, for Chebyshev-Gauss-Lobatto interpolation and any ifJ E lPN, '

II

(2.86) In general, O.:lNv =f. INo",v. But we have the following result. Theorem 2 . 19. For any

v

E H�(A) , r

>

�

and 0 :5

J.t

:5 r,

Proof. We use the transformation (2.73) and denote by IN the corresponding in

terpolation associated with the interpolation points in A· . It is easy to see that (IN V r = INv· and thus Hence th, result for J.t = 0 comes from Theorem 2.4 immediately. For J.t > 0, we have from Theorem 2.14 that

Then we reach the aim, using Theorem 2.15 and the previous result.

•

Theorem 2.19 is cited from Canuto and Quarteroni (1982a) . Another estimate is h t at for r > �, We also refer to Bernardi and Maday (1989), and Maday (1990) . Finally, by Theo1 rem 2.19 and (2.42) , for any v E H�(A) , r > 2 and ifJ E lPN, (2.87)

Chapter 2. Orthogonal Approximations in Sobolev Spaces

55

In the end of this section, we state a close relation between Legendre transforma tion and Chebyshev transformation . Let r(y) be the Gamma function and 1/1 (y) = (y + + 1 ) . Denote by and a pair of (N + 1 ) x (N + 1 ) mat ices with the elements and

r

! ) r - I (y

{I,

AN,i,k =

{

AN,i,k � 1/1 2 (�) ,

� 1/1

0,

AN BN,i,k ,

(9) (�) , 1/1

BN

if 0 = j � k < N + 1 and k is even, if 0 < j � k < N + 1 and j + k is even, otherwise,

BN,i,k = 2",(;- k)(i, + �) 1 ( k-i - 2 ) 1 ( k+i-I ) ' (k+i+1 ) (k -i) 2 2 �

. ,

. ,

'I'

'I'

0,

if j = k if 0 < j

;

= 0, = k < N + 1,

if 0 � j < k < N + 1 and j + k is even, otherwise. -

Now, suppose that v(x) has a finite Legendre expansion of the form v(cos y)

N

= L a / L / ( cos y) . /=0

Then it also has a finite Chebyshev expansion of the form

where

a

= ( ao, . . . , aN)

v(cos y ) and

b

N

/=0

= L btTt(cos y)

= ( bo, . . . , bN)

(2.88)

( 2 . 8 9)

are related by the equation

Conversely, if v is a function given by (2. 89) , then it may be expressed in the form of (2.88 ) , where Based on the above fact , Alpert and Rokhlin (1991) developed the Fast Legendre

values at N + 1 Chebyshev points in A with a cost proportional to (N + 1) In(N + 1 ) .

Transformation (FLT ) . For a Legendre expansion of degree N, FLT produces its at N + 1 Chebyshev points. The cost of this algorithm is roughly three times that of Similarly, FLT produces the Legendre expansion of degree N from the values of v

FFT of length N + 1 , provided that the calculations are performed to single precision

56

Spectral Methods and Their Applica tions

accuracy. In double precision, the ratio is approximately 5.5. FLT saves a lot of work in Legendre spectral methods and Legendre pseudospectral methods , and makes them more efficient and applicable to numerical solutions of partial differential equations.

2.6.

S om e O rt hogonal Approxim at ions in Infinit e Int ervals

A number of physical problems are set in unbounded domains . The first remark with spectral methods on these problems, is that polynomials are not integrable on unbounded domains. So the idea is to work with weighted Sobolev spaces where the

( )=

weight must be of exponential type in order to avoid restrictions on the degree of spaces Lr, (A) , p � 1 , W':'I'(A) , H: (A), C ( a , bj W':'I'(A)) , H" ( a, bj W':,P(A)) and their the polynomials. We first consider the case A = (0, 00) and

w x

e -X •

Define the

semi-norms and norms in the same way as in Section 2.3 . Also we set the inner product ( . ·) w of L! (A) following the same lines as in that section. Clearly for any ,

bounded interval A*

c

A , the restrictions to A* of all functions in H: (A) belong to

H: (A* ) . We quote some essential properties of H: (A) . They state that (i) for any r � 0, the space Coo (ii) for any r � 0, the mapping: onto. H' (A)j (iii) for any 0 :::;

(iv) for any r

II

< r and 0 <

e

(1\.)

is dense in H: (A)j

v(x)

--+

e- � v(x)

is an isomorphism from H� (A)

< 1,

> � , the space H: (A)

is contained in C

(1\.) , and for any v E H� (A),

For technical reasons , we also need another Sobolev space associated with a positive integer

a,

This space is provided with the associated natural norm

II v llr,w,a = II v( l + x) � Ilr,w .

Chapter 2. Orthogonal Approximations in Sobolev Spaces

57

The Laguerre polynomial of degree I is

' (x ' e-") . £ , (x) = .!.e"o l! "

(2.90)

It is the I-th eigenfunction of the singular Sturm-Liouville problem

= I. Clearly £oCx) = l ' £ l (x) = 1 - x, and they

related to the I-th eigenvalue A, satisfy the recurrence relations ( I + 1)£ ' +1 (X)

=

1 - x)£,(x) - (1 - l )£ ' _l (X), I � 1, £,(x) = o,,£,(x) - 0,, £ 1 + 1 (x), 1 � O. It can be checked that

(21 +

(2.91) (2.92)

£,(0) = 1, 0,,£,(0) = I for 1 � 1, and 1 £,(x) 1 � e � ,

x E A.

(2.93)

The set of Laguerre polynomials is the L�-orthogonal system on the half line A, i.e. ,

By integrating by parts, we deduce that

1 o,,£,(X)O,,£m (X)XW(X) dx = IO m ' "

(2.9 4 )

The Laguerre expansion of a function v E L�(A) is 00

vex) = L: v,£, (x) '=0

with

v, =

lPN

1 v(x)£,(x)w(x) dx.

(2.9.5 )

Now let be the space of restrictions to A of polynomials of degree at most N. Several inverse inequalities exist in this space. To do this, we need some preparations. Let A* = ( a, b), - 00 � a < b � 00, and L (A') be the space of functions, p -th power integrable with the weight x(x) � O. We define the norm II ' IIL� in the usual way. Let

�

58

Spectral Methods and Their Applications

'lh(x), 1 = 0, 1 , . . . , be an orthogonal system such that 'lp, E L � ( A* ) for alI I ::; p ::; and suppose that there exist positive constant Co and some real C such that

00,

(2.96)

For any w E L � (A* ) , its formal expansion in terms of 'lh( x) is 00

w (x) = L WI'lh(x) 1=0

with

WI =

w (x)tPl (x)X (x) dx . IItP li2)( l .

�

Consider the Cesaro mean of order k, k being some non-negative integer. Set A� (,%t�)! . Then the Cesaro mean of the function w is of the form

The system { tPl } is said to be regular, if for Bome k* , there exists a constant such that for any w E L � and 1 ::; p ::; 00 ,

Cl

=

>0

Denote by QN the span { tP l (X) 1 0 ::; I ::; N}. Nessel and Wilmes (1976) proved the following result. Lemma 2 . 9 . Let {tPl } be regular and (2. 96) hold. Th en for any tP

p :5 q :5

E QN and 1 :5

00 ,

where u(N) is the same as in Theorem 2. 5. Proof. For example, we consider the case

C

>

-�. By (2.96) , (2.97)

Chapter 2. Orthogonal Approximations in Sobolev Spaces

A(X)

Let be a smooth function on A* such that for x � 2. Then the mean

A(X)

=

1 for 0 :::;

59 x

:::; 1 and

A(:C)

=

0

possesses the following properties:

LNW E Q 2N for all W E L� (A·) j (ii) LNtP tP for all tP E Q N j (iii) for any W E L� (A·) and 1 :::; p :::; I LN wI l L� :::; clllwl I L� { I y,\:o a;o+I A( Y ) I dyj (iv) for any w E L� (A* ) , (i)

=

00 ,

Indeed, the results ( i ) and (ii) are obvious. The result (iii) follows by a multiplier criterion as given in Butzer, Nessel and Trebels (1972) . The result (iv) is an immediate is of types (1, 1 ) , ( 1 , 00 ) and ( 00 , 00 ) . So consequence of (i) ,(iii) and (2.97) . Thus the classical Riesz-Thorin convexity theorem implies that is of type (p, q ) for all 1 P :::; q :::; 00 . In view of (ii), we get the desired conclusion.

LN

:::;

Theorem

Proof. Let

2. 20.

a =

For any E

0, b =

00

LN

lPN and 1 :::; p :::; q :::;

and x( x )

==

00 ,

1 . Consider the functions

1 = 0, 1 , . . . .

By (2.93) and the orthogonality of C/(x), we know that (2.96) is valid and 6 Since {lPl ( x ) } is regular, Lemma 2.9 leads to the conclusion of this theorem. Theorem

2.21.

For any

E

lPN,

=

O. _

Spectral Methods and Their Applications

60

Proof. By (2.92) ,

and so l I a3;q)I I � =

This gives the result.

� C�l �lr

The L�-orthogonal projection PN ; L�(A) v E L� (A) , or equivalently, PNV (X) = 2. 1 0 .

any v E H�(A, a ) ,

L emma

Let r � 0 and

a

�

E �� ( N

--+

-

P

)' •

IP N is such a mapping that for any

N

Ev,C,(x), 1=0

be the largest integer for which

Proof. We have

a

< r + 1 . Then for

00

EN l vr

I= + According to the Sturm-Liouville equation, we set

Il v - PNv l l � =

It is a continuous mapping from HJ, + 2(A, (3 + 2) into HJ,(A, (3) where ,, (3 � 0 and , is an integer. When r = a is an even integer, we have from integration by parts that

whence II v - PNv l l � � N-ct

A� V (X ) C l (x ) e-3; dx ) 2 ( r iA +l

f:N

I=

< N-ct I I Af v l l � � N -ct l l v ll ;,w, ' ct

Chapter 2. Orthogonal Approximations in Sobolev Spaces z _ a;,

LA a;' v (x).c, (x ) e-'" dx i{A a", (A a-' v (x) ) a",.c, (x)xe - X dx .

61

When r = a is an odd integer, we have from integration by parts again that z - !ttl 2

VI By (2.94) ,

I l v - PNv l l � ::; N-CX

::;

a ' 2 a", (A ; v (x) ) ax.c, (x)xe - '" dx ) f: ( 1 A I=N l

(L (ax.c, (X) )2 xe-x dxr 1 a ' N -cx ll a", (A ; v ) x � I I � ::; N - cx ll v l l � ,w,cx ' +

The general result follows by an interpolation argument between the spaces a + 1). 1 ) and

H�( A,

Theorem

2. 22.

For any

vE

H;(A, a )

and

wh ere a is th e larg est integer such that a p.

We observe from (2.92) that

Thus

<

0 ::; r+

p.

H�+ l (A,

0+ •

::; r,

1.

be an integer. We prove the result by induction. Clearly it is = 0 by Lemma 2 . 1 0 . Assume that it holds for p. - 1. We have

Proof. Firstly, let

true for

p.

(2.99)

.....

62

Spectral Methods and Their Applications

2.21 and (2.94), we obtain that I i O,,£N+ I I i IL-I ,W ::; cNI'- l l i o,,£N+l l i w ::; CNIL-�. If r i s an integer, we have from (2.98) and (2.99) that IIPNO"v - o"PNv l i lL- l ,w ::; CNIL-� l i v l i r,w,c> ' Using Theorem

If r is not an integer, then we still get the above result by the space interpolation. Combining this line with (2.100), we complete the induction. When f.L is not an integer, the result is proven by an interpolation argument between the two inequalities 1 ]+1 l!:l.±.!.

II v - PNV llllL]+1,w $ cN IL - 2 II v II Ir+1] ,w,a+ 1 , l i v - PNv llllL] ,w cNIIL] - � l i vl i r,W,"'+1' <

•

Theorem was given by Maday, Pernaud-Thomas and Vandeven (1985). Some times, we take 11. = (0, (0 ) and e" . The choice between the two types depends on the behavior at infinity which is enforced on the solution of the exact problem. In practice, we evaluate by various asymptotic approximations, see Bateman (1953). For instance, for 0 ::; x < 00 and large I,

2.22

w(x)

=

C/ (x)

C/(x)

"fi

_1_ e � x - t l - t cos

(2 v1x _ �)4 .

We now consider the case A = ( - 00, 00 ) and w ( x ) = e-,,2 . We define L� ( A ) , p � 1, and their norms in the same way as in Section 2.3. The Hermite polynomial of degree I is '"

H':;(A)

H/(x) = ( _ 1) l e,,2o; (e-,,2 ) .

(2.101)

a" (e -,,2 a"v(x) ) + Ae -,,2 v(x) = 0, related to the l-th eigenvalue A l = 21. Clearly Ho(x) = 1 , HI (x) = 2x, and they It is the l-th eigenfunction of the singular Sturm-Liouville problem

H/+I(x) - 2xH/(x) + 21H/_I(x) axHI(x) 21H/_I(x), I � 1 .

satisfy the recurrence relations

=

=

0,

1 � 1,

(2.102) (2.103)

Chapter 2. Orthogonal Approximations in Sobolev Spaces

63

It can be checked that (2. 1 04 )

where C2 1 .0 8 6435. The set of Hermite polynoimals is the L!- orthogonal system on the whole line A, i.e. , '"

(2. 1 0 5 )

from which and (2. 1 03), (2. 10 6 )

The Hermite expansion of a function

v

E L!(A) is 00

vex) = L: v, H, (x) '=0

with

v, = 2'zt.;;r L v(x)H, (x)w(x) dx.

Now let IPN be the space of polynomials of degree at most N. Several inverse inequalities exist in this space. Theorem 2 . 23. For any r/J E IPN and 1 :::; p :::;

Proof. Let a = functions

- 00 ,

b =

00

and

X(x) = 1

q :::;

00,

in Lemma 2.9. Consider the Hermite

t/J,(x) = (- I)' ( 2 'l!.;;rr t e� a� (e-",2 ) .

By Hille and Phillips (1957) , (2.9 6 ) is valid and 0 = 12 Since the system {t/J, } is regular, e.g. , k* = 1 , Lemma 2.9 implies the conclusion of this theorem. • -

Theorem 2 . 24. For any r/J E IPN,

.

64

Spectral Methods

Proof. By

(2.1 03 ) ,

and so

v

Their Applications

N O:c

IH,-1 ( X ) N-l N 2 lI o:c�+1 :::; c I I
The L�-orthogonal projection E L� ,

(A)

and

•

PN : L� (A) -+ IP N is such a mapping that for any

or equivalently,

N PN v(x) = L 1=0 vIH,(x) . Lemma 2 . 1 1 . For any v E H�(A) and r :::: 0, •

Proof. We have

00

Il v - PNv l � :::; I=NL+l 2'lly'7rvr

According to the corresponding Sturm-Liouville equation, we set It is a continuous mapping from where I is any non-negative into integer. When r = Q is an even integer, we have from the integration by parts that

HJ,+2 (A)

f iA

HJ,(A)

2 dx. Q v(x)H,(x)e-X2 dx = (2zt2Q iAf A2v(x)H,(x)e-:C

Thus When r = Q is an odd integer, we have

Chapter 2. Orthogonal Approximations in Sobolev Spaces

65

(2.105) and (2.106), I Iv - PN v l l� < I=Nt+ 1 21+0I+1 l!+1 1 l Vi (iAf 8", (A a;' vex) ) 8", H,(x)e-",2 dxr � (2N) - 0I1 I 8", (Aa;, v ) I ! � (2N ) - 0I II v ll�,w '

Thus by

.

7r

•

Theorem

2.25.

For any

v E H�(A)

and

Proof. It suffices to consider any integer

2.11. By

0 � I' � r,

1"

Clearly, it is true for I' = Assume that it is true for I' - 1. We have

0 by Lemma

(2.103),

Using Theorem

2.24 and (2.105),

Moreover Therefore from which the induction is complete.

Sometimes, we take A = ( -00 00 , ) and = In practice, we calculate For example, for by various asymptotic ap·proximations, see Bateman 00 < < 00 and large

H,(x) -

•

w(x) e",2.

x

1,

(1953).

Spectral Methods and Their Applications

66

(1990).

For the applications of Hermite approximation, we refer to Funaro and Kavian So far, we have considered the spectral approximations based on algebraic polyno mials. However, we can also use other kinds of orthogonal systems. Christov Boyd and Weideman used rational functions. For instance, let A = = and +

(1987) (1992) 1 (-oo, oo),w(x) (x 2 1 )R/(x) = cos(l arccot x), 1 = 0, 1, . . . . The set of { R/(x) } is an orthogonal system, i.e. , L R/(x)R.,.(x)w(x) dx = � CI' II" t5/,m where = 2 and c/ = 1 for I ?: 1. The expansion of a function v E L!(A) is v(x) = :L /=0 v/(x)R/ ( x)

(1982),

Co

00

v/(x) = � 71'C! iA( v(x)R/ (x)w(x) dx .

with Let

IRN = span {R/(x) I 0 � I � N } . The L! -orthogonal projection PN : L!(A) -t IRN is such a mapping that for any v

E L! (A) ,

or equivalently,

N PNV(X) = :L /=0 v/(x)R/(x). W.e can also estimate the error II v - PNv l w ' 2.7.

Filterings and Recovering the Spectral Accuracy

Spectral methods have the convergence rate of "infinite order" , also called the spectral accuracy, and so often provide good numerical solutions of partial differential equa tions. But this merit may be destroyed by two facts. The first is the instability of

Chapter 2. Orthogonal Approximations in Sobolev Spaces

67

nonli near computation. The second is the lower accuracy caused by the discontinuity of data. In order to remedy these deficiencies, various techniques have been devel oped, such as filterings, non-oscillatory polynomial interpolat ions and reconstructions of orthogonal approximations. We first discuss the improvement of stability. As we know, pseudospectral approx imations are preferable in practice, since they are only needed to evaluate unknown functions at interpolation points and it is much easier to deal with nonlinear terms. But they are usually less stable than the corresponding spectral approximations, due to aliasing interaction in calculations of nonlinear terms and derivatives, such as the term VO",V in the Burgers' equation. Kreiss and Oliger first proposed the fil tering operator RN for Fourier approximations. The simplest one is based on Cesaro mean. Let A = (0, 211') , tP E VN and �I be the Fourier coefficients of tP , �_ I = � I ' Then the filtering is given by

( 1 979)

If

RN tP(X) = � i 1 i :S N

( 1 - 1�1 ) �Ieil"'.

tP(x) has the error � ( x), then RN IN (�(x)o",�(x)) weakens the effect of high fre

quency components of �(x) and thus improves the stability. On the other hand, if v E C (X), then RNPNV(X) converges to v(x) uniformly in X, while it is not so for standard Fourier approximation. The drawback of this approach is that the conver gence rate of RNPNV ( X) in LOO-norm is limited to 0 (k) , no matter how smooth the function v( x) is. Kuo proposed another filtering operator RN ( ex , fJ ) based on a generalization of Bochner summation, i.e. ,

( 1 983)

( 1 _ 1�· la) fJ �Ieila;,

(2. 107) 1. This approach is analyzed in Ma and Guo (1986) , Guo and Cao (1988), and Guo and Ma ( 1 99 1 ). The key point is to. improve the stability and keep the same accuracy as

.

RN (ex , fJ ) tP ( X) = � ili SN

the standard pseudospectral methods. Theorem 2.26. For any

tP E VN and 0 � r

-

J.I

�

ex,

ex , fJ ;:::

68 If

Spectral Methods and Their Applications

in addition

J1-

�

0, then

Proof. Let

1/;(x) = I - ( I - x) '\ O S x S 1. Then 1/;(0) = ° and ° S o",1/;(x) = jJ (1 - x)iJ- 1 S jJ for ° S x S 1. Therefore I RN( a, jJ )¢ - ¢ I � S C O 1 ¢1 1 2 O !, then by Theorem 2 .4, (2.109) For fixed value of a, the accuracy of filtered Fourier approximation is still limited. as N However we can take a = a(N) and a (N) In this case, for any o S S r and suitably large N, (r-l-') I 2 (c>(N) -r+l-') 2 I 2 2 a jJ � S ( 1 RN r I ( (N), )INv - vll c(3 O

J1-

J1-

--+ 00

Ix-x' l $ p

--+ 00 .

•

Chapter 2. Orthogonal Approximations in Sobolev Spaces

69 and wm(Vj p) = w(8;'vj p) for positive integer m. For (3 = 1, Cheng and Chen (1956) showed that if a > m + 1 or a = m + 1, m being even, then for any v E C;' (X) ,

a = m + 1, m being odd, then for any v E C;' (X), RN(a, l)PNv(x) - v(x) = 0 C�� Wm (Vj �) ) . A general filtering for if> E VN is of the form RN¢>(X) = 2:N uN,I �, eil'" (2.110) 1 1 1 :5 where UN, I = UN, - I are real numbers, which decay smoothly from one to zero when I II goes from zero to N. Let v w be the convolution between v and w, defined as (2.111) v w(x) = 2� i v (y)w(x - y) dy. Then (2.110) i s equivalent to (2.112) and if

*

*

with

KN(X) = 2:N uN,l eil", . 1 1 1 :5 The coefficients UN, I could be produced by a function u ( x), 0 :5 x :5 1 such that u( O ) = 1, u( l) = 0 and 8;u(0) = 8;u(1) = 0 for 1 :5 k :5 m. Set UN,I = U ( lkJ) in (2.110). Then for smooth function v(x) , RNPN V (X) converges to v(x) pointwise in the order of 0 ( N�+ l ) ' see Vandeven (1991). " In actual computations, we could take u (x ) = e - Ot"" " where a is chosen so that UN,N is the machine zero and 2')' is called the order of the exponential filtering. Majda, McDonough and Osher (1978) discussed

the effects of discontinuous data on the stabilities and the convergences of numerical schemes for hyperbolic equations, and used the filtering of exponential type. Harvey also proposed a technique to control the oscillations of expansions of functions.

(1981)

(2.110)

Spectral Methods and Their Applications

70

Mulholland and Sloan (1991) discussed the filtering (2.107) with f3 = 1 and (2. 1 1 0 ) . They also used the filterings for the discrete Fourier transformations of derivatives of functions ¢> E VN . The corresponding filtering operator RN is defined by RNO': ¢>( X) = L (il) m),, / J/ei/x (2. 1 13) I / I$ N where ),, / are non-negative real numbers, ),,0 = 1 , ),, / = ),, _ / and ),, / decreases as 1 1 increases. Several kinds of ),, / are taken, such as ),, / = e-cx[2 , ),, / = (sin ;;�1 ) C;;�1 ) 2 and ),, / = * ( 1 + cos C;;�1 )) ' In particular, let {ap(N)} be a series , ao(N) = 1 and ap(N) decays as N and p increase. Set

Then for

v E Cp (A)

00

L ap(N)( il)2P • p=o and all interpolation points xU), ),, / =

RNINo':v (x Ci » ) = o':v (x Ci » ) + f ap(N)o,:+2Pv ( xU» )

p=1

.

If we use it for the problem

where LU = i.e. ,

d

L bq

q =1

{ U(x o U, �

Ot U + LU = 0, + 21l', t) = U(x, t),

x E A, 0 < t � T, o � t � T,

then it is equivalent to solving an artificial viscosity problem, Ot U + LU + LAU = 0

where

LA =

� (� apO%+2p) . bq

. In the early work of Gottlieb and Turkel (1980) , the filterings are coupled with the mesh size T of time t for revolutionary problems. Let U = U ( 1 ) , U(2) , . . . , u( m ) and A be an m x m matrix. Consider the problem

{t

O U = Aox U,

U(x, t) = U(x + 21l', t), U(x, O) = Uo(x),

{

x E A, 0 < t � T, o � t � T, x E A.

}

Chapter 2. Orthogonal Approximations in Sobolev Spaces

71

The corresponding approximate problem is U(x, t + T ) = U(x, t - T ) + 2TA8.,U(x, t).

Further, we use discrete Fourier approximation in the space and denote by UN (X, t) the numerical solution. The last term in the above formula is evaluated as

Gottlieb and Turkel ( 1980) replaced the above term by

where

'

sin ( laT ) , a = ap(A) , a � 1 , a and p(A) is the spectral radius of matrix A. For hyperbolic systems, there ex ists a matrix B such that BAB- l = D, D = [dl , d2 , , dm ] . Let sin ( ITD ) = [sin lTdl , . . . , sin ITdm ] . We can take A l = B-1 sin ( lT D)B. An alternative is to re place IT A by a Pade approximation. In this case, the term IT A;;(t) is replaced "I

_

_

• • •

(

)I

by eZ, and (I + a2 12 T 2 A2) el = IT A� ) 1 ' If A = I, we can take A l = 1,; (e� ITi - 1) . We next consider the recovering spectral accuracy. The usual Fourier expansion of a function with discontinuity converges slowly. For example, consider a sawtooth-like function for x :$ x', -Ax, f(x, x', A) = (2. 1 1 4 ) for x > x' 21l"A - Ax,

{

where x' is the location of the discontinuity and A = [1]." = the jump of f (x, x', A) across x'. Let VN (X) = PN v (x). Then

L

fl (x', A) ei l.,

A e - i lx

I II � l.

fN (X, x', A) =

III$ N

with fo (x', A) = A1l" - Ax', and

fl (x , A) = "

'

Z -' i-

-

f(xl+,xl,A)2;(XI-,xl,A)

IS

(2. 1 15 )

72

Spectral Methods and Their Applications

-+

We can see that i, (x', A) only decays like (t) as 1 00. As a result, the conver gence of will only be of the first order, and moreover, the Gibbs oscillations near x' will be of the order of In order to get rid of them, several techniques have been proposed. Cai, Gottlieb and Shu provided a method based on the property of sawtooth like function Assume that vex) is a periodic piecewise Coo function with one discontinuity at x' and A = [v]." . Let x" and A" be the approximations to x' and A respectively. Define

0

(2.115)

0(1).

(2.114).

(1989)

VN ( X ) =

E

i ., v,e '

+

E �"z l

e

i / (.,-., O)

.

Since the last sum is actually f (x, x" , A ' ) - fN ( X , x" , A") , we have VN ( X ) =

E

I'I� N

I'I� N

I'I> N

(v, - i, (x' , A")

)

" e "

(2.116)

+ f (x , x", A") .

(2.116) yields an essentially nonoscillatory approximation provided that the approx

imations for the location x' and the jump A are quite accurate. Indeed, it is shown that V (vN ) :5 V(v) + c

ln N

N

+ cN ln N I x ' - x" 1 + e ln N IA - A" I ,

Il v - VN I Ll :5 c l�� + e ln N l x' - x' l + c I A - A" 1

where. V ( v) is the total variation of v . Moreover, both I x' - x" 1 and I A - A" I are of if x ' and A" are determined by the following system the order of

0 (�2 ) '

{

-i(N+1).,* A* i(N+1) e i(N+ 2 )"'* A* i(N+ 2 ) e

= =

VN+1 , •

•

VN+2 '

A more efficient approach is related to the idea of Harten, Engquist, Osher and Chakravarthy ( ) Let xU) be the same as in and vex) have only one discontinuity at x'. Define a polynomial interpolant of degree m for vex) at point xU) , i.e. , Qm xU) ; v = v xU» , 0 :5 j :5 2N, Qm(x ; v) = qm .j+ � (x; v ) , for x(j ) :5 x :5 xU + 1) ,

1987 .

(2.19)

( ) ( )

Chapter 2. Orthogonal Approximations in Sobolev Spaces

73

qm ,i + � (Xj V ) interpol �tes v(x) at (m + 1 ) successive p oints x ( ,x), A m (j) � A � A m (j) + m. The stencil of these (m + 1 ) points will be chosen according to the smoothness of the data v ( x ( ,x» around xU). A recursive algorithm to define A m (j ) starts with

)

(2. 1 1 7) It means that ql,j + � (x) is the first degree polynomial which interpolates v( x) at x (i ) and xU + I ) . Assume that qp ,i + � (x) i s the p-th degree polynomial which interpolates v (x) at the points (2.1 18) Then we need one additional point for determining qp+ l ,j + � (X) . That point may be the nearest one to the left of stencil (2.1 18), i.e., x (,x"U) - l ) , or the nearest one to the right of stencil (2. 1 18), i.e. , x (,x"U) +P + I ) . The choice will be based on the absolute values of the corresponding (p + l )-th order divided differences, namely

[

if v x (,x,,(i ) - l ) , . . . , x (>',,(i ) +p) otherwise.

] < v [x (>',,(i)) ,

•

• •

]

, x (>'"U) +P+ I ) ,

(2.1 19) The piecewise polynomial Q m (Xj v) gives uniform nonoscillatory approximation to vex) up to the discontinuity. In fact, it can be shown that for 0 � k � m, (2. 120) except for the cell containing x'. Using sub-cell resolutions in Harten ( 1989), this is also correct in the shocked cell. Cai and Shu (1991) provided a uniform approximation by combining the idea in the previous paragraph and the filtering (2. 1 10). Assume that the discontinuity has been detected within an interval [x; , x�] . Denote all the points inside the interval [x;, x�] as xU'), . . . , x (ir) . Then we define a piecewise m-th order polynomial ¢(x) which interpolates the function v( x) at the points xU) , jl � j � j. ,

[

if x E x (i) , xU + I ) if x E [0, x;] , if x E [x� , 2'1l'] ,

]

n

[X I x�] for some j, '

Spectral Methods and Their Applications

74

{

where qm ,i + t (x) is defined by (2. 1 1 7) - (2. 1 1 9 ) , and PI (X) and Pr (x ) are both m'-th order polynomials on the interval [0, x l ] and [x � , 2 11" ] respectively, m' = 2 m + 1, and for 0 $ k $ m, a!PI (x l ) = a! qm,il - t (x l ) , a!Pr (x � ) = a! qm,ir+! (x � ) , (2. 12 1 ) a!PI (x � - 2 11" ) = a! qm,ir+! (x � ) , a!Pr (X l + 2 11" ) = a!qm,il - ! (x l ) . This condition ensures that 'I/;(x) is at least globally em continuous. There are

2m + 2 = m' + 1 constraints on the m'-th polynomials PI (X) and Pr (x) respectively. Therefore, they are uniquely defined. By (2.120 ) ,

( ) j, $ j $ j. , m 'I/; (x) - v(x) 0 ( N- - 1 ) , x E [x ; , x �l . ( )

'I/; xW = v xCi » ,

(2. 122) (2. 1 23)

=

Now we consider the difference w(x) = v(x) - 'I/;(x) . w(x) is a e m function in A. except at x', and [w]." = 0 (N- m -l ) . Moreover w xCi » = O, j, $ j $ jr ' Let O'N, I

=

where

exp ( - 0 1:k1 2'l)

( )

and

-- L: ( ( ) ( L: ( ( )

( ))

2N 1 v xCi » - 'I/; x U » e - il.,w 2N + 1 i=O 1 v xCi » - PI xCi » e- il.,W + v x Ci » - Pr x U» __ 2N + 1 i<;1 i>ir

( ))

Finally we define the uniform approximation by

L: ( (

)

( )) e- il.,(j)

PN v(x) = 'I/;(x) + WN,q ( X ) .

The derivatives of v(x) will be approximated by those of PNV(X), i.e. ,

It is proved that

)

(2.124)

.

Chapter 2. Orthogonal Approximations in Sobolev Spaces

75

which establishes the uniform (m + 1 Hh order convergence of the approximation P;'v to v in L 2 -norm. In the regions outside [xl, x�l, (2.124) has the convergence rate of "infinite order" . Another uniform approximation is based on the one-sided filtering. Suppose that the function v(x) is analytic over [0, 271") , but v (O) #- v(271"). Vandeven (1991) consid ered the two-sided filtering (2. 1 10) with (2p I ) ! i. I f y ( 1 - y )I P- 1 dy , p = Ni , 0 < c < 1 , UN,1 - 1 «p 1 )!) 2 io -

-

_

(2. 125)

and proved that where 8 = Ne - l , and A is a certain constant independent of N. Cai, Gottlieb and Shu (1992) defined the one-sided filtering as RNfjI ( x) = L: N �

1 1 1� N

where

u ,I l eil:r: ,

V fjI E VN

(2. 126)

m L...- ( -l)pH p I(. m ! p) 1. exp (,.p iNe -l ) , m = Ni , UN,1 = UN, 1 � = p1 *

_

and UN, 1 is given by (2.125). They proved that if v(x) is periodic, analytic in [0, 271"), but v(O) #- v(271"), then for any 0 < c < � ,

where 8 = 2N� e- l . The above filtering i s naturally labelled "right-sided" filtering �ince it can recover the spectral accuracy up to the discontinuity x = 0 from the right. The "left-sided" filtering can be constructed similarly. But a very large N is needed for the reasonable size of m in (2. 126). Cai , Gottlieb and Shu (1992) used a least square procedure to create uN, 1 for 8 � N � 32. The numerical experiments showed the advantages of this approach. Recently Gottlieb, Shu, Solomonoff and Vandeven (1992) developed a new method for recovering the spectral accuracy. The main idea of this method is to reconstruct

76

Spectral Methods an.d Their Applications

the Fourier expansion by using Gegenbauer polynomials. The Gegenbauer polynomial of degree 1 with parameter >. � ° is given by

>. CI,>.(X) = GI,>. ( 1 - x 2 ) � - OxI ( ( 1 - x 2 ) 1 + >'-� ) where

(-l) l r (>. + �) r(l + 2>.) GI , >. = 1 2 l!r(2).)r ( 1 + >. + �) , (_1) 1 y0r for 1 � 1 G 1,0 - 2 1 1 1r - (1 + �) ,

for >. >

(2.127)

0,

and

Go,o = 1. B y this standardization , Co,o( x) = 1 and for 1 � 0 , CI,o (x) = I2 TI(x),

(2.128)

The Gegenbauer polynomials satisfy the recurrence relations Ox ( ( 1

- x 2 ) >'-� CI,>.(X) ) = G Gl ',.\ 1 ( 1 - x 2 ) >'-� CI+1, >. - 1 (X) 1+1 >'-

(2.129)

and

(2.130)

It is shown that CI,>.( l ) = and I CI,>.(x) 1 � CI,>.(l) for I x l � 1. Let A = (-1, 1) 2 and w>.(x) = (1 - x ) >'-� . The set of Gegenbauer polynomials i s the L� A -orthogonal system in A, i.e. ,

��g�)

(2.131) where for >. >

0,

There exists a constant

Co

hi, >. =

y0r CI,>.(l)r (>. + �) (l + >.)r(>.)

.

independent of 1 and >. such that

Chapter 2. Orthogonal Approximations in Sobolev Spaces

77

v L�, (A) is vex) I: VI,>, CI,>. (X)

The Gegenbauer expansion of a function E =

00

with the coefficients

VI,>.

=

(2.132)

1=0

I hI ,>' JAr V(X)CI, >. (X)W>.(x) dx ,

-

1 = 0, 1,

.. .

.

The corresponding orthogonal projection is =

N

VN, >. (X) I:VI, >. CI,>. (x) . 1=0

(2.133)

Now, we consider an analytic, but non-periodic function in A. If it is extended periodically with the period then has a discontinuity at = ±l . The Fourier coefficients of are defined as

vex)

2,

v( x)

VI

=

� r

vex)

x

2 JA v(x) e -il1l"x dx.

The traditional truncated Fourier sum is =

V (X) I: vl eil1l"X . Let (x) be the Bessel function and WI , >. be the coefficients ofGegenbauer expansion of VN(X). Then they can be expressed as >' VP' WI,>. OI,OVO + + >.)r(>.) I: JI+>.(p7r ) (2) p7r The reconstructed expansion of v (x) is v'N (x) I:WI , >. CI,>. (x). (2.134) N

1>.

=

Il i �N

il ( l

O< lpl � N

=

M

1=0

c( p) depending only on p ::::: 1 such that max l a;v(x)1 � c( p) k� . (2. 135) P This is a standard assumption for analytic functions. The quantity p is the distance Assume that there is a positive constant Ixl � l

from A to the nearest singularity of

v( x) in the complex plane.

Gottlieb, Shu,

78

Spectral Methods and Their Applications

Solomonoff and Vandeven (1992) proved that if (2.135) holds, A {3 < then max Iv(x) - vN(x) 1 :5 bI N2 qf + b2qf

f,;1re,

1",19

where

bl and b2

are certain positive constants related to 0$/$00 max

ql =

( 21re )

27{3 f3

-

q2 =

< 1,

( )

= M =

{3N and

l tid , and

27 f3 < 1. 32 p

The meanings of ql and q2 are explained i n their paper. (2. 134) gives a uniform approximation to v(x), including the discontinuity, and recovers the spectral accuracy. We also refer to work in Gottlieb and Shu (1994) . All of the methods in the previous paragraphs are also available for Legendre approximation and Chebyshev approximation. For instance, let v E IP N and be the Chebyshev coefficients of v(x) , 0 :5 :5 N. Guo and Li (1993) offered a filtering. Ma and Guo (1994) considered RN (a, {3) and R�(a , {3) for ¢> E IP N , defined by

ti,

1

RN (a, {3)¢> (x) R� ( a, {3) ¢>( x)

� (l - l � rr �/TI (X) , � +

where the constants

( 1 1 rr �,T,(x) a N-ITN-I (X) aNTN (x),

N-2

1-

N

a, {3 � l,

(2. 136)

(2. 137)

a N-I and a N are determined by

R� (a, {3) ¢>(-l ) = ¢> ( - 1 ) , Let

+

a, {3 � l ,

�(a, {3) ¢> (1) = ¢>(1).

w(x) = ( 1 - x 2 ) - ' . It can be checked that 1

( RN (a, {3)¢>, 1/Jl.., = ( ¢>, RN (a, {3) 1/Jl.." V ¢>, 1/J E IP N , ( RN (a, {3) ¢>, tPl.., = (¢>, R� (a , {3) 1/JL , V ¢> E IP N-2 , 1/J E IP N . Theorem 2 . 27. For

any ¢> E IPN and 0 :5 r :5 a,

(2. 138) (2.139)

Chapter 2. Orthogonal Approximations in Sobolev Spaces

Proof. We have

-2 (32 L:1=1N Il-N 1 2'" ¢; :::; c(32N-2r L:1=1N [2r ¢; 2 < c(32 N -2 r t [2 r ¢; :::; c(32 N -2 r [' l o� ( L: ¢ l l I eiI Y ) 1 dy 1=0 I I I $N 2 :::; c(32N-2r L: l o� (� ¢cT! (COsy )) 1 dy :::; c(32 N - 2r ll v ll �,w ' < 1

79

7r

- 7r

Theorem 2 . 28.

For any ¢>

with

E

lP N }

b", ,/3 - ( _

If in addition ¢> with Proof. Let

E

lP � }

then

d"',

(

_

/3

-

+ l) f ( �) (3 ( + 1 + �)

)

2

+ (�) ( + 1 + �)

)

:;

f (2 (3 of 2

f (2 (3 2 ) f of 2 (3

ox RN( O, (3 ) ¢> ( x ) =

al = -C2I

Then

and thus I RN( 0 ,

(3)¢> I � ,w

•

L: p

p=l+l p+ l odd

1

1

L:- acT!(x) .

N 1 1=0

(1 - .I -NP 1 "') /3 ¢>1' A

/3 1 ¢d ) 2 I 1 [ � (� ( _ J"') < cN � [ 2 ( 1 I � I"') 2/3 � ¢; :::; cN3 � 1 � 1 2 (1 _ I � I"') 2/3 � ¢; :::; cN4 11¢>1 I � l y ( 1 - y"' ) 2/3 dy CN4 b!,/3 I ¢>I I � . % �1 ci a ; :::; cN _

2

=

Spectral Methods and Their Applications

80 The second result can be proved similarly.

_

A general form of filtering for Chebyshev approximation is

N L: = UN, ,�m(X),

V ¢> E IPN 1=0 where 0 � UN,I � 1 , UN,I "" 1 for I � N and UN,I � 1 for I "" N. Weideman and Trefethen (1988) used this approach for evaluating eigenvalues of matrix related to the second order differentiations of Chebyshev expansions, and took

RN ¢> (X)

UN,I =

{ I, (_ (l=ilL)4) a

exp

N - 1o

o � I � 10 , '

a

> 0, 10

<

I � N.

In the early work of Fulton and Taylor (1984), the filtering is coupled with the mesh size T for the variable t, for the revolutionary problem

�1)

Let (UN(t» ) be the coefficients of the first order derivative of the corresponding numerical solution by Chebyshev approximation. Then the filtered ones are

where or

A(X) = 8 sin x -6 xsin(2x)

� [1 1 - 18e-" + g e-2., - 2e-3"j .

A(X) = 6

Various methods for recovering the spectral accuracy can also be generalized to Legendre approximation and Chebyshev approximation. For example, Gottlieb and Shu (1995a) developed reconstruction methods for these two approximations. Let l v ( x ) be a function in L (A) , which is analytic in a subinterval [a, b] C and satisfies a condition like (2. 135) . Its Gegenbauer projection VN, I' (X ) is given by (2. 133) . Set c = !(b - a), o = !(b + a) and

A

(2. 140)

Chapter 2. Orthogonal Appr9ximations in Sobolev Spaces

If

p,

�

0 and >.

.,,;,

M

=

(lc:N

with (l

<

81

fT, then

f!t� I V (C:Y + 8) - �WI,>. CI,>' (Y ) I :5 b (q�N + q;N) (J

{J 2 (l 7 ( 2 ) 1, q2 = ( 3227pC: ) < 1, and b grows at most as N1+2/J. The proof is given in Gottlieb and Shu (1995a). (2.140) gives a reconstructed uniform approximation for Gegenbauer expansion. By (2.128), it covers Legendre approximation and Chebyshev approximation. In Gottlieb and Shu (1996), sharper result exists for Legendre approximation. On the other hand, we can also recover the exponential accuracy from the values of piecewise analytic functions at interpolation points, see Gottlieb and Shu (1995b). where

ql =

a

<

CHAP TER 3 STABILITY AND CONVERG ENCE

The theory of stability and convergence plays an important role in numerical solu tions of differential equations. Courant, Friedrichs and Lewy ( 19 2 8) first considered the convergence, and Von Neumann and Goldstine ( 1947) started the study of sta bility for linear problems. Kantorovich (1948) developed a general framework for approximations of linear operator equations in Banach space, and Lax and Richt myer (1956) dealt with linear initial-value problems. For nonlinear problems, Guo ( 1965) , Stetter ( 1966) and Keller ( 1975) provided several kinds of stability, which also imply the convergence sometimes. This chapter is a survey of those results.

3.1.

St ability and Convergence for Linear Problems

Bl

B2 be two Banach spaces with the E Bq and R > 0, the balls are

We consider a general framework. Let and norms II . I I Bt and I I . l iB> respectively. For any defined as

Let L be an operator from operator equation

v

Bl to B2 , and f be a given element in Bz . Consider the LU = f·

( 3.1)

Bl , B2

In order to solve ( 3 . 1 ) approximately, we first approximate the spaces and be finite dimensional Banach spaces L, f. Let N be any positive integer, and with the norms II . IIB " N ' q = 1 , 2. For any v E " N and R > 0, the balls are defined

Bq , N B

82

Chapter 3. Stability and Con,vergence

by

83

S

= {w E Bq ,N I li w - v I i B.,N < R} , q = 1 , 2. Assume that Dim(Bl ,N) = D im( B2,N ) , and Bq,N approximates Bq , q = 1 , 2. Let Pq,N q , N ( v , R)

be continuous operators from Bq to Bq,N such that .

Ji� I i Pl ,N v Ii Bl.N = I lv I i B" . 'V v E Bl , 2,N , 'V v E B2 · = Nlim ......oo I lp v Ii B. . N I i v I I B.

(3.2) (3.3)

Let qN be a continuous operator from B2,N to B2 , satisfying (3.4) We denote by B3 another Banach space with the norm II · Ii B3 ' UN is a continuous operator from Bl ,N to B3 , while w is a continuous operator from Bl to B3 • Suppose that there exist positive constant Co and positive integer No such that for all N � No, (3.5) and for any v E Bl ,

= UN ,N Nlim -+oo Ii Pl V - wv l i B3

O.

j

(3.6)

In this section, we assume that all operators L, Pl ,N , P2,N , qN, UN and w are linear.

B'

/

�

L

B',

' B2

---

L-l

"'oN

LN

"" N,

qN

B1-,N----------------L-7l----------------N

Figure 3 . 1 .

Let L N b e an invertible operator from B1 ,N t o B2,N. We approximate (3. 1 ) by (3.7)

Spec tral Methods and Their Appli cations

84 The approximation error is defined as

(3.8)

f

Let D be a subset of solutions of (3. 1 ) , and F be the set of all related to U E D. Assume that F is dense in The approximation (3.7) is said to be consistent with (3. 1 ) , if for . any v E D ,

B2 •

(3.9) IT for all U, the solutio�� of (3. 1 ) and holds

UN, the corresponding solutions of (3.7) , there

then we say that (3.7) is convergent. Usually the data possesses certain error, denoted by It induces the error of denoted by If there exists a positive constant Cl such that for any j and

UN ,

UN.

N 2: No ,

f

1.

(3.10) then we say that (3.7) is stable. Since

LN and UN are linear, it is equivalent to (3. 1 1 )

Since

UN L'ilP2 ,Nf , i t i s also equivalent to =

(3. 12) Theorem

3. 1 .

If L - 1

exists, and (3. 7) is consistent with (3. 1), then the stability is

equivalent to the convergence. Proof. IT (3.7) is convergent, then for all

l I uNLf/P2 , N f Il B3

f,

Thus is bounded uniformly for N. By the Uniform Boundedness Theorem, (3.12) is valid.

.

Chapter 3. Stability and Conyergence

Now assume that (3.7) is stable. For any mate solution

UN,

85

U E D and the corresponding approxi

The first term on the right-hand side tends to zero by (3.6) . Since (3.4) implies

B2 ,N ,

IIUNUN - UNP1 ,N U I B3

IIUNLiV1 p2 ,NI - uNLiV1 p2 ,N QNLNP1 ,NU I B3 I uNLiV1 p2 ,N (LU - QNLNP1 ,NU) I B3 ::; IIUNLiV1 p2 ,N I I RN(U) I B2 '

By (3.9) and (3.12), the above right-hand side tends to zero as N any E D,

U

U

LNP1 ,NU

-+

00.

E

(3. 14)

Hence for (3.15)

If is a solution of (3. 1 ) , but UED, then there exists a sequence to the data such that

{ /(I) }

{ U (I) } E D, related (3.16)

ulj}

Let be the corresponding approximate solutions of (3.7) with the data we have

1(1) . Then

IIUNUN - WU I B3 ::; I UNUN - uNuIj}I I B3 + l I uNulj} - wU(l) I B3 + I w U (I) - WU I B3 ::; I uNLiV1 p2 ,N I I J - 1(1) li B. + IIUNulj} - WU(l) I B3 + I w 111l L - 1 11111 - 1 ( I) II B2• (3. 17) By (3. 12) , (3. 15) and (3. 16), the above right-hand side goes to zero as N, This completes the proof.

l -+

00. •

Theorem 3 . 1 can be found in Guo (1988a) . We now turn to the convergence rate. Theorem 3 . 2 . II

L- 1

exists, and (3. 7) is stable and consistent with (3. 1), then

Spectral Method� and Their Applications

86

Proof The conclusion comes immediately from (3.13), (3. 14) and (3. 17).

•

If w is an isomorphism between and then the above estimate also holds for IJW- 1 UNUN - II B, . = = qN, UN and w are the identity op If = 1 , 2, then erators. If in addition D = then Theorems 3.1 and 3.2 turn to be the Basic Convergence Theorem . of Kantorovich (1948) . Besides, N could be replaced by any real parameter. It is also noted that in spectral methods, are certain subspaces are some projections from = 1 , 2, usually. In this case, to of and = we can take and II · IIB'o N = II · liB. , = 1 , 2. Therefore UN and w are the identity operators, and so

Bl

U B3 Bq Bq , N , q

Bq ,

3.2.

B2 , Pl,N , P2 ,N ,

B1 ,

Bq Bq ,N, q

Pq , N B3 B1

Bq , N

q

qN,

Generalized Stability for Nonlinear Problems

In this section, we deal with nonlinear problems. We still consider problem (3. 1 ) , but have the same meanings as and may be nonlinear. The notations and could be nonlinear. Let before. They also fulfill (3.2) and (3.3) , but and approximate and respectively. Consider the approximate problem

L

LN

IN

L

I

Bq, Bq,N Pq,N Pl,N P2 ,N

(3.18) The approximation error is defined as = Let D be the same as before. If is a solution of (3. 1 ) , then The approximation (3.18) is consistent with (3. 1 ) , if for any v E D and I E F ,

U

If for all

RN(U) IN-LNPl,N U ,

U, the solutions of (3. 1 ) and UN, the corresponding solutions of (3. 18) ,

Chapter 3. Stability and Convergence

87

then we say that ( 3.18 ) is convergent. How to define the stability is a problem. The simplest way is to follow ( 3.10 ) . E and It means that there exists a positive constant Co such that for any N � No, ( 3.19 )

v, w B1,N

Such kind of stability is called linear stability. It is an alternative expression to the stability in the previous section. But nonlinear problems usually do not possess this property. Indeed, this stability is very strong so that it is essentially suitable for linear problems and some specific nonlinear problems, such as sine-Gordon equation. Firstly, we see that the constant Co is uniform for all whereas the corresponding constant depends on the exact solution of ( 3.18 ) for most nonlinear problems. Next, ( 3.19 ) is However, the feature of nonlinear problems is that valid for any only when it lies in certain region, the difference between and is controlled. But may grow rapidly, once exceeds some critical bound. Thus ( 3,19 ) cannot describe the essence of nonlinear problems properly, Finally, Clearly this is not possible for nonlinear problems E ( 3,19 ) holds for all with several solutions, These facts motivated us to look for new definitions of stability, Strang ( 1965 ) served the weak stability for linear problems, which is also used in spectral methods, see Gottlieb and Orsag ( 1977 ) . Stetter ( 1966, 1973 ) developed the nonlinear stability applied to initial-value problems of nonlinear ordinary differential . equations, while Guo ( 1965, 1982 ) provided several kinds of generalized stability used for numerical solutions of nonlinear partial differential equations, All of these works are suitable only for problems with unique solutions, Keller ( 1975 ) generalized the stability in a different way and proposed the local stability, which has been applied successfully to boundary-value problems of nonlinear ordinary differential equations. In 1985, Guo unified the above work, see Guo ( 1988a, 1989 ) . We shall present this general result below, We first int roduce some terminologies. The operator is said to be stable in the ball R) , if there are positive constants R and CR such that

UN ,

I LNv - LNw I i B2 ,N '

I LNV - LNW I B2 ,N

I l v - wI i B1 ,N

v

w

v, w B1 ,N '

Sl (v ,

L

( 3.20 )

Spectral Methods and Their A,pplications

88

L

If U is a solution of (3.1) and there exists a positive constant R such that is stable in 81 (U, R) , then we say that U is a stable solution. Obviously such a solution is ( unique in 81 (U, R) . Assume that the data in (3. 1 ) possesses the error - which induces the error of. solution, denoted by If the operator is linear and stable in - f. 0 and U' - = R $ the ba1l 81 (U, R); then • . Indeed, let Then

I I U�II B, $ R and

II U IIB, cR ll f I lB

U.

f

L ( U + U- ') = f + II URlB, . II I U' II B, - cR1l�JIU II1BIIB2 ,

By (3.20) ,

L II U I B,

f

U · U II IIB,

<

and the desired result follows. Another kind of solution is called isolated solution. If there exists a linear continuous operator such that

L'(v) L(v + w) - Lv = L'(v)w + E (v, w)

and

II E (v, w) I IB. 0, I w llB, then we say that L is Frechet differentiable. L' (v) is called the Frechet derivative of L at the point v. Further, if L'(v)w = 0 implies w = 0, then we say that L'(v) is nonsingular at v . Moreover, if U is a solution of (3.1) and L'(v) is nonsingular, then lim

IIwIl B, -+O

=

we say that U is an isolated solution of (3. 1 ) . There is a close relationship between stable solutions and isolated solutions. Keller ( 1975) proved that

L'( U) exists, then U is an isolated solution; (ii) if U is an isolated solution of (3.1 ) and L'(U) is Lipschitz continuous in 81 (U, R) , (i) if U is a stable solution of (3. 1 ) and then U is a stable solution. Now, we give the definition of the stability for the approximation (3.18). If there are positive constant eo( N) and non-negative constant C1 N) such that for all N 2:: No and E 81,N ( R) , the inequality

w

v, v,

(v,

(3.21)

Chapter 3. Stability and Convergence

89

implies (3.22) then we say that (3.18) is of generalized weak stability for v in the ball Sl ,N( v , R) . Especially, if for all N 2:: No and E Sl ,N( v , R) , the inequalities

w (q)

(3.23) imply

I w(l ) - w(2) IIB" N :::; Cl (V , N) I LNW(l ) - LNw(2) IIB2 , N '

(3.24)

then we say that (3.18) is of generalized uniform weak stability for v in the ball Sl ,N(V, R) .

We have the following result. Theorem 3 . 3 . L et L N be a linear operator. The approximation (3. 1 8) is of linear

stability, if and only if R =

00,

and eo(v, N) and cl (v, N) are independent of v and

N. Proof. If (3.18) is of linear stability, then (3.19) holds for any N 2:: No and any v, w E B1,N' Thus R

=

oo , cl (v, R) == eo and eo(v, R) is arbitrary. Conversely,

assume that R = oo , eo(v, R) ==

Co

and cl (v, R) == Cl . For any

q) w (q) = I LNv (1 ) -CovL(Nv ( 2 ) I Bo , N '

q=

v (q ) E B1,N, let

1 , 2.

Then and so by (3.22) ,

By the linearity of

LN, we reach (3.19) with eo

=

Cl .

•

Another s�fficient and necessary condition of the linear stability for linear problem (3. 18) is that R = 00, and ea(v, N) and Cl (V, N) are independent of v and N. We now give a sufficient condition ensuring the generalized uniform weak stability. Let denote the Frechet derivative of L N .

L'rv

Spectral Methods and Their. Applications

90 Theorem 3.4. Assume that

(i) LN is defined an¢ continuous in Sl ,N( v, R) ; (ii) for all w E Sl ,N(V , R) such that LNw E S2,N (LNv , cQ (v, N)) , the inverse (LN (w) ) - l exists and l ( 3.25 ) II (L�(w) ) - ll � c� (v , N) . Then (3. 1 8) is of generalized uniform weak stability for v in Sl ,N( v , R*) , 0 < R* � R. Proof. Let

(

: )

By the Implicit Function Theorem ( see Schwartz ( 1969 )) , for any w E Sl ,N (V , R) such that LNw E S2,N(LNv , r (v , N) ) , the inverse L;/ exists in an open neighborhood of LNw. Moreover, r (v, N)

=

min c� (v , N) , Ci

( N) '

We have from ( 3.25 ) that

( 3.26 )

Because of the convexity of the ball S2,N (LNv, r(v, N) ) , there exists the inverse (LNl ) ' (g(A)) satisfying ( 3.26 ) . Therefore it follows from the Mean Value Theorem that 2 II LNl ( LN W (l) ) - LNl (LN W ( ) ) I I BI,N 2 � c� (v, N) I I Livw ( l) - LNw ( ) I I B2,N '

We next turn to the existence of solutions of ( 3 . 1 8 ) .

Thus the conclusion follows.

Lemma 3 . 1 . Suppose that

•

Chapter 3. Stability and Convergence

91

(i) LN is defined and continuous in Sl,N( w, R) i (ii) for a!l v ( q)

E Sl,N(W, R)

such that LNV ( q)

E S2,N (LNW, r),

2 l l 2 I I v ( ) - v ( ) I I B1.N ::s: O" II LNv ( ) - LNv ( ) I I B2,N ,

( �) . Then for any

Let ro = min r,

9

E S2,N (LNW, ro) ,

we have

0" >

O.

the problem

LNv = g

possesses a unique solution in the ball Sl ,N( w, R) .

Proof. Since condition (ii) implies the uniqueness of L i/ in S2,N (LNW, rO) nLN (Sl ,N( w, RY it suffices to prove that S2,N (LNW, ro) C LN (Sl,N( W, R) ) . Now assume that g ' E S2,N (LNw, ro) , but g ' ELN (Sl,N(W, R)) . Let g(A)

and

x

=

(1 - A)LNW + Ag ' ,

A�0

(3.27)

{

sup { A ' � 0 1 g(A) E LN (Sl ,N(W, R) ) } , (3.28) 0, if the above set is empty. If X > 1, then g ( l ) = g ' E LN (Sl, N(W, R) ) and the conclusion follows. If X = 0, then y = g (X) = LNw E LN (Sl,N(W, R)) and the inverse ii = Li/Y = w. If 0 < X ::s: 1 , then for any e < X , =

and s o the element L "i/ (g (X - e )) exists. B y condition (ii), there exists the limit ii such that ii = �� L;\i (g ( X - e ) ) . Moreover, we have from condition (ii) and (3.27) that II L;\/ (g ( X - e ) ) - w I I B1 ,N ::s: O" ll g ( X - c) - LNw IIB •. N

< roO"

LN (ii) = g .

and thus

ii E

(X - c) :::; (1 - c)R

=

0" ( X - c) 119' - LN W I I B2,N

Sl,N ( w , R) . By the continuity of L N in the ball Sl , N( w, R) , we have

Spectral Methods and Their,Applications

92

Now we choose a closed ball (ti, e) c w, such that (ti, e)) c (ti, e) and (ti, e)) . is a one-to-one mapping between N W , r) . Then Dim Since Dim (ti, e)) contains an open neighborhood of g. This is a contradiction ,to the definition of X by (3.28) . The proof is complete. _

S2 ,N (L

(B1, N ) =

Theorem 3 . 5 . L et

(i)

LN

LN

U

S1, N

S1,N ( R)

(B2 ,N) , LN (S1,N

S1,N

be a solution of (3. 1) . Assume that

is defined and continuous in

S1,N (P1, N U, R) j

(ii) (3. 18) is of generalized uniform weak stability for (iii)

II RN(U) 1 I 2 ,N < r(N) r(N)

=

and

min

W =

P1, N U and ro

=

P1,N U in S1,N (P1, N U, R) j

(eo (P1,NU, N) , C1 (P1,RN U, N) ) .

Then (3. 1 8) possesses a unique solution Proof. Put

LN (S1,N LN (S1,N

UN in the ball S1,N (P1, N U, R).

r(N) in Lemma 3 . 1 . Then L'il exists in the ball

S2 ,N (LNP1, N U, r(N) ) . On the other hand, condition (iii) implies Thus

fN E S2 ,N ( LNP1,N U, r(N)) and so the result follows.

-

We can evaluate the solution of (3.18) by the Newton procedure

UN(m+1) - UN(m) - (L'N (UN(m) )) -1 (LNUN(m) - fN) , _

m � O,

(3.29)

m � O.

(3.30)

or the simplified Newton procedure

UN(m+1 ) - UN(m) - (L'N (UN(0) )) - 1 (LNUN(m) - fN ) , _

It can be verified that both (3.29) and (3.30) are convergent with the same limit as the solution of (3.18), if the following conditions are fulfilled: (i) L� exists in

S1 ,N (u�) , p) , and for all v(q) E S1 ,N (u�) , p)

and certain

dN > 0,

Chapter 3. Stability and Convergence

(iii)

=

1JN aNbNdN < -21 and

p

�

93

1 - y'1 - 2r/N

aNdN .

We now explain the relation between the stability defined in the above and the which induces stability in actual computation. For instance, let be the error of the error of denoted by Then we have

UN,

iN

UN.

Therefore (3.21) and (3.22) mean that for any in the ball +

Sl,N (UN, N),

UN UN

iN

N � No and the disturbed solution

provided that If Ljil

exists, then (3.18) has a unique solution and R = 00 . If, in addition, = then we get the best stability for usual nonlinear is arbitrary and problems. It means that there is no restriction on the data error. In this case, this stability is nearly the same as the linear stability. But they are still different, since = = then the depends on and If error of solution is bounded by the data error provided that the data error does not exceed a critical value. Such stability was also investigated by Stetter (1966) . = Further, if and = then it turns out to be � the generalized stability of Guo (1982). It is lso called g-stability. The infimum of such values s' is called the index of g-stability. On the other hand, if is arbitrary and = then it is a generalization of the weak stability of Strang (1965) . Finally if R < = and is arbitrary, then it is equivalent to the local stability by Keller (1975) . It means that in the ball R) , the approximation (3.18) is locally stable. Under some conditions, the above stability implies the convergence.

Cl(UN)

CO (UN, N)

Cl (UN, u) Cl(UN),

UN . co ( uN, N) CO ( UN)

CO (uN, N) co(N) N- "

Cl(UN, N) Cl(UN),

Cl(UN, N) cl(N) ,

Cl (UN, N) cNq , oo, Cl(UN, N) Cl(UN)

Sl, N ( UN,

CO ( UN, N)

co(uN, N)

Spectral Methods and Their Applications

94 Theorem 3.6. Let

U

and respectively. Suppose that

UN E Sl,N (Pl,N U, R)

be the solutions oj (3. 1) and (3. 18)

(i) (3. 1 8) is oj genemlized weak stability Jor

Pl ,N U in Sl ,N (Pl ,N U, R) ;

Then Proof. Let

UN UN - Pl,N U. Then LN (Pl,N U) IN - RN(U), LN (Pl,N U + UN) IN. =

=

=

Hence We know from Theorem 3.6 that (3.18) is convergent, if -+ (X). In particular, if (3.18) is of the generalized stability with the index s and > max ( s, 0), then •

o as

Cl(Pl,N U, N) I RN(U) 1 I 2 ,N

N -+

I RN(U) I ! B2 .N :::; c2 (U)N- q , q I l uN - Pl,N U I B, . N :::; c2 (U)N-Q .

3.3.

Initial-Value Problems

We focus on initial-value problems in this section. Let Bl and B2 be two Banach and B � B2 . spaces with the norms and are two differential operators with respect to the spatial variables, mapping v E B into B2 . For simplic ity, assume that is a linear invertible operator. Denote by B3 the third Banach whose element is a mapping from the interval (0, T] to space with the norm B T> Let J(t) be a given element in B3 and be a given element in B1 • describes the initial state. Consider the operator equation

I . I BI

Q

2,

O.

I . l i B" l

I . I B3'

Q

L l

Uo

{ 8U(O)t (Q U(t))Uo.

=

=

L U(t) + J(t),

0 < t :::; T,

Uo

( 3 .31

)

Chapter 3. Stabilit,y and Convergence

95

We define a genuine solution of (3.31) as a one-parameter family U(t) such that U(t) is in the domain of Q and L for each t E [0, TJ , and

· I � (QU(t + s ) - QU(t)) - LU(t) - f(t) t,

�

0,

as s --+ O.

Let Dl and D3 be such subsets of Bl and B3 that for any Uo E Dl and f E D3 , (3.31 ) has a unique solution. Suppose that Dl and D3 are dense in Bl and B3 respectively. The corresponding subset of solutions is denoted by D 2 • Now let N be any positive integer and Bq,N be finite dimensional Banach spaces with the norms II · II B " N ' approximating the spaces Bq , q = 1 , 2, 3. In addition, Dim(B1,N) = Dim(B2 ,N) . Denote by Pq,N the continuous operators from Bq to Bq ,N, satisfying the following conditions: (i) for all v

E Bl , Pl,NV = P2 ,NV i

(ii) for all v

E Bq , I l pq,Nv ll

--+

IIv ll q as N --+

00, q

=

1 , 2, 3.

Next, let r = r (N) > 0 be the mesh spacing of the variable t, and Dr v be the forward difference quotient with respect to t, defined in Chapter 1 . We now approximate QU(t), LU(t) , f(t) and Uo by QNUN(t) , LN(UN(t) , UN(t + r)) , fN(t) and UN,O respectively, and then consider the approximate problem

{

Dr (QNUN(t)) UN(O) = UN,O.

=

LN (UN(t) , UN(t + r ) ) + fN ( t),

(3.32)

Suppose that for each t, UN(t) is determined uniquely by (3.32) . The simplest case is that Q ,/ exists and LN (UN(t) , UN(t + r)) is independent of UN(t + r) . The approx imation error for t > 0 is defined as RN(v(t))

=

- P2 ,N 8t Qv(t) + DrQNPl,NV(t) + P2 ,NLv(t) - LN (Pl,NV (t) , Pl,NV (t + r)) + P2 ,Nf(t) - fN(t) .

If U(t) i s the solution of (3.3 1 ) , then RN(U(t)) i s reduced to

Spectral Methods and Thejr, Applications

96

The approximation (3.32) is said to be consistent with (3.3 1 ) , if for any fE ::; T, ( and 0

::; t

D 2 , D3

If for all

Uo E. D1, U E

U(t), the s�lutions of (3.31) and UN(t), the corresponding solutions of (3.32) , I p1, N U(t) - uN(t) I B1 ,N - 0 , 0 ::; t ::; T, as N -+ 00,

then we say that the approximation (3.32) i s convergent. We now turn to the stability. We first consider the linear stability. It means that there is a positive constant eo(T) such that for all N � and 0 ::; T,

uW

No

::; t

u��o

fiJ ) , q =

where are the solutions of (3.32), corresponding to and 1 , 2. Following the same line as in the proof of Theorem 3 . 1 , we can prove the following Lax Theorem ( see Richtmyer and Morton ( 1 967 ) ) . Theorem 3 . 7 . If (3. 31) is a well-posed linear problem and (3. 32) is consistent with

(3. 31), then the linear stability is equivalent to the convergence. Moreover for all

0 ::;

t ::; T,

We now consider nonlinear problems, e.g. , see Guo ( 1 994 ) . If there are positive constant N, T) such that for all N � N, T) and non-negative constant C1 No, the inequality

eo( uW ,

(u�) ,

I uN,(1 )O - uN,(2 )O I B" N + Il fN(l ) - fN(2) I Ba ,N ::; eo (UN(1) , N, T)

(3.34)

implies

l I u�) (t) - u�) (t) I Bl N ::; C1 (uW, N, T) ( 1 I u��o - u��o I l Bl,N + Il f� ) - f� ) I Ba , N ) ,

o ::; ::; T,(3.35)

t

Chapter 3. Stability and Convergence uW (t) being the solutions of

97

( 3.32 ) related to u ��o and j
=

1, 2, then we say

that the approximation ( ) is of generalized weak stability at u �) . There is a close relationship between the generalized weak stability and the con vergence, which extends the result of Lax Theorem.

3.32

Theorem 3.8. Let U(t) and UN (t) be the solutions of {3.31} and {3.32} respectively. If {3.32} is of generalized weak stability at Pl ,NU (t) and then for all 0 � t � T,

Proof. We have

{

D-r QNPl ,NU(t) = L N (Pl ,NU (t), Pl ,NU(t + T )) + fN(t) + RN ( U (t) ) , Pl ,NU(O) Pl ,NUO .

=

0 < t � T,

(

3.36 ) By comparing ( 3.36 ) with ( 3.32 ) , the desired result comes immediately from ( 3.35 ) . •

The simplest case of generalized weak stability is also called g-stability, see Grif fiths ( ) In this case,

1982 .

Co

(UN, N T)

=

Co

(u N , T) N -" ,

Cl ( UN , N, T)

= Cl (uN , T) .

The infimum of such values s' is also called the index of g-stability, denoted by s . Clearly, i f UN,O Pl , NUO , II Pl , NUO - Uo I I B" N 0 (N - q ) , I I RN (U) II B3,N 0 (N - q ) , > o and s < then ( ) is convergent with the accuracy 0 (N - q ) . I n derivation of generalized stability, we often meet some nonlinear discrete in equalities. Let R-r(T) be as in ( )

q,

,

=

=

3.32

=

q

1.21 .

Lemma 3 . 2 .

Let E(t) be a non-negative function and pet) be a non-decreasing func tion on R.,. (T) . The constants d > 0, A > 0 and c � o. Assume that

Spectral Methods and Their Applications

98 (i) if E (s)

:::; d

for all s E

R,.(t - r ),

then

FE (t - ,r ) FE ( E (O) , E ( r ), · · · , E(t - r )) :::; cE(t - r ), GE (t) GE ( E(O), E ( r ), · · · , E (t)) � A E(t)j =

=

GE (t) :::; P(tl) + r L: FE (s)j .•

Then for all t E

Proof. Let

.,p(t)

ER� (t-'T)

R,.(tl), be such a non-negative function that

�

.,p(0) P l) =

A.,p(t) P(tl) + cr L: .,p(s) .

and for all t

:::; tl ,

=

•

Then

ER�(t-'T)

�

.,p(t) :::; P l ) est :::; d. Clearly E (O) :::; .,p(0). Assume that E (t) :::; .,p(t) for t E R,.(tl t :::; tt ,

) Then for any

r .

A E (t) :::; P(tl) + cr L: E (s) .E� (t-'T)

:::; p(tl) + CT L: .,p(s) A.,p(t) =

.ER�(t-T)

which implies the conclusion. In many cases,

GE (t) = E (t)

FE ( t ) = Co In addition, if for

_

and

(1 + � NaIEbl (t)) E (t) + H(E (O), · · · , E (t)) ,

s :::; t , E ( s) :::; CNd

and c >

0, then

H(E(O), · · · , E (t)) :::; o.

hi �

O.

Chapter 3. Stability and Convergence

Therefore, if

99

(

)

p ( t 1 ) e cott(M+ l ) <- min min N-� ' eNd , l�I� M then for all t E R.. (t l ), ( 3.37 )

In particular, if FE(t) = eoE(t) and H(E(O) , · · · , E(t)) $ 0, then ( 3.37 ) holds for all p ( t ) and t E R'T(T). A more general result is given by Theorem 4.17 of Guo ( 1988a) . In that case, if E(s) $ for all s E R'T (t T ) , then

d

-

It is more suitable for numerical analysis of implicit spectral schemes for evolutionary problems of nonlinear partial differential equations.

- I ',

CHAP TER 4 SP EC-TRAL METHODS AND P SEU D OSP ECTRA-L METHO D S

In this chapter, we apply various orthogonal approximations in Sobolev spaces to numerical solutions of partial differential equations. Particular attention is paid to nonlinear problems arising in fluid dynamics, quantum mechanics and some other top ics. We use Fourier spectral methods and Fourier pseudospectral methods for periodic problems, Legendre spectral methods, Legendre pseudospectral methods, Chebyshev spectral methods and Chebyshev pseudospectral methods for non-periodic problems. Some filterings are adopted in pseudospectral methods for the improvement of stabil ity. Several spectral penalty methods are introduced for initial-boundary value prob lems with inhomogeneous data on the boundary. We also describe vanishing viscosity methods for simulating shock waves. Finally, we discuss the spectral approximations to isolated solutions of nonlinear problems.

4.1.

Fourier Spectral Metho ds and Fourier Pseudospectral Meth o ds

The aim of this section is to take Benjamin-Bona-Mahony equation ( BBM equa tion ) as an example to show how to construct Fourier spectral schemes and Fourier pseudospectral schemes. We shall also analyze the errors strictly and present some numerical results providing the evidences of theoretical analysis. 100

Chapter 4. Spectral Methods a.nd Pseudospectral Methods

{

101

BBM equation describes the movement of regularized long waves. It is of the form x E JR.\ 0 < t � T, x E JR.\

Ot U(x, t) + do",U(x, t) + U(x, t)o",U(x, t) . -80t o�U (x, t) = f (x, t) , U(x , O) = Uo(x) ,

(4. 1 )

where Uo a.n d f are given functions, a.nd d � 0, 8 > O. Moreover Uo a.n d f possess the period 211' for the variable x, a.nd so does the solution U. Let A = (0, 211') a.nd denote the space C(O, Tj H;(A)) by C(O, Tj H;) for simplicity. Also 11 · 11 . = 1I ' lIw ( Al etc .. It ca.n be verified that if Uo E H�(A) and f E L 2 (0, Tj L�) , then (4. 1 ) has a unique solution U E LOO (O, Tj H�). It is commonly admitted that a reasonable numerical algorithm should preserve the features of the genuine solution of the original problem as much as possible. In fact, the solution of (4. 1 ) possesses some conservations, such as

l U(x, t) dx l Uo(x, t) dx + II f (x s) dxds, =

II U(t) 11 2 + 81 U(t) l�

,

=

Il Uo l l 2 + 81 Uo l� + 2

l ( f (s), U(s)) ds.

(4.2) (4.3)

We are going to construct the Fourier spectral scheme. Let N be any positive integer a.nd VN be the subspace of all real-valued functions of VN by (2.17) . Denote by PN the L 2 -orthogonal projection from L�(A) onto VN . For convenience of analysis, let 1 1 (4.4) J(v(x) , w(x)) = 3 w(x)0",v(x) + 3 0", (w(x)v(x) ) . If v, w, z E

H�(A), then (O",(wv), z) + (wo",z, v) (WO",v, z) + (O., (wz) , v)

0, =

O.

Hence (J(v , w), z) + (J(z, w) , v)

=

O.

(4.5)

In particular, (J(v, w), v) = O.

(4 . 6)

Spectral Methods and Their ,Applications

102

On the other hand,

(J(v, w), w) (J(v, w ), w)

= =

1 1 "32 (w2 , f}:r; v) - "31 (wf}:r;w , v) , "3 (w2 , f}:r;v) + "3 (w f}:r; w , v)

from which (4.7)

¢i, t/J E VN, (4.8) (PNJ(¢i, w), t/J) + (PNJ(t/J, w), ¢i) = 0 , (4.9) (PNJ(¢i, w), ¢i) = 0, (4.10 ) (PNJ(¢i, t/J), t/J) = � (f}:r;¢i, t/J 2) . For discretization in time t, let T be the mesh size and use the notations in Chapter 1, such as Rr (T), R.,.(T) and DT v(x, t), etc . . Let UN be the approximation to U, and be parameters, 0 :5 :5 1 , 1 = 1 , 2, 3. A Fourier spectral scheme for (4. 1 ) is to UN(X, t) E VN for all t E RAT) such

In addition, for

{

that

find

17/

17/

DT UN(X, t) + u1 df}:r; U N( X, t + T) + ( 1 - (1 ) df}:r;UN(X, t ) + U2 PNJ (UN(X, t + T), UN(X, t)) + (1 - (2 )PNJ (UN(X, t), UN(X, t)) - {jDTf}� UN(X, t) = FN(X, t), 1 x E lR\ t E Rr (T), UN(X, 0 ) = UN,O (X), x E lR

(4. 1 1 )

The first formula of (4. 1 1 ) can be rewritten as

DT uN( X, t) + df}:r;U N(X, t) + U1 dTDTf}:r;U N(X, t) +PN J(UN (X, t) + U2 TDTuN (X, t) , UN (X, t)) - {jDTf}�UN(X, t) = FN (X, t) . (4.12) If 171 172 = 173 = 0, then (4. 12) is the explicit spectral scheme in Guo and Manoran jan (1985) . If 171 =f 0 and 172 = 0, it becomes an implicit scheme. But by the =

Chapter 4. Spectral Method� and Pseudospectral Methods

103

orthogonality of Fourier system, we .can still evaluate the, coefficients of UN(X, t) ex plicitly at each time t E R.,. (T). This is t.he advantage of using spectral methods. In (4. 1 1 ) , the nonlinear term is approximated by partially implicit approach and so we only have to solve a linear system for the Fourier coefficients step by step, even if 0"10"2 =f. O. Indeed (4 . 1 1 ) reads UN(X, t) + 0"1dr8",UN(X, t) +0"2 rPNJ (UN(X, t) , UN(X, t - r)) - 08�UN(x, t) "'; GN(x , t - r)

(4.13)

where the term GN(x, t - r) depends only on u N(x, t - r) and rFN(x, t - r ) . We take the inner product of both sides of the above equation with UN(t), i.e. , multiplying UN(X, t) and integrating the resulting equation over A. Then we obtain from (4.9) that If GN(X, t - r) == 0 for x E A, then UN(X, t) == 0 for x E A. Since (4. 13) is a linear system, the above fact ensures the existence and uniqueness of UN(X, t) for all t E R,. (T). We check the conservations of the solution of (4. 1 1 ) . We first take 0"2 = 0, integrate (4 . 12) over A and sum the result for t E fl.,. (T). Then it follows that Next, we let 0"1 = 0"2 = l , take the inner product of (4. 1 1 ) with UN(t + r ) + UN(t) and sum the result for t E R,. (T). We obtain from (4.9) that l I uN(t) 1I 2 + O IUN(t) l �

=

lI uN,oll 2 + O IUN,ol � + r

E

(FN( S ) , UN( S + r) + UN( S ) ) ,

.eR,. (t-.,.)

Evidently the above two equations are reasonable analogies of the conservations (4.2) and (4.3) . We now turn to analyze the errors. Suppose that UN,O and FN have the errors UN,O and FN respectively. Then we get a disturbed solution corresponding to UN,O + UN, O and FN + FN, denoted by U N + UN. For simplicity, we denote the errors UN, UN,O and

Spectral Methods and Their' Applications

104

FN by U, Uo and F . They satisfy the error equation as follows D.,.u(x, t) + do",u(x, t) + U1 dTD.,. O",U(x, t) + PNJ (UN(X, t) + U2 T D.,. U N(X, t), u(x, t)) + PNJ (u(x, t) + U2 TD.,. u(x, t), UN(X, t) + u( x, t)) (4. 14) -oD.,. o;, u(x, t) = F (x, t), x E IR? , t E 1l.,.(T). We take the inner product of (4. 14) with 2u(x, t) and then obtain from ( 1 22 ) and .

(4.9) that

D.,. l I u(t) 1 1 2 +oD.,. l u(t) I � -T I D.,. u (t) 11 2 -oT I D.,. u (t) I � + ;=E3 1 Fj(t) = 2 ( F (t), u(t) ) -�

(4.15)

where

F1 (t) = 2U1 dT (D.,.o",u(t), u(t)) , F2 (t) = 2 (J (UN(t) + U2 TD",UN(t), u(t)) , u(t)) , F3 (t) = 2U2 T (J (D.,. u (t), UN(t) + u(t)) , u(t)) .

� be an undetermined positive constant and take the inner product �T D.,. u (x, t). We have from (4.9) that 6 �T I D.,.u(t)1 I 2 + �oT I D.,.u(t) l � + jE= Fj(t) �T (F (t), D.,. u (t) ) (4. 16) 4

Furthermore, let of (4. 14) with

=

where

F4 (t) = d�T (o",u(t), D.,.u(t)) , F5(t) = �T (J (UN(t) + U2 TD.,. UN(t), U(t)) , D.,. u (t)) , F6(t) �T (J (u(t), UN(t) + u(t)) , D.,.u(t)) . =

Let

e be a positive constant. Putting (4. 15) and (4. 16) together, we obtain that D.,. l I u(t) 1 2 + oD.,. l u (t) l � + T(� - 1 - e) ( 1 I D.,.u(t) 1 2 + o I D.,. u (t) I D (4.17) + E1=1 F;(t) :5 l I u(t) 1 2 + (1 + :rf) II F (t) 1 I 2 . We begin to estimate I Fj(t) l . It is easy to see that 1 F1 (t) + F4 (t) 1 = I dT(� - 2ud (o",u(t), D.,.u(t)) I T � (e : 2(1 ) 2 I u (t) I � . :5 eT I D.,. U (t) 1 2 + (4.18)

Chapter 4. Spectral' Methods and Pseudospectral Methods

Let r >

105

�. By (4. 10) and imbedding theory,

Similarly Further, by Theorem 2 . 1 ,

an d s o b y (4.8) ,

I F3 (t) + F6(t) 1 I T (e - 20'2 ) (J (u(t), UN(t) + u(t)) , Dru(t)) 1 :::; c; T I Dru(t) 11 2 + CT(e � 20'2 ) 2 (1 I uNllc(O,T ;W) l I u(t) lIi + N l u(t) 1 2 I u l i ) .(4.21 ) B y substituting (4. 18) - (4.21 ) into (4. 17), we find that

where

C ( 1 + ( 1 + � (e + (e - 20'2 ) 2 ) ) I UN ll c(O ,T ;W) ) l I u(t) 11 2 + c; ((e - 2 0' }2 + (e + (e - 20'2 ) 2) II UN ll c(O ,T;W) ) l u (t) l � + CTc;N (e 20'2 ) 2 1 I u(t) 11 2 I u (t) l� + ( 1 + ::2) I F (t) 11 2 . 1 Now let e > 1 , 0 < qo < � - 1 and c; 4' (e - 1 qo). Then we deduce that R(t)

t

_

=

-

For describing the average error of numerical solution, we define E(

v, t) Il v(t) 1 2 + 8 I v (t) I� + qOT 2 ER2:=t r (l I Drv(s ) 1 2 + 8 l Drv(s ) I i) . s (- ) =

\

,

Spectral Methods and Tbeir Applications

106 Also for the average error of data, set

p( v , w ,t) = l I v Il 2 + h'lvl� + T t E(u, t) � cp(uo , F, t) + c L:

L:

seR(t-T)

I w(s) 1 I 2 .

By summing (4.22) for E R... ( T) , we obtain r

seR. (t-T)

[(1 +

Il uN llc(O,T; W ») E(u ,

s

) + TN (� - 20'2 ) 2 2 u , s J .

E( )

(1'2 > �, then we can take e = 2(1'2 and so E( u , t) � cp( uo , F, t) + cr E ( 1 + I UN ll c(O,T ;W» ) E(u , s).

In particulaJ, if

BeR. (t-T)

Finally we use Lemma 3.2 with following result.

E (t) = E( u , t) and p (t) = p( uo , F, t), and obtain the

Theorem 4 . 1 . There exist positive constants b1 and b2 depending only on d, h' and

I UN I C(O,T;W) , r > �, such that if for certain tl E R,.(T) , p( uo , F, t 1 ) � :�, then for all t E R... (t 1 ) , E( u , t) � cp( uo , F, t) eb2t • If in addition 0'2 > !, then the above estimate holds for all t E R ( T ) and any p( uo , F, t). T

Theorem 4.1 shows that scheme (4. 1 1 ) is of the generalized stability with the index s 1 0 (N-q ) . If 1, then s O. In this case, provided that the computation is stable when the average error of data does not exceed a critical value cb1 • If then there is no restriction on It means that scheme (4. 1 1 ) possesses the index of generalized stability s We know from the above' analysis that the suitable choice of parameter in the approximation of nonlinear term can improve the stability essentially. We next deal with the convergence of (4. 1 1 ) . Let and 0 for

= - q,

0'2 > !,

T=

q=

=

p( uo , F, t). = - 00 .

UN = PNU

0'3 =

Chapter 4. Spectral" Methods and PseudospectraJ Methods

0

1 7

( ), DTUN(X, t) + doxUN(x, t) + U1 dT DToXUN( X, t) +PNJ (UN(x, t)s + U2 T DTUN(X, t), UN(X, t)) - 8DTo;UN( X, t) = FN(x, t) + L G;(x , t), x E lR? , t E RT( T), ;=1 UN(x, 0 ) = PNUO (X), x E lR \

simplicity. By 4 . 1

where

G1 (x , t) = DTUN(X, t) - OtUN(X, t), G2 (x , t) = U1 dT DToXUN(X , t), G3 (x , t) = PNJ (UN(x , t), UN(x, t)) - PNJ (U(x , t) , U(x , t)) , G4 (x, t) = U2 T PNJ (DTUN( X, t), UN( X, t)) , Gs(x , t) = 8ot o;UN(x, t) - 8DTo;UN(X, t). Further, let UN(X, t) = U N (X, t) - UN(X, t) , also denoted by U(x, t) for simplicity. By (4. 1 1 ) , it follows that

DTU(x, t) + doxU(x, t) + U1 dT DToXU(x , t) + PNJ (UN(x, t) + U2 T DTUN( X, t), U(x, t) ) +PNJ (U(x, t) + U2 T DTU(x , t), UN(X, t) + U(x , t) ) - 8DTo;U(x, t) = - j=Ll G;(x , t). ( 4.23) In addition, U(x, O) = O. By comparing this error equation to (4. 14) , we find that it suffices to estimate the terms Gj (x, t), 1 � j � 5. Since T (4.24) Otv(x , t) - DTV( X, t) = - -Tl 1.t t+ (t + T - s)o; v(x , s) ds, S

the property of Bochner integral leads to

T ER(tL T I G1 (s) 1 2 � CT 2 11 U 1 �2 (O,t ;£2 ) ' s -) Clearly,

T ER(tL T I G2 (s)1 1 2 � CT2 1 1 U 1I �' (O,t;H' ) ' s -)

Spectral Methods and Their Applications

108

Moreover for any theory,

v E H; (A) with r > �, we deduce from Theorem 2.3 and imbedding

I l v8",v - PNv8",PNvi l Thus

I v8",v - V8",PNVi l + I v8",PN V - PNv8",PNVi l :5 I v il Lool v - PNv i l + 1 8",PNV il Lool i v - PNvi l l :5 cN - r l v ll� · <

7 e1't.Lt T I Gs(s) I 2 :5 cN2- 2r I U I &(o,t ;H" ) ' s (- )

It is not difficult to verify that

8 eR. (t- T )

+ I 8",UN I &(o,t;Loo ) I UN I kl (O,t;L2 » ) :5 C7 2 1 I U I &(O,t;H") I U I �, (O,t ;H' ) .

We have

1 7 seR.L(t-T) (Gs(s), [r(s)) I :5 07 .e1't.L(t-T) 1 (Dr8",UN(S) - 8t8",UN(S), 8",[r(s)) 1

07 .eR.Lt T l O (s) l : + C072 1 I U l ko (o,t;Hl ) ' (- ) It is noted that 7 2 + N 2 - 2r O ( �) for 7 o (N - t) and r > � . 7 argument as in the proof of Theorem 4.1, we obtain the following result. :5

=

=

Let 0 in (4.11). If U E C(O, Tj H;) H2 ( 0 , Tj HI ), r 7 :5 bsN- , then for all t E R,. ( T),

Theorem 4 . 2 .

O's =

t

n

So by an

>

�

and

where ba, b4 and bs are positive constants depending on d, Il U llc(O,T;H; ) and I U I H2 (O,T;Hl). If ! , then the above estimate is valid for any O. If we take � and approximate the nonlinear term, by �4 PNJ(UN(X, t + 7) + UN(X, t), UN(X, t + 7) + UN(X, t)) , then E(UN - UN, t) 0 (74 + N2 - 2r ). 8,

0'2

>

7>

'

0'1 = O's =

=

Chapter 4. Spectral Methods and Pseudospectral Methods

109

We now present some numerical results. For describing the errors, let AN be as in (2. 19), and

IU(x,::-: t) 1 --, t) :-:- - v(x, ( v, t) xmax EAN -'---'-IU(x, t)1-,.':---''-' where U is the solution of ( 4. 1 ) , and v is. the corresponding numerical solution. For 2� E

_ -

. comparIson, we aI so use a finite difference scheme to solve (4. 1 ) . Let h and

vx(x, t) h1 (v( x + h , t) - v(x, t)) , vx(x, t) h1 (v(x, t) - v( x - h , t)) , vx(x,t) '21 vx(x, t) + '21 vx(x, t).

= 2N + 1

=

=

=

Define the finite difference operators as

MhV(X, t) '61 v(x - h, t) + '32 v(x, t) + '61 v(x + h, t), Jh(V(X,t), v(x,t)) � v(x,t)Vx(x,t) + � (v 2 (x, t) ) x ' �hV(X, t) vxx(x, t). =

=

{

=

A finite difference scheme for (4. 1 ) is

DTMhUh(X, t) + dUh'X (X ' t) + Jh (U h( X , t), Uh(X, t)) - 6DT�hUh(X, t) f(x, t), x E AN, t E RT(T), x E AN, t E RT(T), Uh(X, t) Uh(X +. 2�, t), x E AN. Uh,O (X) Uo(x) , =

=

=

We take the test function

U(x , t)

=

A exp ( B sin + wt ) .

x

( 4.25 )

( 4.26 )

The calculations are carried out with = = O.OOl , A = B = O. l , w = O . l , N = 5 and T = 0.01. Table 4.1 shows the errors N for scheme ( 4 . 1 1 ) with 0"1 = 0"2 = 0"3 = 0, and the errors for scheme ( 4.25 ) . We find that (4. 1 1 ) gives quite accurate numerical results even for small N and large T. It is also shown that (4. 1 1 ) provides better numerical results than ( 4.25 ) .

E (Uh, t)

d 1, 6 E (U , t)

Spectral Methods and Their Applications .

110

E (UN , t) and E (Uh , t). E , t (UN t) E (Uh, t)

Table 4. 1 . The error� 0.5 1.0 1.5 2.0

6.719E-3 8.832E-3 6.675E-3 1 . 027E-2

8.931E-2 1 . l50E-2 7.574E-2 7.791E-2

As i ¥ dicated before, it is not easy to deal with nonlinear terms in spectral methods usually. So we prefer Fourier pseudospectral methods in actual computations. Let be the Fourier interpolation points, the discrete interval and the and Fourier interpolant in (2.19) and (2.20) . The discrete inner product and the discrete norm are given by (2.22) . Clearly

xU) , AN .

IN

( . , · )N

I · liN

We can approximate (4. 1 ) directly and derive a scheme similar to (4. 1 1 ) , in which But such a scheme cannot , provide good nu the projection is replaced by merical results. This is caused by two things. Firstly, the conservations (4.2) and (4.3) are simulated in spectral scheme (4. 1 1 ) automatically. However, it is not so in pseudospectral methods, since interpolation is used. In order to preserve the global properties of genuine solution of (4. 1 ) , we have to deal with the nonlinear term care fully. For this purpose, we define the operator

PN

Let

cP,

IN.

IN(V, w)(x) = 31 IN(wo:r; v )(x) + 31 0",!N( WV)( X ).

'I/J E VN. By (4.27) and (4.28), ( IN( cp , W ), 'I/J ) 31 (wO:r; ,p , 'I/J )N - 31 (w ,p , o", 'I/J )N, ( IN( 'I/J , w), ,p) 31 (wo",'I/J , ,p )N - 31 (w'I/J , O",CP)N. =

=

Consequently,

( IN( ,p , W ), 'I/J ) + ( IN( 'I/J , W ), ,p) = 0 , ( IN( ,p , w),
(4.29 ) (4.30)

Chapter 4. Spectral Methods and Pseudospectral Methods

111

Therefore IN( ¢>, w) preserves the property of ¢>, w) . This technique i s called skew symmetric decomposition of nonlinear convective term. We shall see below that it enables the numerical solution to keep the conservations. On the other hand, the nonlinear computation may destroy the stability. Mainly, this trouble is due to the higher frequency terms. To remedy this, we can use the filtering RN( CY., (3) in (2. 107) , denoted by RN for simplicity. The use of filtering for nonlinear terms brings some difficulties in error analysis. To simplify the statements, we introduce the circle convolution as (4.31) ¢> * ¢ = :L :L �p "j; , _ pe i'x , V ¢> , ¢ E VN .

J(

It is easy to see that ¢> * ¢ = ¢ * ¢>.

111� N lp l� N

Lemma 4 . 1 . For any ¢>, ¢ E

VN and X E VN ,

IN( ¢>¢ ) = ¢> * ¢,

( ¢> * x , ¢ ) = ( ¢>, X * ¢ ) ·

Proof. For the first conclusion, i t i s sufficient t o prove that

In fact,

� �1 eipx(J) � .7. i( l -p)x(i) � o/l - p e � p

L �, ipx

Ip l:S N

e

)¢

111� N

Ip l� N

(j

(xU»)

=

¢> (xU») ¢

(x U») ,

and so the desired result follows. Next, by (4.28) and the first conclusion, we deduce that

Lemma 4 . 2 . For any ¢>, ¢

•

E VN, W E H; (A)

and r

> �,

112

Spectral Methods and THeir Applications

{,

Proof. Let WI be the Fourier coefficients ofw. Define 1 - p + 2N + 1 , if / - p < - N, ( 4.32 ) if 1 / - p i � N, '"(I,p = 1 - p, l - p - 2N - 1 , if / - p > N. Clearly '"(- I , -p = -: '"(I ,p . We have (8x 4> * RN"p, w) = (8x¢> * RN.,p, PNW) = 27ri L it, L ( 1 - I � l a r '"(I,p J,-p¢P , .(¢> * RN8x.,p, w) = (¢> * RN8x.,p, PNW) = 27ri IL5N it, pLl 5N (l - I �D,B pJ, -P ¢P · 11 I �

m�

;

��

Putting these two formulas together, we get that

(8x¢> * RN.,p, w) + ( ¢> * RN.,p, w) = 27ri IL5N (1 + I / l ) it, pLl 5N ).. " p�,-p ¢p I 11 where

O" p = (l - I�D (p + '"(I,p ) ( 1 + 1 1 1 ) -1 . Set f(x) = ( 1 - xa ) ,B - a,8(l - x). Clearly f( l ) = 0 and for 0 � x � 1 , 8xf( x) = a ,8 ( 1 - xa- 1 (1 - X a ) ,B- l ) � o .

Let

Therefore for 0 and s o for

� � 1, x

I I I , Ip i � N,

(1 - xa l

� a,8(1 - x),

We now estimate only. If 1 1 - �

I O"p l . Since O_ I, _p = - O" p , it suffices to consider the case 0 � p � N p i N, then I O',p l = I I I 1( 1+-I IN)I � 1 . If 1 - p -N and s o 0 � N - p - I I l l , then (N - p)( 2 N + 1 + 1) 1 / 1 (2N + 1 - I l l ) 1 0 I ,p I = N(1 + I l l ) - N(1 + I l l ) - 2. <

<

=

<

<

4.

113 If 1 - p > N and so p ::; O. It is contrary to p � O. Therefore I I,p l ::; 2afJ for all Illthat, lp l ::; N. Furthermore let be a sufficiently small positive constant. We deduce Chapter

Spectral Methods and Pseudospectral Methods

A

e

I (oxtP * RN'I/J , w) ( tP RNox'I/J , w) 1 ::; caf3 LN (1 + I l I ) l wtl L 1�,-p l l �p l III � Ip l �N ::; caf3 L�N (1 I l I ) l wtl ( LN 1 �I_p I 2 ) " ( LN l �p I 2 ) " Ip l � III Ip l � ::; caf3l 1 tP l I l I 'l/J l (I'LI �N(1 1 1 D -1-2<). " (I'LI�N(1 1 1 D3+2
*

1

1

+

1

1

+

Lemma 4.3. For any

Proof.

We have

tP

•

E

w) =

VN, w

E

H; (A) and r >

(OxtP RNtP, (OxtP RNtP, PNw) (tP RNO:c tP, w) (tP RNOxtP, PNw) *

=

*

Thus where Let

*

*

(OxtP RNtP, w) - (tP RNOxtP, w) *

+

*

�,

211'i E�N WI E P ( 1 - l 'YIVI P I Ol) /3 tPptPl-p , I ' I Ip l � N = 211'i E it, E P (1 - I � 1 00 r �p�,-p, I I I � N Ip l � N A

=

_

=

211'i EN(1 + I l I }it, EN A',p �p �,_p I'I� Ip l �

A

Spectral Methods and Their Applications

1 14 Then for

I l l , Ip l � N,

We now estimate If

I � P � l'{

+

I (h,p l . It suffices to consider the case 0 � P � N. If 0 � P � I, then

I and so 0

< p - I � N, then 1 0I,p 1 -< N(lpi+l i I l l ) -< 1 If P > N + I and so 1 - p < -N, O � N - p < - I I l l , then 1 01,p l � Ip - (2N1 ++ 1I I+I 1 - p) 1 1 2 (N -1p)+ +I I I 1 - 1111 2(N - p) + 1 I l l ) � 2. � max ( 1 + I II 1 + I II Hence I A I ,p l � a,8 1 01 ,p l � 2a,8. The rest of the proof is clear. .

=

'

•

The combination of the above two lemmas implies the following result. Lemma 4.4.

For any

r

and >

�,

4.2-

Lemma 4.1 and Lemma Lemma 4.3 with ,8 = 1 were established in Ma and Guo (1986) . A Fourier pseudospectral scheme for (4. 1 ) is to find for ali R.,.(T) such that

{

UN(X,t) E VN

tE

DTUN(X, t) + dRN8",uN(x, t) + u1 dTRNDT8",UN( X, t) + RNJN (RNUN(X, t) + U2T RNDTuN(X, t), UN(X, t)) - ODT8� UN(X, t) FN(x, t), E lRl , t E RAT), UN(X, O) UN,O (X), x E =

IR \

=

x

(4.33)

Chapter 4. Spectral Methods and Pseudospectral Methods

Clearly for any

l15

fj) E VN,

(RNO",fj), fj)) (o",Rk fj) , Rk fj)) By (4.29) , for any fj), 'I/J E VN, =

=

(4.34)

O.

(4.35)

In particular, (4.36)

If we take

(J'2

=

0 in (4.33) , then

(J'l (J'2 = !, then it follows from (4.34)-(4.36) that I UN(t)1 I 2 + 6 I u N(t) l � I UN,O I 2 + 6 I uN,0 1 � + T L (FN( S) , UN(S + T) + UN(S)) .

If

=

=

aE14 (t-.,.)

The above two equalities are reasonable analogies of (4.2) and (4.3). We next analyze the errors. Let and F be the errors of and respec tively, which induce the error of denoted by U. Then we have the following error equation

UN,

Uo

UN,O

FN

DTu(x, t) + dRN O",U(X, t ) + (J'l dT RND.,. o",u(x, t) + RNJN (RNUN( X, t) + (J'2 T RND.,. U N(X, t), u( x, t )) + RNJN (RNU(X, t) + (J'2 T RND'T u( x, t), UN(X, t) +u( x, t)) - 6D'To�u(x, t) F (x, t) , x E t E R,. ( T). IR \

=

By an argument as in the derivation of (4. 1 7) , we obtain from (4.34) - (4.36) that for

e > 0,

D'T l I u(t)1 1 2 + 6D'T l u (t)l � + T(e - 1 - ) (I I DTu(t) 1 I 2 + 6 I D,.u(t) I D + t Fj (t) ::::; l I u (t)1 I 2 + ( 1 + ::2) IIF (t) 11 2 e

where

(4. 3 7)

Spectral Metbods and Their Applications

116

F2 (t) Fs(t) F4 (t) Fs(t) F6(t)

= = = = =

(RNJN (RNUN(t) + 0'2 7 RNDTUN(t), u(t)) , u(t)) , 2 0'2 7 (RNJN (RNDTu(t), UN(t) + u(t» , u(t)) , de7 (RNO",U(t), DTu(t» , e7 (RNiJN (RNUN(t) + 0'2 7 RNDTuN(t), u(t)) , DTu(t)) , e7 (RNJN (RNU(t), UN(t) + U(t)) , DTu(t» . 2

By using Lem� � 4. 1 , Lemma 4.4 and (4.34) - (4.36) , we can establish the estimates for They are exactly the same as (4.18) - (4.21) . Therefore an argument as in the latter part of the proof of Theorem 4 . 1 , yields the following result.

F;(t).

a,

Theorem 4 . 3 . There exist positive constants b6 and b7 depending only on d, 6, (3 and I UN I C(O,T;W) , r > �, such that if for certain tl E R,. (T) , p (uo, P, t 1 ) $ �, then for all t E R (td,

If in a ddition 0'2 > 21 ' then the above estimate holds for all t E RT (T) and any p (uo, !, t) . We now turn to the convergence of (4.33). Let UN PNU, U UN - UN and Us = 0 in (4.33) . We obtain the error equation as follows DTU(x, t) + dRN 0", U( x, t) + 0'1 d7 RND.,.o",U(x, t) + RNJN (RNUN(X, t) + 0'2 7RND.,.UN(X, t), U(x, t» ) + RNJN (RNU(X, t) + 0'2 7RND.,.U(x, t), UN(X, t) 6 (4.38) +U(x, t » ) - 6D.,. 0� U(x, t) - j=1 L Gj (x, t), where G1 (t) and Gs(t ) are the same as in (4.23), and G2 (t) 0'1d7 RN D.,. 0", UN (x , t ), Gs = RNJN (RNUN(X, t), UN(x, t)) - PNJ (U(x, t), U(x, t)) , G4 (t) 0'2 7 RNJN (RNDTUN(X, t), UN(x, t)) , G6 PNf(x, t) - RNINf(x, t). =

=

=

=

=

=

Chapter 4. Spectral Methods and Pseudospectral Methods

117

B y Theorem 2.4, Theorem 2.26, Lemma 4.4 and imbedding theory, we obtain that for a 2: r > � ,

T ERrL T) I G2 (s) 1 2 :s CT2 1 U 1 ;'1 (0,t ;Hl) , s (tT L I G3 (s) 1 2 :s cN2-2r l U l t 0, T ; W ) ' ( sERr (t-T) T L T) I G4 (s) 1 2 :s CT 2 1 U 1 �(0,T;W) I U I ;'1 (0,t ;Hl ) , sERr (tT SERrL T) I Gs(s) 1 I 2 :S cN2 - 2r llfll�(0,t ;W_l) . (t-

Finally by an argument as in the proof of Theorem 4.2, we obtain the following result.

Theorem 4.4. Let (j3 0 in {4.33}. If U E C(O, T; H;) n H2(0, T; HI ) , Uo H; ( A) , f E C (O, Tj H;-I ) , a 2: r > � and T :S bs N -!, th e n for all t E flT( T ) , =

where

E

are positive constants depending on d, 11 U 1 I C(o,T;H; ), I U I H2 (O,T;Hl), I l Uo ll r and Il f llc(o,T; W -' )' If (72 ) ! , then the conclusion is valid for any O. b6 - bs

8, a ,

(3,

T>

0

' Scheme (4.33) with (71 = (j2 = (j3 = was investigated in Guo and Cao (1988) . We have seen in the derivation of (4.37) and (4.38) that since the nonlinear term the effects is approximated by a skew symmetric operator of the leading nonlinear error terms in both (4.37) and (4.38) are cancelled, i.e. ,

UoxU

RNJN (RNUN, UN),

(RNJN (RNU(t), u(t)) , u(t)) = 0, (RNJN (RNU(t), U(t)) , U(t)) = o. Otherwise, we shall require that p( uo . io , td :S bs /N in Theorem 4.3.

It indicates that the suitable approximation to the nonlinear term, not only enables the scheme to keep the conservations, but also strengthens the stability and raises the accuracy.

Spectral Methods and Their Applications

118

' It also shows that there is a close relationship between the conservations and the stability. This idea was first proposed for finite difference methods by Guo (1965) . Finally, we present som� numerical results. The meaning of E(v, t) is the sanie as before. We take the test function (4.26) with A = B and w = 0.5. In calculations, 0'1 = 0'2 = 0'3 = 0, fJ = 1 , N = 8 and T = 0.01. We check the effect of the filtering RN (a, 1 ) , and fi� d the following phenomena. (i) If the vibrations of genuine solution of (4. 1 ) is small (e.g., A = 0 . 1 ) and d = 0, then the effect of RN(a, 1 ) is not clear, see Table 4.2. ,

(ii) If 8 is large (e.g. , 8 = 1), then the filtering is not important even though the vibration of genuine solution is large, see Table 4.3. (iii) If the vibration of genuine solution is large and is small (e.g., A = 1 , 8 10-4 , 10-6 ) , then the effect of filtering is clear, see Tables 4.4 and 4.5.

8

=

(iv) The smaller the parameter 8, the more important the linear term do",U. In this case, the filtering plays an important role. Tables 4.2 and 4.5 tell us that its effect is not clear for d = 0, A = 0 . 1 , but very clear for d = 2, A = 0 . 1 . (v) The value of a i n the filtering must b e chosen suitably. For every large a, the filtering is weak. But for too small a, the accuracy is lowered. So far, there is no criterion for the best choice of a. Numerical experiments show that a better choice is 3 :::; a :::; 10. But the best ones depend on N. The best one in Table 4.5 is a = 3. Table 4.2. The errors a=5 8 = 1.0 8 = 10- 2 = 10-4 = 10-6

8 8

2.36857E-3 2.29817E-3 2.28399E-3 2.28242E-3

E ( UN, 5.0), d = 0, A = 0.1. a = 10

a = 50

2.36493E-3 2.29622E- 3 2.28281E-3 2.28203E-3

2.36577E-3 2.29651E-3 2.28183E-3 2.28193E-3

Chapter 4. Spectral Methods and Pseudospectral Methods

119

Table 4.3. The errors B ( U N , 5 0) , 6 = 1 , A = 1 .0. a=5 a = 50 a = 10 d = O 2.31011E-3 2.15183E-3 2 . 16537E-3 d = 2 1 .441 1 8E-3 1 . 18726E-3 1 . 15333E-3

Table 4.4. The errors E(UN , 1 .0 ) , d = 0 , A = 1 . 0 . a=5 0 a = 10 a = 50 6 = 10-4 3.29499E-3 3.55039E-3 2.35286E- 1 6 = 10-6 2.83234E-2 1 . 24055E-2 1 . 70894E- 1

Table 4.5. The errors B (U N , 2.0) , d = 2, A = 0 . 1 . 6 = 10-4 6

=

10-6

a=3 1 . 55333E-3 1 . 56715E-3

a = 50 a = 10 a=5 1 .63698E-3 2.43057E- 2 7.02116E-1 > 10 1 . 86465E-2 1 . 1 1043E-2

Scheme (4. 1 1 ) and scheme (4.33) are of first order in time. In order to match the spectral accuracy in the space, we should use more accurate temporal discretizations as discussed in Gazdag (1976) . The Fourier spectral methods and the Fourier pseudospectral methods have been widely used for numerical solutions of various partial differential equations. We refer to some early work. For instance, Shamel and Elsasser (1976) , Fornberg and Whitham (1978) , and Abe and Inou ( 1 980) used them for Korteweg-de Vries equation (KdV equation ) , while Guo (1981a) proposed the spectral schemes for KdV-Burgers equa tion with numerical analysis. Pasciak (1982) considered Fourier spectral scheme for generalized BBM equation. Kriess and Oliger (1979) applied them to hyperbolic equations with filterings. Cornille (1982) used Fourier approximation in numerical study of shock waves. Canosa and Gazdag ( 1977) used it for KdV-Burgers equation and found shock-like waves.

Spectral Methods and Their Applications

120

4.2 .

Legendre Spectral Methods and Legendre Pseudospe<;tral Metho ds

For non-periodic problems, it is natural to use Legendre spectral methods or Legendre pseudospectral methods. Because of the appearance of the Fast Legendre Transfor mation, more attention .has been paid to these two methods recently. We shall take the nonlinear Klein-Gordan equation (NLKG equation) as an example to describe these two methods and set up the strict error estimations. The numerical results will show their advantages. NLKG equation plays an important role in quantum mechanics. Let A = ( 1 , 1 ) The initial-boundary value problem of NLVG equation is as follows

{

-

8lU(x , + U(x , + U3(x , U( - 1 , = U( l , = 0, 8tU(x , 0) = U1 (x ) , U(x , O) = Uo ( x ) ,

t) t)

t) t)

t) - 8�U(x , t)

=

f(x,

t),

x E A, ° < 0 < :$ T, x E X, x E X.

t

.

t :$ T, ( 4.39)

It is shown in Lions ( 1969) that if Uo E HJ (A) n L4 ( A) , U1 E L 2 (A) and f E P ( O, T j L 2 ) , then (4.39) has a unique solution U E LOO ( 0, Tj HJ n L 4 ) n w 1 •00 ( 0 , T j L 2 ) . Let

E*( v, t) 11 8tv (t)1 I 2 + I v(t) ll� + 21 I 1 v(t) l i =

The solution U satisfies the conservation E* (U, t)

E * ( U, O) + 2

•.

l (f(s ) , 8. U( s ) ) ds .

(4.40)

For Legendre spectral approximation, let N be any positive integer and lPN, lPl\, be as in Section 2.3 and (2.56) . Denote by PN the L 2 -orthogonal projection from L2 (A) onto lPN• The HJ-orthogonal projection p]/ is given by (2.57) . Also let T be the mesh size in and use the notations R'T T ) , R'T ( T ) , D'Tv ( x, D'T v(x , t) and D'T'Tv(x , in Chapter 1 . We try t o construct a scheme preserving the conservation (4.40). To d o this, define

t)

=

(

t

t),

G (v(x, t)) = -41 � vJ ( x, t + T ) V -J ( x, t - T ) . w j=O

.

3

.

(4.41)

Chapter 4. Spectral Methods and Pseudospectral Methods

D.,.v(x, t)G(v(x, t)) = -.!:..8T . (I v (x, t + T)1 4 - I v (x, t - TW)

Clearly

(G(v(t)), D.,.. v (t)) = "41 D.,.. ll v(t) I i. ·

and so

121

(4.42)

UN be the approximation to u. We approximate the nonlinear term U3 by PNG (UN ) instead of PNU� . A Legendre spectral scheme for (4.39) is to find UN( x, t) E IP � for all t R.,. (T) such that (D.,..,. U N(t) + UN(t) + G (UN(t)) , ;P) + (O:r; UN(t), O:r;;P) = (J(t) , ;P) , V ;p E IP�, t E R.,. (T) , (4.43) D.,.UN(O) UN,l , · UN(O) = UN,O Let

{

E

=

UN,l PNU1 + �TPN (o�Uo - Uo - ug + 1 ( 0») .

The first formula of (4.43) stands for =

We check the conservation. Putting (1.23) , (1 .24) and (4.42) that

v = 2D.,.UN(t)

in (4.43) , we obtain from

. D.,. 1 f>.,. U N(t) 1 2 + D.,. II UN(t) l I � + �D.,. II UN(t) l I i. = 2 (J(t), D.,. U N(t») .

(4.44)

Let

E; (v, t) = 1 D.,.- v (t) 1 2 + 21 I1 v(t) lI � + 21 l1 v(t -· T) I � + "41 I v(t) II i. + "4l1 1 v(t - T) l I i. and

F* (v,t) T I: (J(S), D.,. UN(S») . The summation of (4.44) for t E R.,. (T) yields =

BeRr(t-.,.)

( 4.45 )

122

Spectral Methods and Their Applications

Clearly it is a reasonable analogy of (4.40) . We next derive a priori estimation for Assume that = 7' < 00 . By ' (4.45) , it suffices to bound the initial energy E; and estimate the summ'!:tion From (4.43) and the proof of Theorem

q N2 UN(X, t). (UN, 7') 2.8, I F* (UN, t) l . I UN( 7' )t $ 2 I UN,O I 2 + 21' 2 I UN,l ll 2 , I U�( 1' ) I � $ 2 l 1 uN,o l � + 2a�1' 2 N4 11 uN, l 1 l 2 $ 2 l1 uN,o ll� + � q2 I 1 UN'l I l 2 .

Furthermore By Theorem

2.7 ,

It is easy to see that

L: ( 1I.D1' UN( 8 ) 1 I 2 + lIi ( 8 ) 11 2 ) . I F * (UN ' t)1 $ 7'1I.D1' UN(t) 1I 2 + 2 1' 8e]l,.(t-1') Hence (4.45) implies that

- 1 I 2 + 1 1 I uN(t) l I 2l + 1 11 UN(t - 7' ) 1 21 + "41 I1 UN(t) 11 14 1' ) I D1'UN(t) 2 2 1 + "4 I1 UN(t - 7')I I L4 $ po (UN,O , UN,l , !, t) + 21' 8eRr(t-1') L: I D1'UN( 8 ) I 2

(1 -

4

(4.46)

where

Po (UN,O , UN,! , I, t) ( II UN,o ll� + II UN,o I l 1. + II UN,l ll 2 + 1'3 11 uN, l 1l 4) +C1' BeAr(t) L: 11 /( 8 ) 11 2 . = c

Let By applying Lemma 3.2 to (4.46), we know that E (4.43) is an implicit scheme. Let

(UN, t) $ po (UN,O , UN,l , I, t) .

a (v, w) = (8",v, 8",w) + ( 1 + :2 ) (v, w), b( v, w) = � (v3 , w) , d(v,g, w) � (gv2 , w) + � (g 2 v, W) . =

Chapter

4.

Spectral Methods and Pseudospectral Methods

123

t

Then at each time E R., (T), we have to solve a nonlinear equation

uN(t-2r) and j (t-r) . We now investigate the existence

F(t)

where depends only on and uniqueness of solution of (4.47) .

Let 'H be an m-dimensional Hilbert space with the inner product ( ' , · hi and the norm 1 I · 1 I 1{ . A is a continuous operator from 'H into 'H, and (Ae, e) 1{ � 0 for all e such that l I e l 1{ = r > O . Then there exists eo such that l I eo ll 1{ ::; r and Aeo = o . Proof. Let S( O, r) = { e I l I e l 1{ < r} and A*e = I-AerAell 1{ ' e E 8(0, r ). If Ae :f:. 0 i n the closed ball 8(0, r ) , then A* i s a continuous operator and I A*e l 1{ = r. B y Brouwer Theorem, there exists a fixed point for A*, say eo . S o we have - rAeo I Aeo ll 1{ = eo Lemma 4 . 5 .

E 'H

from which

It is contrary to the assumption and so the proof is complete.

We now use Lemma 4.5 for (4.47) . Let in !P\\,. The space consists of all vectors e

'H

Let

Ae = {T,,} and

{(Pl} be the normalized orthogonal system = { eo, . . . , eN } such that -2 N-2 edJI (x ) . UN(X, t) = L 1=0

A is a continuous operator. Moreover, (Ae, e)1{ = l I uN(t) lI� + r22 1 I uN(t) 1 2 + "21 I UN(t) 1 1 1. + d (UN(t), UN(t - 21') , UN(t)) - (F(t), UN(t)) .

It is easy to see that

•

Spectral Methods and Their Applications

1 24

According to the a priori estimate, it follows from Theorem 2.7 that

I (ut(t), UN(t - 7")) I :::; I i UN(i)l I i,l l uN(t - 27") i l L' :::; C I l UN(t) I L' :::; cNt I UN(t )l l , I (U�(t) , U�(t - 27") ) I :::; I UN(t) l I i, l uN(t - 27") I i i, :::; cN I UN(t) 1 where c is a positive, constant depending on the a priori estimate. Clearly, Since 7"

(A� , �hi solutions

qN- 2 , we know that for suitably large N and = r, we have > O. This implies the existence of solution of (4.47) . If (4.47) has two

I l uN(t) 1

=

uW(t) and uW (t), then let UN(t) uW(t) - u�\t). By (4.47) , I UN(t) l I � + !oI i UN(t)) 1 I 2 + b (uW(t), UN(t)) - b (u�) (t), UN(t) ) +d (uW(t), u� (t), UN(t - 7") ) 0 =

=

where

( 4.48)

d (u�) (t), u�) (t), UN(t - 27") ) ) d (uW(t), UN(t - 2 7" ) , UN(t)) - d (u � (t), UN(t - 2 7" ) , U N(t)) . =

Clearly Moreover, it can be verified that

to Il UN(t) 11

= 0 for suitably large N . Thus the solution of (4.47) Finally, (4.48) leads is unique. We can use the Newton method or the simplified Newton method in Section 3.2 to resolve (4.47). We now discuss the choice of basis. As we know, Legendre polynomials are or thogonal in L 2 ( A ) . But it is not so for the basis of We could take the basis of as the set of Ifol (X) 2 :::; 1 :::; N where

IP�

,

IP j!" .

if 1 is even, if 1 is odd.

Chapter 4. Spectral Methods and PseudospectraJ Methods

125

Unfortuna.tely this basis leads to a linear system with a full matrix whose elements are M1,m = ( (PI , rPm ) . Moreover, the number condition of the matrix M* with the elements M1:m = ( {)x rP/ , {)x rPm ) is very bad. Shen (1994) proposed another basis, i.e. , 1

d1 v'41 + 6 ' _

0 :::; 1 :::;

N - 2.

It can be checked that + 2, otherwise. l = m,

1=m

In particular, Therefore this basis leads to a triangular matrix and an identity matrix. It saves a lot of work in computation and improve the condition number of the related matrix, which in turn strengthens the stability of algorithms. For error estimations, we need some preparations. For simplicity, let (t = ) in the derivations of the following lemmas.

v ) v(x, t

Lemma 4 . 6 . We have

provided that the related norms exist. Proof, The results come from some expressions like (4.24) and the property of Bochner

integration. Lemma 4.7. If v

•

E C(O, Tj C) n Hl (O, Tj L2 ) ,

then

I G( v (t )) - � V3(t + T) - � V3(t - T) I :::; cT � l v ll�(O,T;G)l l v I l Hl (t_'T,t+'T;L2).

Spectral Methods and Their Applications

126 Proof. We have

G (v(t)) - -21 v 3( t + r ) - -21 v:f( t - r ) 1 1 = 4"V2(t + r)(v(t - r) - v(t + ) ) + 4"V2(t - r)(v(t + r) - v(t - r)) r

from which the conclusion comes. Lemma 4 . 8 . If

If,

•

v E C(Q, T; HJ), then

in addition, v E C(Q, T; HJ n W ) and r � 1, then

Proof. We have - 4G( v(t)) ((P.k;°(t + r)t - v3(t + r)) + ((P.k;°v(t + r)r P.k;°v (t - r) - v(t + r)2v(t - r ) ) + (p.k;O (t + r) (P.k;° ( t - r )f - v ( t + r)v2(t - r)) + ( (P.k;°(t - r)t - v3(t - r) ) .

4G (P.k; °v ( t )) =

By Theorem 2. 1 1 ,

I p.k;°vI 1 1 � I lvliI-

Hence by imbedding theory,

I (P.k;O (t + r) r - v3 ( t + r) 1 � c II p.k;°(t + r) - v(t + r)11 ( 1 Iv(t + r) l I i � + I p.k;°v(t + r) I � � ) . � c I I p.k;°v(t + r ) - v(t + r)l l l Iv(t + r)II�, · If v E C(Q, T ; HJ n Hr) with r � 1 , then using Theorem 2 . 1 1 again, yields I (P.k;O( t + r) r - v3(t + r) 1

�

<

c I P.k;° ( t + r) - v(t + r) l llv(t + r) II�' cN-r ll v(t + r) 1 I 1,·

We can estimate other terms similarly and complete the proof.

•

Chapter 4. Spectral Methods and PseudospectraJ Methods

127

E

Ijv, w C(O, Tj HI) , then G(v( x, t ) + w(x, t)) G(v(x, t )) + G(w(x, t )) + R(x, t ), I R(t) 11 2 ::; d(v) ( 1 I w(t + r) l � + I l w(t - r) l � + Il w(t + r) l i< + I l w(t - r) l i. ) , d(v) being certain positive constant depending only on I l v llc(t-1",t+1" ;H' ). Lemma 4 . 9 .

=

Proof. We have =

4G(v(x, t ) + w(x, t)) 4G(v( x, t )) + 4G(w(x, t)) + jL= l Rj (x, t ) where

4

3V2 (X, t + r)w(x, t + r) + 3 v( x, t + r)w2 (x, t + ) vex, t - r)w(x, t + r) (2v(x, t + r) + w(x, t + r)) + ve x, t + r)w(t - r)(v(x, t + r) + 2w(x, t + r)), vex, t + r)w(x, t - r)(2v(x, t - r) + w(x, t - r)) + ve x, t - r)w(x, t + r)(v(x, t - r) + 2w(x, t - r)), 3v 2 (x, t - r)w(x, t - ) + 3v( x, t - r)w2 (x, t - r). r ,

R3 (X, t)

r

Since

Il v(t + r)w2 (t + r) 1 1 2

::; II v ll � (t_1",t+1";LOO)ll w(t + r) lli. ::; cllv ll�(t-1",t+T;Hl ) ll w(t + r) l i. , etc . .

We derive the conclusion. It is noted that for any

v

E

•

HJ (A) , the Poincare inequality is of the form

I v 11 2 ::; 42 I v l � . We now analyze the generalized stability of (4. 43 ) .

(4. 49 )

1["

Suppose that UN, a, UN, l and PNj have the errors ua , UI and 1 respectively, which induce the error of UN, denoted by UN or U for simplicity. Then

{

(b�1"u(t) � fi (t ) + G (t) ,
D1"u(O) = UI , u(O) = Ua

E

CR t ),
E

R,. (T) ,

(4.50)

Spectral Methods and Their Applications

128

where

G(x, t) = G (UN( X, t) + U(X, t)) - G (UN(X, t) ) = G(u(x , t)) + R(x , t ) , and by Lemma 4.9,

I R (t ) 1 2 ::; J(UN) ( l I u(t + r) l I � + l I u(t - r) l I � + l I u(t + r) I 1.. + I l u(t - r)I I 1..) .

(4. 5 1 )

(UN)

According to the a priori estimate, d is bounded. Now, we take tP (4.50) and sum up the resulting equality for t E R,. (T) . We obtain that

= 2 D-r u in

I D-ru(t) 1 2 + � l I u (t) ll� + � l I u(t - r) ll� + � l I u(t) I 1.. + � l I u (t - r) I 1..

L: (R(s), D-ru(s) ) l I ull 1 2 + �l I u( r ) l I � + � l I uoll� + �ll u(r) 1 I 1 +2r eRr(t-.,.) L: (f( s), D.,. u( s) ) . + � I l uo ll 1. + 2r eRr(t-.,.) =

•

•

•

By (4.51 ) , the above equality leads to (1 -

r) I D.,.u(t) 1 2 + (� - r d(uN) ) l u (t) l � + �· l I u(t) 1 2 + (� - r d(uN) ) I l u(t) I 1.. ::; Pl (t) + r (c + d (UN)) L: (1 I D.,. u (t) 1 I 2 + I l u(s ) II� + I l u(s )1 1 1..) .eRr(t-.,.) ( 4.52)

where Pl (t)

= (1 + r d(UN,O )) ( l uo l � + l I ul l 1 2 + l I uo l 1 1. + r 4 1 I ul I 1 1.) + 2 r

•

L: I 1< s) 1 2 .

eAr (t)

Finally we apply Lemma 3.2 to (4.52) , and obtain the following result. Theorem 4 . 5 . Let

q = rN2

< 00

and

N be suitably large.

where b1 is a positive constant depending only on

Then for all t

l I uN I c(o,T;Hl) .

E

R.. (T),

Scheme (4.43) possesses the index of generalized stability s = 00 We next discuss the convergence of (4.43). For the Fourier approximation, the L 2 -orthogonal projection is also the best approximation to the norm But it is -

.

I . Ii I .

Chapter 4. Spectral Methods and Pseudospectral Methods

129

UN

not true for Legendre approximation. If we compare the numerical solution to then the leading term in the error equation is It does not vanish, since #This term lowers the convergence rate essentially. Thus, the ' to in order to derive the optimal error estimate, we should compare HJ-orthogonal projection of upon and then from = To do this, let (4.39) ,

PNU,

where

(8",(U - PN U), 8",tP).

8",PN PN8",.

{

UN P!,/U, UN P]/U

IP\(., .

U

(DTTUN(t! + UNY ) + G ( UN(t)) , tP) + (8",UN(t) , 8",tP) = (J(t) + f; Gj (t), tP) , V tP IP\(." t RT(T), DTUN(O) = pJ/U1 + � PRo (8;Uo - Uo - ug + J(O) ) + Gs, E

E

G1 (x, t) = DTTUN(X, t) - 8?U(x, t), G2 (x, t) UN( X, t) - U(x, t), G3 (x, t) = G (UN(x, t)) - G(U(x, t)), G4 (x, t) G(U(x, t)) - � U3 (X ' t + r) - � U3 (x, t - r), Gs = DTUN(X, 0) - 8tUN( X, 0 ) - � 8?UN(X, 0 ). Setting U = UN - UN, we have from (4.43) that (DTTU(t) + U(t) + (UN(t) + U(t) ) - G (UN(t)) , tP) + (8",U(t), 8",tP) = - J=l t (Gj (t), tP) , V tP E IP\(." t E � (T) , o = PN (U1 + � 8� U ( 0 )) - PR (U1 + � 8� U(0)) - Gs, DTU(O) -U(O) = PNUo 0 1 - P,.; Uo . By comparing this error equation to (4 . 50 ), we deduce a result similar to Theorem 4.5. But the terms I DTu(t) ll , I l u(t) 111 and l I u(t) I L4 in that theorem are replaced by I DTU(t) I , I U(t) 1 1 and I U(t) I L< respectively, while P1 (t) is replaced by P2 (t) = c(r + d(UN(O))) (i l U(O) I : + I DTU(O) r + I U(O) l I i< + r4 I1 DTU(0) l i. ) + cr j=L:l seRrL:t T) I Gj (s) 1 I 2 . (=

=

G

5

Spectral Methods and Their Applicati.ons

130

2 (t )

For the convergence rate, we only need to bound P . Let Theorem 2. 1 1 , Lemmas 4.6 -4.8 and imbedding theory that

r ::::: 1 . We obtain from

I G1 (t W :S cN-2r ll U lI�2 (O ,T;W) + cr3 I 1 U(t ) 11 1-' (t -.,.,t+.,.;£2 ) , I G2 ( t) 1I 2 :S cN -2r l U ll�(O ,T;W) ' I IG3 (t W :S cN-2r l U l � (O ,T;W) ' I G4 (t W :S cr 4 1 U 1 �(O,T;W) I U I �' (O,T;£2 ) '

Also by Theorems 2.7, 2.9 and 2 . 1 1 , and Lemma 4.6, we find that

r 4 I1 D.,. U( 0 ) 11 1, :s cr4N2 1 I D.,.U( 0 ) 1 I 4 :s cr4 N2 -4r l Ul l � + cr8 N -2 I 1 a;u(0) I � + cr 1 2 N 2 1 U 1 �3 (O , T ;£2 ) ' IIU (O) II� :s cN- 2r ll Uo l I �+ �2 , etc .. Thus Finally we get the following conclusion.

Theorem 4.6. Let r N2 < and r ::::: 1 . IfU E C2 ( 0, T; HJnHr ) nH4 ( 0, T; L 2 ) , Uo E W+ ! ( A ) and U1 E W ( A ), then for all t E R.,.(T), 00

where b2 and b3 are positive constants depending on the norms of U and f in the spaces mentioned above. If we take

UN,O = p]/ Uo, and

then we can remove the restrictions on o (r 4 + N- 2r ).

Uo and U1 in Theorem 4.6, and E (UN - UN, t)

=

Chapter 4. Spectral Methods and Pseudospectral Methods

131

We now present some numerical results. For describing the numerical errors of scheme (4.43), let t) - v ( t) 11 ( 4.5�) E(v , t) = II U( I I U( t) I I where U is the solution of (4.39) , and v is the corresponding numerical solution. For comparison, we also use a finite difference scheme and a finite element scheme to solve I and (4.39) . Let h = N

{

Ah = A h U { - I , l } .

Ih = {x = j h, -N + 1 :S j :S N - l } ,

( 4.54)

The finite difference scheme is

G

D-rr U h ( X , t) - Uh ( X , t) + (Uh( X , t)) - LlhUh ( X , t) u h ( -I , t) = u h ( l , t) = 0 , D'TUh( X , O) = Uh,l ( X ) , Uh( X , O) = Uo (x),

where Next, let A h ,j

Uh , l ( X ) = U1 (x) +

=

!h

= j (x, t),

x E Ah, t E R'T (T) , t E R'T (T) , x E Ah, X E Ah ( 4.55)

� (a;Uo (x) - Uo(x) - ug (x) + f (O) ) .

{x I j h :S x :S jh + h, -N :S j :S N - I } ,

= {v

I v i s a linear function i n A h ,j , -N :S j :S N - 1 } .

Set Sh = Eh n HJ (A). The finite element scheme is

The values D'T u h (x, O) and u h ( x, O) are evaluated as in ( 4.55) . For describing the errors of (4.55) and (4.5 6 ) , define

Eh ( v, t)

( L: = (

I U (x, t) - v(x, tW

""'hc--____ EA -'x..:..

)

1

---;-,--"--

L: I U (x, tW

xEAh

The meanings of U and v are the same as in (4.53).

)

2'

2'

Spectral Methods and Their Appiicatipns

1 32

We take the test function U(x ,

t)

=

A

(X 2 1 ) cos ( Bx + Bt) e>.t . -

(4.57)

Table 4.6 shows the errors of the scheme (4.43) with A = 0.5 and = ). = 1 . 0 . Obviously, this approach serves very accurate numerical results even for small It can be seen that 'if increases and T decreases proportionally, then the errors decay quickly. This -agrees with the theoretical analysis. Table 4.7 lists the errors for scheme (4.55) and for scheme (4.43 ) , for scheme (4.56) . It is indicated that scheme (4.43) gives much better numerical results than schemes (4.55) and (4.56).

B

N

E (uh ,t)

E (Uh,t)

Table 4 . 6 . The errors E( UN , t)

N = 8, T = 0.005

N = 16, T = 0.005 N = 8, T = 0.001

N = 16, T = 0.001

E(U h , 1 .0) E(u h , 1 .0)

E (UN, t)

.

t = 0.2

t = 0.6

9.50890E-7

1 . 39561E-6

t = 1.0 6.31079E-6

9.41326E- 7

1 .31907E-6

6.22905E-6

1 . 64741E-7

4.94624E- 7

1 . 03551E-7

9.08240E-8

1 . 08537E-7

8.43561E-8

Table 4.7. The errors o f schemes (4.43) , (4.55) and (4.56 ) .

E(UN, 1 .0)

N.

A = B = 1 , >' = 1

A = B = 1 , >' = 2

A = B = 2, >. = 2

N = 16, T = 0.001

2.53742E- 6 3.39940E-7

2.97078E-5 1 . 33918E-6

2 . 1 7966E-4 9.98687E-6

1 . 34663E-2

2. 65475E-3

3.87233E-3

N = 8, T = 0.005

3.38�77E- 3

6.58751E-4

1 . 1 0229E-3

1 .32584E-3

1 . 80903E-2 4.42965E- 3

3.02362E-2

N = 8, T = 0.005

N = 8, T = 0.005

N = 16, T = 0.001

N = 16, T = 0.001

3.28798E-3

7.71182E-3

The Legendre spectral methods for the generalized NLKG equation were consid ered in Guo, Li and Vazquez ( 1 996) . Legendre pseudospectral methods are much easier to be implemented. Let xU) and wU) be the Legendre-Gauss-Lobatto interpolation points and weights in Section

Chapter 4. Spectral Methods and Pseudospectral Methods 2.4. Denote by

AN the set of x U ) , 0 ::; j ::; N.

{ (t

133

We introduce the discrete LP-norm

associated with the interpolation points as

IIVIlLP,N

=

v (:z:; W(i ) ; , =O i m� v ( x ) ,

:Z:EAN

)

i r

l l

for 1

::; p <

for p =

00 ,

00 .

Also define the discrete L 2 -inner product by

V N

( , W)

=

N

I > (x U » ) W (x U » ) w U) .

i =O

There is a close relationship between the discrete norms and the usual norms. The and is (2 . 60) . Some other results are first relation between the norms given in Li and Guo ( 1 996) .

I . liN

I .I

Lemma 4 . 1 0 . For any E

IPN

and p � 2,

Proof. By virtue of Theorem 2. 7,

By (2.60) ,

•

Lemma 4 . 1 1 . For any E PJ!v and 1

::; p <

c(p) is a positive constant dependent of p.

00 ,

Spectral Methods and Their Appiicati ?ns

134

Proof. Let M be any positive integer and IPM be the set of all polynomials on A

e(j) p(j)

p. (x)

of degree at most M. Let be a Jacobi weight. are the nodes and and the weights of Gauss quadrature associated with the Jacobi weight Let m be proved that positive integer. Nevai (1 979) ( also see Szabados and Vertesi a ( 1992)) for any E IP mM ' any �acobi weight and 0 < < 00 ,

7f;

,

p.(x).

p J(x) M P L: 1 7f; (e U ) ) r J (e (j ) ) p U ) c(p) 1 1 7f;(x) I J (x)p.(x) dx. =j 1

On the other hand, ( 2.44 ) implies that

:'5

A

p.(x)

( 4.58 )

- x2 •

So {o,.L, (x) } is an orthogonal system with respect to the weight = 1 As a result, we have that the interior nodes of the Legendre-Gauss-Lobatto quadrature with N + 1 nodes coincide with the nodes e(il , 1 :'5 j :'5 N - 1 of the Gauss quadrature with N - 1 nodes, related to the weight i.e. , = e(il , 1 :'5 j :'5 , we have By ( 2.37 ) , · E IP N 1 . Furthermore, for any E IP

x U)

-

p.(x), x U) ¢1(x)( l - x2) 2N-1 •

2N-3

¢1

w U ) ( (e (j ) ) 2 ) - 1 Let J(x) X2 )-1 and p.(x) x2 in Then we find that N N1 1I¢11 1 L.,N 2:= 1 ¢1 (xU) ) I " w (j) 2:=-1 1 ¢1 (x (j) ) I " w (j) jO j 1 3t=1 1 ¢1 (x U) ) I" ( - (e(j)r- r c(p) L 1¢1 (xW ( 1 - X 2) 1 ( - X 2 ) dx c(p) I ¢1I I L. . whence

=

= (1 -

p (il ,

1-

=

1-

1 :'5 j :'5 N - 1 .

(4.58) .

=

=

p
1

:'5

1

=

It is noted that by ( 2.60 ) and (4.49) ,

1 ¢1l l t :'5 :2 (2 + �) l I o,. ¢1l l t ,

•

V

¢1

E

IP�.

(4.59)

Cbapter

4.

Spectral Metbods and Pseudospectral Metbods

INv

v

(A) ,

135

Let be the Legendre interpolant of E C related to the Legendre Gauss-Lobatto interpolation points. Clearly for all E , and for all Let be the same as before. It is easy to E C see that (4.60)

(K) , (V, W)N = (INv, INw) w

V, W

G(v)

IPN, IN =

UN

Now, we construct the Legendre pseudospectral scheme for (4.39) . Let ap proximate U, and The scheme is to find approximate U3 instead of E for all E such that

{

INuJ-v . ING(uN) UN(X, t) IP R, t Rr(T) (DrrUN(t) + UN(t) + G (UN(t)) ' E IPR" t Rr(T), DrUN(O) = UN,I , UN(O) = UN,O

By

E

(4.61

)

(2. 37), (OxUN(t), ox
E� N (V, t) = I l DrV(t)Il ;' + � l v (t) I � + � l v (t - T) I � + � ll v(t) l ;" + � l v(t - T) I ;' + �ll v(t) ll i"N + �l l v(t - T) ll i"N' We derive from (4.60) and (4.61) that

E�,N (UN, t) = E�,N (UN, T) + 2T BERTL:t r) (J(S), DrUN(S) ) N (which is also a reasonable analogy of (4.40). For error analysis, we need some preparations. it is not difficult to prove the following lemmas.

Spectral Methods and Their Applications

136 Lemma 4 . 1 2 .

We have

I v(t) - V( t) II N � CT �I I VIl H� (t-T,t+T;C)' I DTTV(t) � al v(t) I N � CT � I VIl H4 (t_T,t+T;C)' II DTV(i) - Otv(t) - iolv(t) l i N � CT �I I V Il H3 (t -T.t+T;C) ,

provided that the. related norms exist. Lemma 4 . 1 3 . For any v C(O, Tj C) n H1 ( 0 , Tj C), E

Lemma 4 . 1 4 .

E

For any v, w C(O, Tj IPN) ,

G(v(x,t) + w(x,t)) = G(v(x,t)) + G(w(x,t)) + R(x,t)

with

I R(t ) l I � � d(v) (l I o,.w(t + T)I I � + I l o,.w(t - T) I I � + I l w(t + T) 1 I 1.,N + I l w(t - T) 1 1.,N) ' d(v) being the same as in Lemma 4.9. We now turn to the generalized stability of (4. 6 1 ) . Suppose that UN,O , UN, l and INf have the errors uo, U 1 and j respectively, which cause the error of UN, denoted by u. Then (D:TU(t) � fl (t) + G (t), <1» N + (o,.fl (t), 0,.<1» (i(t), ) N ' v E IP�, t E � (T) , DTu(O) = Ul, u(O) = Uo (4 . 62)

{

where and

=

G(x, t) = G (u(x, t)) + R (x, t)

Cbapter 4. Spectral Metbods and Pseudospectral Metbods

137

By an argument as in the derivation of a priori estimate for the solution of (4.43) , we know that is bounded. By putting = in (4.62 ) and summing the resulting equality for E R,.(T), we obtain

d(UN)

cP n./u (t)

t

I DT u (t ) l : + G - r d(uN)) (1 I u(t) ll � + l u(t) l i + lI u(t)l I i.,N) :::; P3 (t) + r (c + d(uN)) .ERT(tL T) ( I DTU(s) I � + l u (s)li + ll u(s)l I i.,N)

with

(c + r d( UN,O )) ( l uoi l i + I Ul Il 2) + (� + r d (UN,O) ) l I uo l i.,N + 2 r4 I1 ul ll i.,N + 2r ERT(t) L I J(s) 1 I 2 .

P3 (t)

•

Let

EN(V, t) = I DTv(t) l I � + I v (t) l i + I v(t) l I i< ,N '

Following the same line as in the latter part of the proof of Theorem 4.5, we reach the following result. Theorem 4.1.

Let q = rN2 < and N be suitably large. Then for all t E RT(T) , 00

b4 being a positive constant depending only on I UN l c(O,T ;H')'

with INU. In this We now consider the convergence. We could compare case, the optimal error estimate fails, since #But the derivation of error bound is simpler. Moreover the technique used is available also for more complicated problems, for instance, the nonlinear term U3 is replaced by a being any non negative constant. For this reason, we first let UN = By (4.39 ) ,

INa; a; IN '

UN

l u l " u,

INU. (nrTUN(t ) + UN(t) + G (UN( t)) , cP) N + (aXUN(t), axcP) 3 = L (Gj (t) , cP ) N + ( J (t), cP) N , If cP E lP Rr , t E RA T), j= l DTUN(O) = IN U + � IN (a;uo - Uo - ug + f (O) ) + G4 , UN(O) = INUo I

Spectral Methods and Their Applications

138 where

, ' , t), G1 (X, t) = DTTUN(X, t) - at2 UN(X G2 (x, t) = IN (G (UN(x, t)) - � U3(X t + r) - � U3(X ' t - r) ) , G3 (x, t) = INo; U(x , t) - o;INU(x, t), G4 (x , t) = DTUN(X, 0 ) - Ot UN(X, 0 ) - � O; UN(O) . Put U = uN - UN. Then ( 4.61 ) leads to (bTTU(t) + U(t) + G (UN(t) + U(t)) - G (UN(t)) , E IP Rr , t E RT(T), ;=1 DTU(O) = -G4 , U(O) = O. Evidently we could derive an estimate for U(t), similar to Theorem 4.7 in which the norms I DTu(t) II N , l u (t)\I and lI u (t) ll u ,N are replaced by I DTU(t) II N , I U(t) 11 and 1 ! U(t) ll u ,N respectively. Also P3 (t) is replaced by P4 (t) = cllG4 1 2 + 2r4 I1 G4 I 1 i',N + cr j=L3 l SERrLt T) I Gj (s) 1 1· ('

According to Lemmas 4.12 and 4.13,

I G1 (t) 1 I 1 :::; cr3 I UN Il t-, (t_T,t+T;C) :::; cr3 I1 U I l t-' (t_T,t+T;C) ' I G2 (t) I 1 :::; cr4 11 U I � (O T ) By virtue of (2 6 0 ), (2 63 ) and Theorem 2 . 8, I G3 (t) I N :::; cIlG3 (t) 1 I :::; cIlINo;U(t) - o;U(t) I 2 + c ll o;U(t) - o;INU(t) I 2 :::; cN 2r Il U(t) II:+ 2 + cN4 II U(t) - INU(t) lI� < cN -2 r ll U l I � (O , T ; Hr+ 3) . '

.

.

-

By Lemma 4.12,

,

;C '

Chapter 4. Spectral Methods and Pseudospectral Methods

Thanks to Lemmas

139

4.11 and 4.12, Theorem 2. 7 and ( 2. 60 ),

(T4 + N-2r ) and we have the following result. Theorem 4.8. Let T N2 < and r � 1. If U C(O, Tj HJ Hr+3) H4 ( 0, Tj C ) , then for all t R,. ( T),

Hence

P4 (t)

=

0

E

E

00

n

n

bs and b6 being positive constants depending on the norms of U in the spaces men tioned above.

UN with pJ/U. Put UN pll u . In this case, ( 4.39 ) yields (DnUN(t) 9 + UN + G (UN(t)) , E IPIj.. , t E R,. (T), jEG D.. UN(O) pJ/ (U1 + j8lU(0» ) + GlO , UN(O) pJ/Uo

We next compare

=

=

=

=

where

G1 (t) (D.... UN(t) - D.... U(t),
=

=

=

=

T

=

=

=

T

T ,

Spectral Methods and Their Applications

140

- UN, we obtain from (4.61 ) th�t (DnU(t) + U(t) + G (UN(t) + U(t») - G (UN(t» , O. Since (Drr U(t),
U

UN

=

=

=

=

r

=

Theorem 2. 1 1 that

I G1 (t) 1 < I Dn UN(t) - DnINU(t) I N 1I
By ( 2.64 ) ,

By Lemma 4.6,

I G3 (t) 1 $ cll
It is not difficult to verify that

IG4 (t)1 I Gs(t) 1

$ $

I j UN(t) - INU(t) I N 1I
Furthermore, Lemma 4 . 1 3 , Theorem 2 . 1 1 and imbedding theory lead to

I G6(t) 1

$ c ll
+ � T4 1 I UN l\t(t -T,t+T;C) I I UN II�' (t_T'HT;C)

c�+ � T4 1 1 U111:r1 (t-T'HT;W) .

Imbedding theory, Theorem 2. 1 1 and ( 2.63 ) yield $

I G7(t) 1

+ � (I \ U3 (t + T ) - ulv (t + r )l \ � + II U3 (t - r) - Ulv(t - T)I \ �) c l \

$ cll

-

$

Chapter 4. Spectral Methods and Pseudospectral Methods

141

By using imbedding theory and (2.64) again,

Clearly

I Gg (t ) 1 S e ll 4>1 1 2 + ::'e N- 2T ( l I f (t + T) I � + I f(t - T) lI n .

By Theorems 2.7 and 2. 1 1 , Lemmas 4.6, 4 . 1 1 and 4.12, (2.60) , (2.63) and imbedding theory,

I D7' U(O) r = CT4 1 U 1 I �3 (O,T;C) ' I D7' U(O) I :',N S Il D7' U (0) 1 1. S CN2 I 1 D7' U (0) 1 I 4 S N2 -4T ( I I U1 1 I : 1 U1 1 �' (O,T;W) ) +CT4N-2 11 U 1 �2 (O,T;Hl ) + CTs N2 1 U 11 �3 (O,T;C3 ) ' c

c

+

Besides, by Theorem 2 . 1 1 and (2.63),

The above statements lead to the following convergence result.

Theorem 4 . 9 . Let T N2 < and r 2:: 1 . If U E C2 ( 0 , T; HT) C3 ( 0 , T; C ) n H4 (0, T; L 2 ) , Uo E HT+ 1 (/\. ), U1 E W (/\') and f E L2 ( 0, T; W), then for all t E Rr ( T), 00

n

b7 and bs being positive constants depending on the norms of U, Uo and f in the spaces mentioned above. Finally, we present some numerical results. For describing the errors of scheme (4.61 ) , let )

- uN(t)I I N ' EN (UN, t) = I U( tI U(t) IN We also use scheme (4.55) with the relative error Eh (Uh , t) as before. The test function is given by (4.57) . In Table 4.8, the calculation is carried out with A 0.5, B A 1.0 , N = 8 and T 0.005. The errors EN(UN, t) of scheme (4.61) are compared to the errors Eh( Uh, t) of scheme (4.55) . Clearly (4.61 ) provides better numerical results =

=

=

=

Spectral Methods and Tbeir Applications

142

than scheme (4.55), and has very high accuracy even for small N. Table 4.9 shows the numerical errors with A = = 1.0 and A = 2.0. We find that if N increases and decreases proportionally, then the errors decay very quickly, and that the solution of (4.61) converges faster than that of (4.55).

B

T

Table 4.8. The errors of schemes (4.55) and (4.61 ) .

EN(UN ,/1 -

t t t t t

N

4

8 16

0.2 = 0.4 = 0.6 = 0.8 = 1.0

=

E,, (Uh , t)

v

9.21281E-7 5.91656E-7 1 .52947E-6 3.82931E-6 6.61214E-6

1.23154E-3 4.34661E-3 8.66063E-3 1 .36452E-2 1 .88353E-2

Table 4.9 The errors of schemes (4.55) and (4.61) . Uh , 1 . 0 UN , 1.0

0.005 1 .52081E-3 T

=

2 .95840E-5 2.94326E-5

)

EN (

0.001 1 .54 167E-3 T

=

1 . 16568E- 6 1 . 1641E-6

0.0005 1 .54159E -3 T

=

2.97163E- 7 2.93173E-7

T

0.005

8.79735E-3 =

2.65475E-3 6.76090E-4

Eh (

)

0.001 8.78729E-3 T

=

2.63820E-3 6.58751E-4

T

0.0005

8 . 78698E-3 =

2.63768E-3 6.58222E-4

The generalized NLKG equation is of the form

a:u + U + g ( U) - a�u

Il

=

f

where g ( z) = z "' a, a � O. In this case, define G(v ( x , t))

=

l g (yv (x , t + T) + (1 - y ) v (t - )) dy ,

ING(uN) ,

T

and approximate g (U ) by see Li and Guo (1997) . The numerical solution is compared to IN U when a is not an integer. Legendre spectral methods and Legendre pseudospectral methods have been used for numerical studies in various fields. For instance, Maday and Quarteroni (1981) used it for the steady Burgers ' equation. Canuto and Quarteroni (1982b) studied the stability of semi-discrete Legendre spectral and Legendre pseudospectral schemes for linear hyperbolic equations. They also considered Jacobi spectral approximations in that paper.

UN

Chapter

4.3.

4.

Spectral Methods and Pseudospectral Methods

143

Chebyshev Spectral Met hods and Chebyshev Pseudospec tral M ethods

Chebyshev spectral methods and Chebyshev pseudospectral methods are also used widely for non-periodic problems. Since Chebyshev polynomials can be changed into trigonometric polynomials by a transformation of independent variab le, the Fast Fourier Transformation can be used in actual computations. But the existence of the weight function brings some difficulties in theoretical analysis. In this section, we take the Burgers' equation as an example to show how to build Chebyshev spectral schemes and Chebyshev pseudospectral scheme, and how to analyze the errors pre cisely. The filterings are applied to improving the stability. The numerical results presented show the effects of filterings. Burgers' equation is one of the important models describing the movement of unsteady one-dimensional fluid flow. Let A = ( - 1 , 1 ) . The initial-boundary value problem of Burgers' equation is the following

{

=

<

8tU(x, t) + U(x, t)8xU(x, t) - 1t8;U(x, t) f(x, t), x E A, 0 t S T, ' U(- I , t) U( I , t) 0, 0 < t S T, (4 .63) xEA U(x, O) Uo(x), where Uo(x), j(x, t) and > 0 are the initial state, the body force and the kinetic viscosity respectively. It is known that if Uo E L 2 (A) and f E L2(0, T; L 2 ), then (4.63) has a unique solution U E L 2 (0, T; H I ) n L= (O, T; L 2 ) . The solution of (4.63) =

=

=

It

possesses the conservation

(4.64) I U(t) 11 2 + 21t l l U(s ) I � ds Il Uo l 1 2 + 2 l ( 1(s), U(s)) ds. Let w(x) ( 1 - x2) - ' . The space L� (A) is defined as in Section 2.5, equipped with the inner product (', · )w and the norm 1 I · l I w ' The spaces LE(A) , H;:: (A) , Hr;:j A) and their norms I . I Ll:" I . I l m,w are also the same as before. Let N be any positive =

=

1

integer and PN be the L�-orthogonal projection from L�(A) onto lPR,. The H� orthogonal projections P/v , fil'/ and pi'/ are given in Section 2.5. The mesh size in and time is taken to be T . The notations t) are the same as before. When we approach (4.63) by the Legendre spectral approximations, we can get a scheme whose solution fulfills a conservation simulating (4.64) . But this property is

t

Rr(T ), fJ (x, t)

Drv(x,

Spectral Methods and Their Applications

144

destroyed by the weight function in Chebyshev spectral approximations. In Section 4. 1 , we constructed the schemes with the a�curac f first order in time discretization, unless the nonlinear term is approximat ed by a fully implicit technique. If we use three-level scheme, then we can obtain the accuracy of second order in temporal discretization. While the unknown solution can still be evaluated explicitly by the variable transformation and the orthogonality of Fourier system. We shall adopt this trick. be the approximation to U. A Chebyshev spectral scheme for (4.63) is to Let find such that E for all E

y/�

{

UN t Rr(T) UN(X, t) lP� (Dr UN(t), 1>t - ! (u� (t), a",(1>w)) + IL ( a", U N(t), a,,, ( 1)w)) (J (t), 1>t , V 1> E lP�, t E Rr(T), UN( r) UN,! , UN(O) UN,O =

=

( 4.65)

=

The scheme (4.65) stands for

which is easy to be implemented in practice. (4.65) is an implicit scheme. At each time E we need to solve a linear system

t Rr (T), F(t)

uN(t - r), uN(t - 2r) UN(t)

i(x, t).

and where depends on By Lemma 2 . 1 , it possesses a can be calculated explicitly as explained above. unique solution. Furthermore, In actual calculations, we can take the basis functions as

( ) { T/(x) T/(x) - To(x), - Tl (X),

1>/ x

=

l is even, I is odd.

But these functions lead to a linear system with a full matrix. A better choice ( see Shen ( 1995)) is to take

Chapter 4. Spectral Methods and Pseudospectral Methods

Then

{

and

145

l = m, I = m - 2 or m + 2,

otherwise,

21r ( m + 1 )( m + 2 ) , l = m , (o:z: '1h , O:z:( 1/Jmw)) = 41r ( m + 1 ) , I = m + 2, m + 4, . . . , 0, 1 < m or 1 + m odd. We now consider the gener alized stability of (4.65 ) . Let uo , U l and J be the errors of UN,O , UN,l and PN j , which induce the error of UN, say U. From (4.65 ) , (b'T u(t), E IP ltr , t E R,. (T), (4.66 ) where

F( vex, t), w(x, t)) = vex, t)w(x, t) + �W2 (X ' t).

= t1 in (4.66) , we get from ( 1 .23) and Lemma 2.8 that � bT ll u(t) ll � + � 11t1 (t) II�,w :5 (J(t), fi(t)t + (F (UN(t), u(t)) , O:z: (fi (t)w)) .

By taking ¢>

(4.67 )

We next deal with the right-hand side of (4.67) . First of all, we know from Theorem and the proof of Lemma 2.8 that

2.13

1 4 (F (UN(t), u(t)) , O:z: (� (t)w)) I :5 � 1 � (t) I : ,w + � I F (UN (t), u(t) ) I ! :5 � 1 � (t) I : ,w + dl(UN , p ) ( 1 I u(t) l I ! + N l I u(t) I !)

dl(v, u) is a certain positive constant depending only on I v ll c(O,T;LOO)

where Clearly, Let

1 4 (l
r < � and

E(v,t) = ll v(t) I � + � pr 2: 1 � (s) I :,w ' Pl(t) = 2 l 1 uo l ! + 2 l 1 ul l I ! + 4r � 1 1(s) I : . seRr(t-T)

seRr(t)

(4.68 )

and

p.

(4.69 )

Spectral Methods and Their App1ica�ions

146

t

By substituting (4.68) and (4.69) into (4.97) and summing up the result for E Rr (T) , we obtain

E (U , t) ::; PI( t) + 7' . R2.:t-.,. d ( UN , I') (E(u , s ) + NE2 (u , s ) ) . _

e r

( )

Finally we use Lemma 3.2 to conclude that

Let 7' be suitably small. There exist positive constants bI and b2 depending only on l I uN l c(o,T;LOO) and such that if PI(t I) ::; � for certain t l E Rr (T), then for all t E R.- ( tI), Theorem 4 . 1 0 .

1',

PNU, P]. U Pj/ U . (U - pj/ U)

We now deal with the convergence. We can compare UN t o or But in these cases, the leading error terms cannot be cancelled. It means that the terms (0", , o",(¢w)) 0",(¢W)) and (0", , o",(¢w)) , (0", They shall lower the convergence rate. Therefore we com do not vanish for ¢ E pare UN to UN = pj/ U, where the HJ-projection pj/ is defined as in (2.82) . By

(U - PNU)

(4.63) ,

lP�.

(U - P]. U) ,

�

(b.,.uN (t), ¢L - ( U1(t), 0",(¢W)) + p, (O",UN (t) , O",(¢W)) = (GI ( t) + j(t), ¢L + ( G2 ( t ) + G3(t), 0",(¢W)) , V ¢ E lP� , t E Rr (T ) , with

GI ( x , t) = b",UN ( X , t) - Ot U ( x , t), 1 1 2 X , t), G2 ( X , t) = "2 Y2 ( x , t ) - "2 YN( · .,. U2 ( x , t). G3( x , t) = 41 7' 2 D.,. U = UN - UN , we obtain from (4.65) that (b.,.U(t), ¢L - (F ( UN ( t ), U(t)) , 0"'(¢W)) + I' (o",U(t), 0"'(¢W)) . + (GI(t ) , ¢t + (G2 ( t ) + G3(t), 0"'(¢W)) = 0, V ¢ E lP�, t E R.,. (T) . (4. 70)

Putting

Let

r ;:::: 1 . By Theorem 2.18, 2.: I G ( s ) l I ! < c7' 4 11 U 1 I 13 (o ,T;L� ) + cN- 2r I U I 11 (o, T ; H,!; ) ' seRr(t-.,.) I G2 (t ) l I � < d2 (U)N-2r Il U(t) I �,w ' I G3 (t) l I � ::; 7'4 d3 (U) ( 1 U 1 I �2 (o,T;L�) + 1 ) 7'

I

Chapter 4. Spectral Methods and Pseudospectral Methods

147

I vllc(o,T ;H.!, ) , I vllcl(o,T ;LOO ) '

d3 (v)

and where d2 (v) is a certain positive constant depending only on other the hand, On is a cert�n positive constant depending only on

Let

I be the identity operator. Since U ( -r ) = (PN - I)(Uo + -r Ot U(O)) + (I - pt/)U(-r) + Uo + -r Ot U(O) - U(-r),

we assert that

4.10 to ( 4.70 ) . Theorem 4 . 1 1 . Let :::; b3 N - t . If U E C 1 ( 0 , Tj HJ ,w H�) H3 ( 0 , Tj L! ) with r � 1, then E (UN - 'U N, t) :::; b4 ebs t (-r 4 + N- 2 T ) , b3 , b4 and bs being positive constants depending on I-' and the norms of U in the spaces mentioned above.

Finally we apply Theorem

n

-r

n

Chebyshev pseudospectral methods are more preferable in practical cases. We be the Chebyshev-Gauss-Lobatto shall use this approach for Let and = interpolation points and weights as in Section Set :::; j :::; N . . D.enote by the Chebyshev interpolant of E associated with The are defined by discrete inner product and the discrete norm = for related to the Chebyshev-Gauss-Lobatto points and weights. Clearly, = any ¢J E and for all E Let b e the approximation to In order t o strengthen the stability, we use the = filtering (3), (3 � defined by The Chebyshev pseudospectral scheme for is to find E for all E .R.,. (T) such that

(4.63 ).

x(;)

w(j) 2.5.

AN { xU) I 0 } AN. INv v C ( A ), .I I N,w ( , ' )N,w ( 2.36), IN ¢J ¢J IP N, (V , W)N,w (INv, INw)N,w V , w C ( A). 'UN U. RN RN(ex, ex, 1, ( 2.136 ) . 4.63 'U IPjt,. t N(X,t) ( ) '

148

Spectral Methods and Their Applications

UN,O = INUo and UN,1 = IN (Uo + rOt U(O) ) INUo + rIN (JtO; Uo - � Ox Ug + f( O )) . Scheme (4. 71) stands for DTuN( X, t) + ;/1 NoxRNINuN2 (x, t) - JtINo",2 uN(X, t) = INRNINf(x, t). where

=

A

A

This expression is easier to be performed in actual computations. Now suppose that have the errors and j respcetively, I and which cause the error of From

UN,O , UN, RNINi uo, U I UN. (4. 71), UN, (bTu(t ),
(4.73). By Lemma 2.8, Theorems 2.19 and 2.27, (2.86 ), 1 4 (RNINF(t), o", (ii(t)w)) 1 :$ 8 1 I INF(t) ll w l ii (t) I I ,W :$ � / ii (t) / : ,w + 3: I F(t) I � · Hence it is bounded by the right-hand side of ( 4. 6 8). Clearly

We now estimate the right-hand side of and

Let r <

!, and E(v, t) and d(UN ' Jt) have the same meaning as before. Set

By putting the above two estimates into we find that

(4. 73) and summing the result for t E RT(T),

E (U, t) :$ p2 (t) + r 8ERT(t-T) I: d(UN, jl ) (E(u, s) + NE 2 (u, s)) . Applying Lemma 3.2 to the above 'inequality, we obtain the generalized stability

expressed as follows.

Chapter 4. Spectral Methods and Pseudospectral Methods

149

Let '( be suitably small. There exist positive constants be and b7 depending only on I UN l c(O,T;LOO ) and such that if P2 (t 1 ) $ � for some t l E Rr ( T ) , then for all t E ii. (t 1 )

Theorem 4 . 1 2 .

,

/J ,

(4. 7 1). For the optimal error estimate, we also UN UN U. U UN - UN. Then (4. 63 ) and (4. 7 1) produce th at (fJr f)(t), cf» N,w - (RN INF (UN (t), f)(t)) , a.,(cPw)) + (a,,/J ( t ), a.,(cPw) ) +G1(t) + (G2 (t) + G3 (t), a.,(cPw)) + G4 (t) = 0, V t/> E IP�, t E Rr(T)�4. 74)

We next state the convergence of to compare = PN1 0 Let =

p.

where F(v , w) and

Let

G3 (t) are the same as in (4.70), and G1 (t) = (b., UN (t), t/» N,w - (a/J (t ) , t/» w , G2 (x, t) = � U2 (X ' t) - � RN IN UXr ( x , t), G4 (t) = (i (t), t/>L - ( RN INi(t), t/» N,w ·

r ::::: 1. We have I G1(t) 1 $ I (b., U(t) - atU(t), t/» w ! + l (b., U(t ) , t/>L - (b., U(t), cf» N.w ! + I (b.,U(t) - b., UN (t), cf» N,w ! ·

According to Theorem

2.18 and ( 2.87),

with Moreover, '(

We have

l: A( s ) $ c'(4 I J U I �3 (O ,T ; L� ) + cN -2r I J U II�1 (O ,T ; H.!; ) · ·.ERr(t-.,)

15

�d

0

Spectral Meth s and Their Applications

By Theorems 2.19 and 2.27,

where

d4 (v) is a positive constant depending only on I v l c(O,T;H�) ' We also have

Besides,

iI U(O) i I w :::; cN-r l Uo l r,w , I U(r) 1 ! w :::; cN-r ( I U(O) l I r,w + I U(r) l I r,w + r l otU(O) l I r,w ) + cr 2 11 U Il c2 (O,T ;L� ) '

Finally, we apply Theorem 4.12 to ( 4.74 ) .

E

n

n

E

Let :::; bs N-i . If U Cl (O, T; HJ,w H�) H3 ( 0, T; L�), f C (O, T; H� ) and 1 :::; r :::; then E (UN - UN, t) :::; bg eb, o t (r4 + N-2r ) , bs , bg and bl O being positive constants depending on and the norms of U and f in th� spaces mentioned above.

Theorem 4 . 13.

r

a,

p.

In order to save work, we sometimes use explicit Chebyshev pseudospectral schemes = for ( 4.63 ) . For instance, let a, (3) , a, f3 ;::: 1 be the filter defined by ( 2. 137 ) , and consider the following Chebyshev pseudospectral scheme

R� R�(

DTuN(X, t ) + � INRNO:r: (RNINu'fv(x, t ) ) - p.INRN{J�R�uN(x, t) = INRNINf(x, t ) . ( 4.75 ) 4 0 then we can derive the same result of stability and convergence r (N- ), as in Theorems 4.12 and 4.13. The condition on r is much more severe than the condition for the stability in the corresponding Fourier pseudospectral schemes, in which r 0 (N- 2) . However, the usefulness of the filters RN and R� can improve If

=

the results in practical computations. We check the effects of the filters in ( 4.75 ) by numerical experiments. For description of the accuracy, let =

Chapter

4.

Spectral Methods and Pseudospectral Methods

151

We take f = 0 and so (4.63) has a solution

2x - t ) . U (x,t) = -21 (1 - tanh -81-'

Set I-' = 0.005, T = 0.0005 and N = 64. To test the effects of the filters, various values of a and /3 are carried out. For a = 00 or /3 = 0, (4.75) becomes the usual Chebyshev pseudospectral scheme. In this case, B2 ( UN , 0.25) = 1 . 302 and Boo ( UN , 0.25) = 3.907. The numerical errors with other choices of a and /3 are presented in Tables 4.10 and 4. 1 1 . Evidently, the filters improve the accuracy. For N =64, a better choice appears to be a = 6 '" 8, /3 = 2 '" 3. Table 4.10. The errors E2 a = 10 a = 8 /3 = 1 0.883 0.787 /3 = 2 0.393 0.016 /3 = 3 0.016 0.018

0.25 ) . a=6 0.023 0.018 0.021

UN ,

Table 4. 1 1 . The errors Boo ( UN , 0.25) . a = 10 a = 8 a = 6 /3 = 1 3.409 3.283 0. 102 /3 = 2 2.596 0.065 0.073 /3 = 3 0.065 0.071 0.082 The Chebyshev spectral methods and Chebyshev pseudospectral methods for gen eralized Burgers ' equation are given in Ma and Guo (1988). The filtered schemes can be found in Ma and Guo (1994) . The Chebyshev spectral methods and Chebyshev pseudospectral methods have been used successfully for various partial differential equations. For instance, Maday and QuarterOIii (1981) used it for steady Burgers' equation. Gottlieb (1981) first applied the properties of Gauss-type quadrature to studying the stability of Cheby shev pseudospectral methods for linear hyperbolic and parabolic equations. Gottlieb, Orszag and Turkel (1981) considered hyperbolic eq� ations with variable coefficients.

Spectral }.(etho:ds and Their. Applications

152

4.4.

Spectral Penalty Methods

We can deal with inhomogeneou s boundary conditions in several ways. One of them is the spectral penalty method in which the boundary conditions become a part of equation, see Funaro and Gottlieb (1988,1991) . In the last section, we introduced the Chebyshev pseudospectral method associated with the Chebyshev-Gauss-Lobatto interpolation points. Those points are chosen, because they allow the use of FFT in computations. For inhomogeneous boundary conditions, we can use Chebyshev penalty method as described below. Consequently, we adopt the weighted norm in numerical analysis. However, this is not a natural norm and complicates the analysis. A more natural way is to use Legendre-Gauss-Lobatto interpolation and the corresponding penalty method. As those points are not given explicitly, their evaluation for large is not robust due to roundoff errors. In this section, we focus on a new approach, implementing Legendre method on Chebyshev points, enjoying the advantages of both Legendre and Chebyshev methods. It is named the Chebyshev Legendre method by Don and Gottlieb (1994) . We begin with a simple problem. Let A = ( - 1 , 1 ) and consider the hyperbolic equation = E A, 0, (4.76) 0, = = E

N

{

8tU(x, t) - 8",U(x, t) f(x, t) , x t > t> U(l, t) g(t), Uo(x), x A. U(x, O)

x(j) and w(j) be the Chebyshev-Gauss-Lobatto interpolation points and weights. AN = { x U) I ° :5 j :5 N}. The usual semi-discrete Chebyshev pseudospectral scheme is to find UN(X, t) E IPN for t 2: ° such that 8tUN( X, t) - 8:cUN(X, t) = f(x, t), x E AN/{l}, t > 0, (4.77) uN(l, t) = g(t), t > 0, = UN(X, O) Uo(x), x E AN . Let

{

Funaro and Gottlieb (1988 ) proposed a penalty method in which, instead of the boundary condition in (4.77) , we require that

>.

where is determined from stability considerations. This approach is stable for the initial data as long as 2: Since are the zeros of the polynomial

>. � N2•

xU)

Chapter 4. Spectral Methods and Pseudospectral Methods

(1

{

- X 2) OxTN( x), this scheme can be written as >'(1 X)8 (x) 8tU � (x, t) - 8",UN(X, t) � "'r) (uN(I, t) - get)) 2 ",TN 1 +

153

I(x, t), x E AN , t > 0 ,

x E AN· (4.78) The main feature of (4.78) is that the numerical solution does not fulfill the boundary condition exactly, but only in the limit as N --. 00 . In fact, the boundary condition is a part of the equation. Another penalty method based on Legendre-Gauss-Lobatto interpolation was served by Funaro and Gottlieb (199 1 ) . Let be the and N nodes and weights of this interpolation. = y (i) = x ( -i) I 0 :::; j :::; N } . Since y (i) . are the zeros of the polynomial ( 1 the penalty scheme for (4.76) is to find t) E for t � 0 such that UN(X, 0)

=

Uo (x),

UN(X ,

{

lPN

8tUN(X, t) - 8",UN(X, t) +

. UN(X, O)

=

=

AN { x2) OxLN(X ),

>'(1 + x)8xLN(X) . (UN(1, t) - get)) 28", LN (I)

w (j )

x(i)

=

I(x, t), x E AN , t

-

Uo (x),

This scheme is also stable for the initial data when A � ! N ( N + 1 ) . The Chebyshev-Legendre penalty scheme for (4. 76) i s t o find t � 0 such that

{

� 0,

x E AN· (4.79)

UN(X, t ) E lP N for

8tUN(X, t) - 8xUN(X, t) +

>'(1 + x)8",LN(X) . (UN(I, t) - get)) 28xLN(I)

=

I(x, t), x E AN , t

-

� 0,

x E AN . (4.80) We now explore the relation between (4.79) and (4.80). The insight can be gained if one compares the differentiation matrix induced by (4.79) and the differentiation matrix induced by (4.80). As in the proof of Theorem 2.6, put UN(X, O)

=

Uo (x),

qc( x) = (1 - X 2) O",TN(X), q£C x) = ( 1 - x 2 ) O",LN(X) . Define the Lagrange polynomials

154

Spectral M�thods and Their' Applications

v C ( A ), the Chebyshev interpolant IN v and the Legendre interpolant iNV are N ( I» (4.81) INv(x) = L ',,;N0 > (X ) 91 (X), (l) (4.8 2 ) iNV( X ) = L 1=0 > (y ) h,(x). Denote by Bc the differentiation matrix in Chebyshev pseudospectral approximation, and by DL the differentiation matrix in Legendre pseudospectral approximation. It means that N (4.83) 8.,!NV (x (l) ) = m=L:O DC,l,m V (x (m» ) , 0 � 1 � N, N (4.84) 8.,iNv (y (l) ) = m=L:O DL,l,m V (y (m») , 0 � 1 � N. For E as follows

By virtue of (4.81) and (4.82),

DC,l,m = 8.,9m (X (I) ) , DL,l,m = 8., hm (y (l» ) , 0 ::; 1, m ::; N. Let A and B be the ( N + 1 ) x (N + 1 ) matrices with the elements A" m = hm (X (I») , B',m = 9m (y (l) ) , 0 ::; 1, m ::; N. Furthermore, denote by F the matrix induced by the differentiation and the penalty in (4.80), and denote by G the matrix induced by the differentiation and the penalty in (4. 79). We have the following results.

We have AB = I and Dc = ADLB. Proof. Since 9m(X) E IP N, it can be expressed as N 9m(X ) = j=O L:9m (y (j») hj (x) . Because 9m (X ( I ») = Ol ,m , N N L: A" j Bj,m O',m = j=O L:9m (y (j» ) hj (x (I» ) = j=O Lemma 4 . 1 5 .

(4.85)

Chapter 4. Spectral Methods and Pseudospectral Methods

whence

AB

=

155

I. Next, differentiating (4.85) yields

N 8.,9m(x) j=O I: 9m (y (j» ) 8., hj (x). =

Since

E

8., hj (x) lPN, it can be expressed as N 8., hj (x) k=I:O 8., hj (y (k» ) hk( x). =

Therefore

N N (j » I: I: 9m (y ) 8., hj (y (k» ) h k (x (l») j=ON k=O N I: I: A/ , k DL, k ,j Bj,m (ADLB) / ,m . j=Ok=O =

•

We have F AGB. Proof. The differentiation matrix G is essentially the matrix DL introduced in (4.84) ,

Theorem 4 . 1 4 .

=

modified to take into account the boundary conditions, imposed via penalty in (4.79). Thus

G/ ,m DL,/ ,m - J.. h/ ,o hm,o . =

(l» ) 8.,LN (x (I») F/ ,m Dc,/,m - J.. ( 1 + 2x8.,LN( hm,o . I)

In the same manner, we obtain from (4.80) that =

(4.86)

By Lemma 4.15, we deduce that

NN (ADL B ) /,m - J.. j=O I: kI:=O hj,ohk,o hj (x (l) ) 9m (y (k» ) Dc,/,m - J.. ho (x (I» ) 9m (y (O» ) . According to the definition of h o(x), a computation produces that (AGB}! ,m

=

=

(4.87)

Spectral Metljods and Their Appliqations

156

yeO) = x (O) = 1 , gm (y (O) ) 8m 0 . Hence " ( 4. 8 7) is equal to the right-hand side of (4.86 ). Then the desired result follows. We now turn to analyze the stability of scheme (4. 80 ). By the theory in Chapter 3, it suffices to verify that the norm of UN depends on the norms of data continuously. Let N 2 > (y (j ) ) W (y Ci) ) w (i) , (V , W)N = j=O and I v l i N = (v, v)Ar · Also set N N ( I) ) W (x ( m) ) x H" mV ( L I v liN = L I=O m=O where N H" m = j=O L g, (y Ci) ) gm (y Ci) ) w Ci) . For any if> E IP N, we have N if>(x) = INif>(x) = L=0 if> (x (I) ) g,(x) =

Since

,

•

1.

1

and so

(4. 88 ) 1 I if> I N = 11 if> I N . Theorem 4.15. If H1 + ) (N2 + N) ::; A ::; (N2 + N) nd > O J th en Il uN(t) ll � ::; en ( I UN(O) I � + � l ( I INf(s) l I � + 4c�l (s) ) dS) . Proof. Since UN(X, t) E IPN for t ;::: 0, (4.80) implies that for all x E A and t > 0, + X )O",LN( X ) OtUN(X, t) - O",UN(X, t ) + A(l 2o",LN(1) (uN(l , t) - get)) = INf(x, t). ( 4.89) By setting x y Ci ) in (4. 89 ) , multiplying it by 2UN (y Ci) , t ) w (i ) and summing the Consequently,

e

result for 0

::;

=

j

::; N,

we obtain that

Cl

a

e

Ot l uN(t) ll � - 2 (UN(t) , O",UN(t)) N + 2AW (0) U�(1, t) 2Aw (0) uN(1 , t)g(t) + 2 (INf(t), UN(t))N ' =

Chapter 4. Spectral Methods and Pseudospectral Methods

By

(2 . 37),

157

2 L UN { X, t)O",UN(X, t) dx u�(l, t) - u�( - 1, t)

and then

otlluN {t) l I � + (2 "\w (O) - 1 - e) u�(l , t) + u�( - 1, t) (.� (O» 2 :::; e I UN(t) l I � + -Ie1 I INf {t) l I � + we ) l(t). Sinc w (O) = : � N { + 1) ' we integrate the above inequality to obtain from which the conclusion follows. Theorem 4.15 shows the linear stability of scheme (4.80) . In particular, if 0, then

f(x, t)

•

=

g {t) = I l uN(t) I � + l (( N (�..\+ l) - l ) U�(l ' S ) + U�(-l, s» ) ds = l I uN(O) I � · Next, we consider the convergence of (4.80) . Let UN = iNU. We get from (4.76)

that

(4.90)

where

G1 (x, t) = iNo",U(x, t) - o",iNU(x, t), G2 (x, t) = iNf(x, t) - INf(x, t), G3 { x,t) = iNUO (x) - INUo (x).

Put (;

= UN - UN. Then we have from (4.80) and (4.90) that for all x E A and t > 0, + X)O",LN{X) U(l, t) -G1 (x, t) - G (x, t). (4. 91 ) OtU(x, t) - o",U(x, t) + ..\ ( 1 2o",LN(1) 2 �

�

,

�

=

Spectral Methods and Thrur Applications

158

1\

.

'

Furthermore, by using (2.63), ,

Similarly,

IIG2 (t) l I � � cI I G2 (t) 1 1 2 � clliNf(t) - f(t )1 I 2 + cIl INf(t) - f(t) I �· By Theorem 2.19 and (2.63),

Also,

U(x, O) = - G3(x), and

Finally we get the following result by comparing (4.91) with (4.89).

Let ). + 1 ) and r > � . ffU E L2 ( O, Tj W+1 ) , Uo E H� ( A ) and f L2 ( 0, then for all t � T,

Theorem 4.16. � �N(N E Tj H� ) ,

where bl is a positive constant depending only on the norms of U, Uo and f in the mentioned spaces.

{

Now we apply the Chebyshev-Legendre penalty method to parabolic equations. Consider the problem

x E A , t > 0, 8tU( x, t ) - 8; U(x, t) = f(x, t), 0:8",U(1, t) + (3U(I, t) = g+ (t), t > 0, (4.92) -'"'(8",U(-I,t) + 8U(- I , t ) = g-(t ), t > 0, XEA U(x, O) = Uo(x), where 0:, (3, 8, '"'( � 0, 0: + (3 > ° and '"'( + 8 > 0. For the Chebyshev pseudospectral scheme, Gottlieb (1 981 ) proved the stability with 0: = '"'( = 0, and Gottlieb, Hussiani

Chapter 4. Spectral Methods and Pseudospectral Methods

159

and Orszag proved the stability with = = 0. In order to describe the Chebyshev-Legendre penalty method, we introduce the operator . as

{3 8

( 1984)

A

Av(x,t) = - o�v(x, t) + R(v, x, t) where

R(v, x, t)

Ao QO ( X ) (ao",v (1, t) + {3v(1, t) - g + (t)) + AN QN(X) (- ,o",v( - 1, t) + 8v( - 1, t) - g- (t))

with The Chebyshev-Legendre penalty scheme for such that

{ UN(X, OtUN(X, t) - O�UN(X, t) + R (UN, X, O) Uo(x),

E

(4.92) is to seek UN( x, t) lP N for t � °

t) =

f(x, t), x E �N ' t > 0, (4.93) x AN. lx:;) (O) and We first consider the stability. For simplifying the statements, let K = a c( a, b) = a�(O) (1 + 2 1C + 2JK2 + K) , d( a, b) = a�(O) (1 + 2K - 2 JK2 + K) . Lemma 4 . 1 6 . Let d(a, {3 ) :5 A o :5 c( a, {3 ) and d(,, 8 ) ::; AN :5 c h, 8). Then for all
E

1 60

Spectral Method's and Their Applications

Therefore

N-l � (OX«p (y (j )) ) \ � (j) + F (I , Q, ,8, 0 ) + F( - 1 , I, 0, N ) j l O) - Aow ( «p(I)g+ (t) - ANw (N) «p( - 1 )g- (t) =

where

For F( I ,

Q, ,8, 0 ) to be non-negative, we need ( QAo w (O) 1 r � 4,8 Ao (w (O) r -

or

Q2 (w (O) ) 2 A� 2 (Q + 2,8w (O) ) w (O) AD + 1 � 0, i.e., d ( Q, ,8) � Ao � c( Q, ,8). Similarly F( - I ' I, o, N ) � 0, if d ( /' o ) � AN � C (/, O) . _

The proof is complete.

fa N2 ( N +

•

0, ,8 1, then the condition in Lemma 4 . 1 6 is reduced to that AD � 1 ) 2. While for Q = 1, ,8 = 0, we require that Ao = � N (N + 1 ) . We now turn to the stability. For simplicity, we only consider the case g + (t) = g - (t) = o . Theorem 4 . 1 7. Let AD and AN satisfy the conditions in Lemma 4 . 1 6. If g + (t) g - (t) OJ then for any > O J If

Q

=

=

=

c:

=

Proof. Since

E

UN(X, t) lPN for t � 0, (4.93) implies that for all

x

E

A and

t > 0, (4.94)

i.e., By Lemma 4.16,

Cha.pter 4. Spectral Methods and Pseudospectral Methods

161

from which the conclusion comes.

•

We next consider the convergence of scheme (4.93) . For simplicity let a =:= '"'I = 0, = = 1 . Let be the HJ-orthogonal projection upon defined by (2.57) , and = Then by (4.92) and (2.37) ,

{J 6 IP},;o IPRr UN P';/ U . (8tUN(t), rp)N + (8",UN(t), 8", rp) N + (R (UN, t) , rp) N GI (t) + G2 (t) + (INf(t), rp) N ' rp E IPRr , t > 0, EA UN(O) pj/Uo ,

'{

'It

=

=

x

where

GI(t) = (8t UN(t), rp) N - (8tU(t) , rp) , G2 (t) (J(t), rp ) - (J(t ) , rp )N (J(t) - INf(t) , rp) N · +

=

On the other hand, (4.94) and (2.37) imply that

Set

fJ

UN - UN . We obtain that { U(O) (8ir (t), rp) N + (8; ? (t), 8", rp) N + (R ( f) , t) , rp) N = -GI (t) - G2 (t), INUo - PN Uo . =

(4.95)

=

By Theorem 2. 1 1 , Theorem 2.19, (2.60) and (2.64) , we know that for r ;::: 1 , +

:5 1 (8t UN(t), rp) N - (8tUN(t), rp )1 1 (8tUN(t) - 8tU(t), rp) 1 < cN-r llrpI l I 1 8tU(t)l I r :5 e llrpll� + �e N- 2r Il 8tU(t)II� , < cN -r llrpll l f(t) l I r,w :5 e l rpll� + �e N -2r l f(t) I �,w .

Also by Theorem 2 . 1 1 , Theorem 2.19 and (2.60) ,

Ii U(O) I N :5 c IW (O) I :5 c (1 1 Uo - INUo l 1 + Il Uo - pkO Uo l I ) :5 cN-r Il Uo l i r .

By taking rp = fJ in (4.95) and an argument as in the proof of Theorem 4 . 1 7, we reach the following aim. Theorem 4 . 1 8 . E

Let

a

=

'"'I

=

O, {J 6 =

=

1 , r ;::: 1

and Ao , AN ;::: fsN2(N + 1 ) 2

U HI (O, t j H' ), Uo E Hr (A) and f E L2 (0, Tj H�) , then for all t :5 T,

.

If

Spectral Methods and Their Applic�tions

162

where b2 is a positive constant depending only on the norms oj U, Uo and J in the mentioned spaces. The Chebyshev-Legendre penalty method can be used for nonlinear problems. Consider the problem Let J(z) be a continuous convex-function and J'(z) =

with

I-'

8zJ ( z ) . 8tU(x, t) 8,,:/ ( U(x, t)) - 1-'8�U(x, t) = J(x, t), x E A, 0 < t � T -jl-

� 0 and the boundary conditions

U(I, t) = g+(t), U( -1, t) = g ( t ), -

if I-' > 0 or I-' = 0, if I-' > 0 or I-' = 0, J'

f' (g+(t)) � 0, (g-(t)) � o .

The corresponding Chebyshev-Legendre penalty method is as follows

with

baCt) = 1 for I-' > 0 or I-' = 0, /' (g+(t)) � 0, and bo (t) = 0 otherwise; and bN (t) = 1 for I-' > 0 or I-' = 0, J' (g - (t)) � 0, and bN (t) = 0 otherwise. Finally we list some numerical results. Let T be the mesh size in t . We discretize (4.78)- (4.80) by the third order Heun-Runge-Kutta method in time. Then for x E AN and t E R,. (T) , uiv (x, t) UN(X, t) + '3T ( DUN(X, t) - AQ (X) (UN(X, t) - g(t))) , U';J(x, t) = UN(X, t) + 2; (DuN(x, t) - AQ (X) (uiv(x, t) - get) - i8tg(t) )) , = � UN(X' t) + �u*(x, t) + 3; (D u';J(x, t) - AQ (X) ( u';J(x, t) . -get) - iatg(t) - � T 2 a:g(t))) where A N = A N , D = Dc for (4. 78) and (4.80), and A iv = AN , D = DL for (7.79) ,

where

and

for (7.78) , for (7.79) and (7.80) .

Chapter 4. Spectral . Methods and Pseudospectral Methods

163

For scheme. (4. 77) , A = 0 and uN( l, t) = g(t) . In all calculations, we take q = T N 2 , A = 2Xw(O) . The test function is U ( x, t ) = sin ( 2 1r ( x + t ) ) .

Table 4.12 shows the L2 -errors between U ( x , t ) and the numerical solution of (4.77) . Clearly large time step T is allowed. Table 4.13 shows the L 2 -errors between U ( x, t) and the solution of (4.80) with X = 2. It indicates the advantage of scheme (4.80). Table 4 . 14 shows the L2 -errors between U ( x , and the solution of (4.80) with dif ferent values of X . It is found that the accuracy with X � 1 is very good. While the computation is unstable for X < 1. It confirms the theoretical prediction.

t)

Table 4.12. The L 2 -errors of scheme (4.77) . q=4 q=1 q=8 N = 16 8.2E-4 1 .0E-4 2.9E-6 N = 32 1.5E-5 1 .8E-6 2.8E-8 N = 64 4.2E-7 4.9E-8 7.2E-10 N = 128 1 . 7E-8 1 .9E-9 2.8E- l l Table 4.13 . The £ 2 -e�rors q=8 N = 16 4.7E-4 N = 32 7.4E-6 N = 64 1.2E-7 N = 128 1.8E-9

of scheme 4.80) with X = 2. q=1 q=4 6.0E-5 2.8E-6 9.3E-7 1 . 5E-8 1 .5E-8 2.3E- 1O 2.3E- 10 3.6E- 12

Table 4. 14. The L 2 -errors of scheme (4.80) with q = 1 . X=8 X=2 X=1 x = 0.9 3. 1E-6 2.8E-6 6.5E-6 8.3E-6 N = 16 . N = 32 1.5E-8 1 .5E-8 1 .5E-8 1 .5E-8 N = 64 2.3E-I0 2.3E-I0 2.3E-1O 2.3E-7 N = 128 3.6E- 12 3.6E- 12 3.6E- 12 unstable

' Spectral Meth�ds and Their Applka:.tions

, 164

4.5.

S p ect ral Vanishing Viscosity Methods

In this section, we study the spectral methods for nonlinear conservation laws whose solutions may develop spontaneous jump discontinuities, i.e. , shock waves. Hence weak solutions must be admitted. There are many weak solutions for the same ' problem usually. So we have to single out the unique "physically relevant" solution satisfying the entropy condition. We shall describe the spectral vanishing viscosity methods for nonlinear conservation laws such that the numerical solutions converge to· weak solutions fulfilling the entropy condition. Such solutions are called the entropy solutions. o} and Q = IR", X IRt . Let D be Let IR", = -00 < < oo } , IRt+ = a set in IR2 . If D l is a compact set in D, then we say that Dl cc D. Moreover, if is defined on D almost everywhere, then we say that is defined on D, a . e . . If a function is defined on D, a . e . , and for any measurable subset Dl CC D, E Lq (Dd , 1 :5 q :5 00, then it is denoted by E Lloc(D) . We can define the spaces Hloc(D) and l¥t�: (D) similarly. Furthermore, if VI is a sequence such that

{x I

v(x, t)

v

lim

1-+00

{t I t >

x

v(x, t)

v(x, t)

v

j'iD( VI ( X , t)w(x, t) dxdt = j'iD( v(x, t)w(x, t) dxdt,

V

w E Ll (D),

v

then we say that VI converges to in LOO(D) weak-star. Besides, let V( Q) be the space involving all infinitely differentiable functions vanishing for sufficiently large

I x l + t.

We now discuss the nonlinear conservation laws. Let = in C l (IR) , and Consider the problem

J( z )

be a convex function

J' ( z ) ozJ(z). { OtU(x, t) ozJ (U(x, t)) = 0, (x, t) E Q , (4.96) x E IR",. U(x, O) = Uo(x), A weak solution of (4.96) is a bounded measurable function U(x, t) such that Jh (U(x, t)Otw(x, t) + J(u(x, t))o",w(x, t)) dxdt + ( Uo(x)w(x, O) dx = O, V W E V(Q) . (4.97) iRs Next let E( z ) be certain smooth function in IR\ and F( z ) = J E'( z )J'( z ) dz. +

Chapter

4.

SpefJtra,J. Methods and Pseudospectral Methods

If for any smooth solution of (4.96) , there holds

OtE(U(x, t)) + o:r: F (U(x, t)) = 0, (x, t)

165 E

Q,

then we say that E and F are a pair of entropy and entropy flux for (4.96). If for any strictly convex entropy E ( z) , a weak solution t) satisfies

U(x, (4.98) OtE(U( x , t ) ) + o:r:F(U(x , t)) ::; 0, (x, t ) Q, then we say that U(x , t) is an admissible solution. It can be verified that if both U(x, t) and Vex , t) are admissible solutions of (4.96), and lim U(x, t) = t--+o+ lim Vex , t ) , in Ltoc(Q), t--+o+ then U (x, t) V (x, t) in Q , a.e . E

.

=

There are fruitful results for the existence of weak solutions, see Lax ( 1972). Tartar (1975) used compensated compactness to study (4.96). The key point is to explore the conditions with which a sequence tends to a weak solution as it tends to in L�c(Q) weak-star, or in other words, tends to For this purpose, the following lemma plays an important role, see Tartar ( 1975) . Let M(D) be a space of Borel measures.

{ VI } f( vd

V

f(v).

C1 (lR.1 ). If VI v weak-star, and for all convex function E(z), otE(vI) + o:r: F (v I ) are in a

Lemma 4.17. L et D be a bounded open set in lR? and f e z ) E

�

in LOO(D) compact set of H- 1 (D) or a bounded set of M(D), then

f(vd -- f(v), f'(vd -- f'(v ) ,

in LOO (D) weak-star, in L9(D) strongly for ali I ::; q <

00 .

Moreover, if there is n o interval on which f is affine, then

VI

-- v ,

i n L9 (D) strongly for ali I ::; q <

00 .

We can take LI (D) as a space of measures. Other two commonly used results from compensated compactness are due to Murat and Tartar, see Tartar ( 1975) . Lemma 4 . 1 8 . L et D be a bounded open set in lR?, and the sequence

{ VI }

be in a bounded set of W-I,OO (D) . If is also in a union of a compact set of H-I (D) and a bounded set of M(D), then it belongs to a compact set of Hi;.! (D) .

{ VI }

Spectral Methods

166

Let v be a function vector and v curl v = 0I V(2) - 02 V(I).

=

and

(V(I), V(2) ) . Set div v

Lemma 4 . 1 9 . Let D be a bounded open set in IR2 . [f vI

=

Their Applications

01 V(I) + 02 V(2) and

-+

-+

V , WI W in L 2 (D) x 2 l L (D) weakly, and div v ; -+ div v , curi wi -+ curlw in H- (D) x H- l (D) strongly, then VI . WI -+ V . W in 'D' (D) .

We first consider periodic problems. Let A = (0, 211") and Q = A x IRt . Denote . by 'DI'( Q) the space involving all infinitely differentiable functions with the period 211" for the variable vanishing for sufficiently large Let D = A x { t 0 < < < Define the spaces L';' (Q) , L;,J.oc (Q) and H;,loc (Q) similarly. The periodic nonlinear conservation laws are of the form

x,

{

I

t.

tl t t 2 }.,

OtU(X , t) + o",j (U(x, t)) = 0, (x, t) E Q, (4. 99) (x, t) E Q, U(x, t) = U(x + 211" , t), x E A. U(x, O) = Uo(x), The weak solutions of ( 4 . 9 9 ) are bounded measurable functions such that J10 (U(x, t)Otw(x, t) + f(U(x, t))o", w (x, t) ) dxdH l Uo(x)w(x, 0) = 0, V w E 'DI'(Q).

( 4. 100) The admissible solutions are also identified by (4.98 ) . In this case, Lemmas 4.17 -4.19 are also valid with slight modifications. For instance, LI'(D) and Wr, q (D) are replaced by L� (D) and W; , q (D) , etc . . We are going t o construct the Fourier spectral scheme. Let b e the L 2 _ orthogonal projection upon as defined in Section 2.2. The standard Fourier spec tral scheme is to find for t � 0 such that

Let

[

{

VN t) UN(X, E VN OtUN(X, t) + o",PNf (UN(X, t)) UN(X, O) = PNUO (x),

PN

=

0,

(x, t � E Q, x E A.

be the identity operator, then this scheme stands for

Let us multiply the above equation by

UN(X, t) . We find after integration that (4. 101 )

Chapter

4.

Spectral Methods and Pseudospectral Methods

167

UN

U,

This yields the existence of a weak solution which is the weak limit of in T; for any > Unfortunately it does not solve our problem. In fact, if weakly. and hence should tend to U is a weak solution, then converges to U If in addition = weakly and so then tends to in T; strongly. Therefore by (4. 101 ) , is also conserved in time. However the inequality (4.98) is valid strictly at the locations of shock waves. It leads to a contradiction. Hence the standard Fourier spectral methods do not work for the problems with shock waves. This fact motivates us to modify the above scheme. Tadmor (1989) proposed the following Fourier spectral vanishing viscosity scheme

L2(0, L2)

t O. PNf(UN) , f( z ) !Z2 , uh L2 (0, L2 )

-+

-+

f (UN) U2 I U(t ) 1 2

M M( ) M ) -+ UN.

where c = c(N) 0 as N 00 , and = N < N, ( N We shall establish some a priori estimates for Let c ( N) '" cN-a ,

We multiply (4. 102) by

UN

M( N ) '" cN(J ,

00

feU)

as N

0 < 2f3 < Q' :::; 1 .

-+

(4.102) 00 .

(4.103)

UN and integrate over A. Since - (PNf (UN(X, t)) , OrUN(t) ) - (f (UN(X, t)) , OxUN(t)) = 0,

we obtain and so

Let C o be a positive constant depending only on

I U(O) 11 2 . Then for any fixed T > 0, (4.104)

< <

11 (1 - PM)OrUN(t) 1I 2 + II PMor UN (t) 1I 2 1 (1 - PM)OxUN(t) 1 2 + M2 1 I uN(t)1 1 2 1 (1 - PM)OxUN(t)1 1 2 + cN2(J ll uN(t) II 2 .

168

Spectral Methods and 'Their Applicatic;ID s

(4.103) and (4.104), we assert that ( 4.105) 1 8",UN l I i2 (0,T',L2) ::;: ceeo . Next, multiplying ( 4.102) by 8tUN and integrating the result over A, we get that 11 8tUN (t) 1 I 2 + 2 8tl l (1 - PM)8",UN(t) 1 2 < 21 1 8tUN(t) 1 I 2 + 21 1 8",PNf (UN(t)) I 2 ::;: 21 1 1 8tUN(t) 1 I 2 + 21 "f' (UN(t)) 8",UN(t)1 1 2 . Then the integration of the above formula from 0 to t yields 1 1 8tUN lli' (o,t;£2) + e l ( 1 - PM )8"'UN(t) 1 2 ::;: e ll (1 - PM)8",UN( 0 ) 1 2 + 1 f'(UN)l I ioo (o,t;A) l I l 8",UN(S) 11 2 ds. (4.106) By using

,

e

Now assume that there exists a positive constant such that

(4. 107)

(4.105 )-(4 . 107 ) lead to 1 8tUN l i2 (O,T;£2) ::;: CA ( � + 1 ) . ( 4.108) We are going to derive the convergence of scheme (4.102) by a compensated com pactness argument. Let D = A x (0, T) and (V, W)D = loT L v(x,t)w(x, t) dxdt, I v ll D = (v, v) b . Then by (4.102), for any W HJ (O , Tj H� ) , and denote by CA a positive constant depending on A. Then

E

where

G1 (W) = (8",( 1 - PN)f(UN ), W) D ' G2 (w) e (8",(I - PM)8",UN, W) D ' By Theorem 2.3, ( 4.103 ) and ( 4.105 ), =

Chapter

4.

Spec�ral Methods and Pseudospectral Methods

169

On the oth�r hand, (4. 104) leads to

I G2 (w)1 ::; e 1« 1 - PM)8",UN, 8",w) v l ::; e l ( I - PM)8", uN l v I l 8",w l v < Co veIl 8",w l v ::; CoN - 'r 1 8",wl l v . Therefore (4. 109)

Since a 1, the right-hand side of the above inequality tends to zero as -+ 00 . It implies that belongs to a compact subset of Hl:� (Q). Next, let be any strictly convex entropy and be the corresponding entropy flux. We repeat the above procedure with Then we deduce from (4. 109) that for any E HJ ( O , T; H:),

N 8tUN + 8:t!(UN) F( z ) E( z ) E'(UN)W. w 1 ( 8t E(UN) + 8",F(UN), w) v l 1 (8tUN + 8:t!(UN), E'(UN)W) v l ::; Co (cA N'r - 1 + N- 'r) 11 8",(E'(uN)w) l I v ::; Co (cA N 'r - 1 + N- 'r ) ( II E"(uN)8:tuN ll v ll w Il Loo (v) + II E'(uN)8",w l v ) . ::;

By using (4. 103) and (4. 105) again, we find that

where

Bl CoCA (cA NOI- l + Co) I wIl LOO (v) , B2 CoCA (cA N 'r -1 + N- 'r) 1 8",wl l v . Thus 8tE( UN) +8",F( UN) is Since ::; 1 , Bl is bounded. Clearly B2 0 as composed of two terms, each of them is in a compact subset of Hl:� (Q) or a bounded =

=

a

-+

N

-+

00.

subset of Lloc (Q) . By Lemma 4.18, it is in a compact subset of Hl:� (Q) . Furthermore Lemma 4.17 ensures that converges to U* , a weak solution of (4.99) in Lfoc (Q) strongly for al1 1 q < 00 . Now, we check the entropy condition. Let a < 1 . We first have that

::;

UN

Spectral Methods ana' Their Applications

170 where

G3 (w) = - ((1 - PN)f(uN), 8:r: (E'(UN)W))D ' G4 (w)' = -c ((I - PM)8:r:UN, 8:r: ( E'(UN)W)) D '

By Theorem 2.3, (4. 103) and (4. 105) , we get that

I G3 (w) 1 ::; �1 I 8:r: UN I D I 8:r:(E'(UN)W ) I D ::; �� 1 I 8:r:(E' (UN)W) I I D ::; CocA N t -1 ( 1 I 8",UNIIDllwIlLOO (D) + I E'(UN)I I Loo (D) 11 8",w Il D ) ' Thus G3 � 0, as N � Furthermore, we have 00 .

with

G�l ) (W) = -c (8",UN, E"(UN)w8",UN) D ' Gi2) (w) = -c (8",UN, E'(UN)8",w) D 1 Gi3) (w) = c (PM8",UN, 8", (E' (UN)W)) D ' By taking w � 0 and using the convexity of E, we know that G�l ) (W) ::; O. By (4. 105) , I G�2) (w) 1 ::; cAyg ll 8",w ll D -- 0, as N -Because PMUN E IP M , we have from Theorem 2.2 and (4. 105) that I Gi3) (w) I ::; c Il8",PMUN II D ( 1 I E"(UN)w8",UNI I D + II E'(UN)8",w I D) ::; cAcM (� I W I LOO(D) + I 8",w l i D) < cA NfJ - t ( l I w Il L OO(D) + 118",w Il D ) . Since 2f3 < I G�3) (w) 1 � 0 as N � Therefore U* fulfills (4.98) in the sense of 00 .

a,

00.

distribution. The above statements lead to the following conclusion.

(.{..103) and (.(..107) hold. Then for 1 ::; q < the solution of (4.102) tends to a weak solution of (4.99) in Lioc ( Q ) strongly. If in addition < 1 ,

Theorem 4.19. Let

then i t is the unique entropy solution.

00,

a

Chapter

4.

171

Spectral Methods and Pseudospectral Methods

For the .. existence of weak solution of (4.99) with j(z) = � , we only need to check that both OtUN + �O",uh and �OtUh + l o",u� belong to a union of a compact be the weak subset of Hj;,! (Q) and a bounded subset of LloAQ). In this case, let limits of Then by Lemma 4.19,

Z2

U;

uiv .

(4. 1 10)

Moreover

(UN - U; )4

--+

+

U; - 4U; U; + 6U; (U; ) 2 - 4 (U;)4 (U; )4 , weakly,

from which and (4.1 10),

( U; - (U;) 2) 2 � 0, weakly. So U; = (U;) 2 . Consequently, UN converges to U; in Lloc (Q) strongly and so it is a weak solution of (4.99) with j(z) = � z 2., see Tadmor (1989) . (UN

-

U;) 4

--+

-

3

In actual computations, we can take the vanishing viscosity term as cO", (qNO",UN) where

qN(X, t) = L Mt)eil'" O:5I I I:5 N

and

c

= e ( N) -+ 0, M = M (N) -+ 00 as N -+ 00. For instance, M ql = A

{ o, 1,

I l l � M, I l l > M. �Z2 and Uo(x)

'"

2 Nt and (4. 1 1 1 )

Tadmor (1990) solved (4.99) with j(z) = = sin x by the Fourier spectral method and the Fourier vanishing viscosity method with = 8, N = 32, e = 0.31 and (4. 1 1 1 ) . They are integrated in time up to t = 1 .5, using the fourth order Runge-Kutta procedure. In the figures, the real lines present the same exact solution of (4.99), while the dotted curves present the numerical ones. Figure 4.1 shows the results of Fourier spectral method, and Figure 4.2 shows the results of Fourier vanishing viscosity method. Obviously the latter provides much better numerical approximation to the shock waves. In Figure 4.3, the improved results are obtained by using COO-viscosity coefficients til connecting the wave numbers in the inviscid region 1 < ( N) and the highest wave numbers N. This technique prevents the propagation of the Gibbs phenomenon into the whole computational domain. Figure 4.4 shows the numerical solutions of Fourier vanishing viscosity scheme after it is post-processed by the spectrally accurate smoothing procedure discussed in Gottlieb and Tadmor ( 1985).

M

III

f'V

M

Spectral MetHods and Their Applications

172

1t �7 .. " " : " .L

1

. . '

. • • • •

o

1 " ·-.r

"

. '

.

,

I '

• • • •

Figure 4. 1 .

Figure 4.2.

Figure 4.3.

• •

•

0 ' , , I '

Chapter

4.

Spectral Methods and Pseudospectral Methods ..L

173

T

.�

·r + .. L .1

t

Figure 4.4.

The total variations of Fourier viscosity schemes and their stabilities and conver gences are discussed in Maday and Tadmor ( 1989), and Tadmor (1991 ,1993a) . In particular, Chen, Du and Tadmor ( 1993) considered multi-dimensional scalar con servation laws, and proved that for certain Fourier vanishing viscosity schemes, the temporal LOO-norms of the solutions depend on the Loo-norms of the initial data continuously. Thus, the assumption I I UN I I LOO(O,T;LOO) :5 A is reasonable. The first as sumption in (4. 107) can also be fulfilled once the initial data satisfies certain Lipschitz condition. Chen, Du and Tadmor (1993) also showed that for fixed t > 0, the norm I I U( t) - uN ( t ) I I L1 (A) is bounded by Ve(N). Now, let A = ( - 1 , 1 ) , Q = A x lRt+ and consider the initial-boundary value problem of (4.96) with appropriate data g(t) E Hl� AlRt+ ) , prescribed at x = ± l . Let PN and IN be the L 2 -orthogonal projection upon IPN and the interpolation associated with Legendre-Gauss-Lobatto interpolation points x U) and weights Let v E L 2 ( A ) and v/ be the coefficients of its Legendre expansion. A viscosity operator q is defined as N

wei).

q v (x) L q/v/L/(x) /=0 =

with M = M( N ) and

{

q/ 0, M4 q'/ -> I - [4 ' =

for [ :5 M,

N. The free pair (e , M) will be chosen later, such for M < [ :5

Let e = e(N) be the small parameter. that e --+ 0, M --+ 00 as N --+ 00 . On the other hand, let

B (UN (t))

=

(A( t ) ( l - x )

+ j.t(t) ( l + x)) OxLN ( X ) .

Spectral Methods and Their Appiic.ations

174

The pair (J., ,, ) is chosen to match the inflow boundary data prescribed at = 1 whenever f ' (UN ( 1 , t)) < 0, and 'll.t = - 1 whenever f ' (UN( - 1 , t)) > 0. A Legendre vanishing viscosity penalty scheme for the initial-boundary value problem of (4.96) was given by Maday, Kaber and Tadmor (1993). It is to find UN(X, t) E IPN for t � ° such that ( Ot U N(t) + o",!N f (UN(t)) , ¢» N + c (q O", U N(t) , O"'¢» N = E IP N, t > 0, (UN(t)) , ¢» N ' (4. 1 12) U N (X, t) = at the inflow boundary points, t > 0, UN(O, t) = INUo(x), xE

{

x

x

(B

V ¢>

g(t),

A.

We shall establish some a priori estimates for U N. First of all, let that for any E IP N , N

¢>

u be a filter such

u ¢>(x) L, 1=0 ul �,LI (x). =

Also let

I v ll � = (uv , v).

Lemma 4.20. For

any

¢>

E

IPN,

N

2 l I o", ¢>I I ! ::; cN2 L, 1=1 IUdl¢>1 I •

.

Pro of Let

J', N

=

{j I I

+ 1 ::; j ::; N, 1 + j odd}.

Then by (2.51), N -I '(1 ) '(1) = (2 o", ¢> (x) = L, 1=0 ¢>, L,(x) , ¢>, 1 + 1) L, ¢>j . •

By (2.48),

2 � ul l �P) 1 2 I 1 LdI 2 � 171 (2 1 + 1) 2 Ij � �j 1 1 Ld 1 2 =

(uo:c¢> , o:c¢»

< < It is noted that when

N

1 ) 17 + 1 ) . 2: l�j I 2 I 1 LjI 1 2 . , 2: (2 1 1 2: I L 'I 1 2 1=0 (N N N 2 I LjI 1 2 L, IUI (N2 - 12) ::; cN2 L, ludl ¢>1I 2 . CL, �j I I 1=0 1=0 j=O

2

N -I

171

J EJ1 , N

3 E JI , N

3

• ==

1 , Lemma 4.20 is reduced to Theorem 2.8.

Chapter 4. Spectral Methods and Pseudospectral Methods Lemma 4 . 2. 1 . For

any

175

¢> E IPN }

= 1 - ii,. Then 0-, = 1 for I � M and 0-, � 'f.' for M < 1 � N. Since 2 = ., 8., 118 ¢>1 1 11 ¢>1 I � + 118., ¢>1 I � , it remains to bound 1 1 8.,¢>1 I � . To this end, let J = log2 � and decompose the function ¢> as M 2i M L ¢>(x) = L + = (x) ,(x) , L ¢>j(x), ¢>j L �,L,(x). � '=0 j=l '>2i-' M By Lemma 4.20, Proof. Let 0-,

J

Therefore

118., ¢>1 I �

� �

2 /1 8., � �'L{ + 2J � 118,, ¢>jll � 2 2CM4 1 � �'L' /l + 4cJM4 � II¢>jI l 2 � cM4 In N I ¢>1 I 2 .

•

To simplify the presentation, we shall deal with the prototype case where one boundary, say = is an inflow boundary, while is an outflow one. Then

x=1 B (UN(t)) = A(t)( I - x)8",LN(x). As we know, x(j ) are the zeros of 8 LN ( x ) 1 � j � N 1 and w ( O) = ;: N( + 1 ) ' N 8",LN( -1 ) � (_I) +1 N(N + 1 ) . Thus (B (UN(t)) , v)N 2( _ 1 ) N+ 1 A (t)V( -1, t). (4. 1 13) Let ¢> 1 in (4. 1 12). Since INf (uN(x,t)) E IPN, we deduce from (2. 37) that x -1,

"

,

-

=

=

==

Consequently,

(4.1 14)

Spectral Methods ' �d Their Applications

176 Further, set and assume that for all N,

' t/J(t) = l >.(s) ds, (4. 1 15)

Then for

t S; T, 1 t/J (t) 1 <- y1Ml2 l uN(t)1 1 + y1Mll2 uN(O)1 I + t max 1 ! (z) l· .

��

(4. 1 16)

We now derive a priori estimates. Let Co be a positive constant depending on c� be a positive constant depending on A, and

I Wo l ,

Assume that

( 4. 1 1 7)

UN in (4. 1 12). In view of (2.37) , (2.60) , (2.63) and Lemma 4.21, we find that � 8t ll uN(t) l � + F*(uN(l, t)) - F* (UN( - 1 , t)) N + c( q8",UN(t), 8",UN(t)) + 2( _l) >'(t)UN( - 1 , t) (8",!(UN(t)), UN(t)) - (8",IN!(UN(t)), UN(t)) N = - «I - IN)!(UN(t)), 8",UN(t)) S; ;1I 8",!(uN(t)) l i 1 8",uN(t)1 I S; � (1 I 8",UN(t) ll� + M4 InN l uN(t) I �) . Thus for any t S; T, Set if> =

==

By (4. 116),

il >.(s)g(s) ds i

i g (t)t/J(t) - g(O)t/J(O) - l 8.g(s)t/J(s) dsi < II g (t) II (1 I uN(t) 1 I + Co + CA) + l I 88g(s) 1 ( l I UN( S)1 I

+

Co + CA)

ds .

Chapter

4.

Spectral Methods and Pseudospectral Methods

177

Putting the above two estimates together, we have from (4. 1 1 7) that By using Lemma 4.21 and (4. 1 1 7) again,

(4.118 ) Next, we set rP OtUN in ( 4 . 1 1 2 ). By Maday (1991), I IN VI I � ci v i l for any v E Hl ( A ). So by (2.37) , (2.60) and (4. 114 ), I l otUN(t)1 1 2 + � Ot I l OXUN(t) l � � cA l oxUN(t) 1 2 + � I OtUN(t)1 1 2 + CA I Otg(t) 1 2 + CA . Temporal integration of the above inequality followed by ( 4.118 ) , implies I OtUN lllfoc (Q) � CA (Co + Il oxuN lllfoc (Q) + Il g l I �l�c (lRt) + 1 ) � c; (Co + Il g ll �l�c (lRt) + 1 ) . (4 . 1 1 9) We now prove the existence of weak solutions. Let D = A (0, T) and define (" ' )D and I · l i D as before. Let (V, W)DN, = Jo( T (V(t), W(t))N dt. For any W E HJ (D) , define WN(X, t) = 1: PN-lOyW(y, t) dy . Clearly WN E lP N and wN(-1,t) = O. Let 1 - q and rP = WN in ( 4 . 11 2 ) . We have from (2. 37) and (4.113) that s COtUN, Ox!(UN), W )D j=2:l Gj (w), V W E HJ C D ), =

x

u =

=

where

G1CW) G2 (w) G3 (w) G4 (w) Gs(w)

= = = = =

COtUN + Ox!(UN), W - WN)D, (Ox!(UN) - oxIN!CuN), wN)D, COtUN + oxIN!CuN), WN)D - COtUN + oxIN!CuN), wN)D,N , E( UOxUN, OxWN)D , - E(OxUN, OxWN)D.

Spectral Method; and Their Applications

1 78

By (4. 118) and (4. 1 19),

CA ll w - wN Ii D :s: CA "a"WID . I G1 (W) 1 :s: yg ygN

According to (2.63) and (4. 1 18),

CA ll a"WN Ii D . IG2 (w) 1 = 1 (f(UN) - IN!(uN), a"wN)DI :S: NC "a"!(UN)IDla,,WNI i D :s: ygN

By virtue of (2.64) and (4. 1 19),

We see from the proof of Lemma 4.21 that (4.120)

Clearly, The above statements with (4.1 17) imply that

at UN a,,!(UN)

It means that belongs to a compact subset of HI�� (Q). Furthermore + let be any strictly convex entropy and be the corresponding entropy flux. We have 5 (4. 121) = 2: +

E

F

(atE(UN) a"F(UN)' W)D j=l Gj (E' (UN)W) .

The previous statements tell us that

It Gj (E'(UN)W)D I

CA ( N� + M2 v'ln N + Vc") ( lI a,"UN I D ll w I LOO (D) + CA lla" w II D ) :s: CA ( I W II L OO ( D) + ( � + c:M2 v'ln N + Vc") lI a" W I D ) . N Thus the term at E(UN) + a"F(UN) can be written as a sum of two term� which :s:

c:

belong to a compact subset of HI�� (Q) and a bounded set of L�oc(Q). Then by Lemma 4.18, it belongs to a compact set of Hi;'� (Q) . Further, Lemma 4. 1 7 ensures

Chapter 4. Spectral Methods and Pseudospectral Methods

$

1 79

that for 1 Cf < 00 , UN converges to U·; a weak solution, in £foc ( Q) strongly. Finally, we check the entropy condition. Let a < 1 . It is easy to see that 3

L I G; (E'(UN)w) 1

<

;=1

$

leN 118", (E'(UN) W )NI ID

leN (1I8",uN II D llwIILoo(D) + II UN II LOO(D) II 8",w l i D )

$ CA (NOI - 1 I1 w Il LOO(D) + N i -1 118",w IlD)

--+

O.

--+

O.

By (4. 1 17), (4. 1 18) and (4: 120),

I G4 (E'(UN )W) I $ cAeM 2 v'ln N 118", (E'(UN )W) li D cAvfcM2 v'ln N O l wl l D + 118", w Il D) Moreover, for w

;:::

$

0,

G5 (E'(UN) W )

-e ( 8",UN, PN - 1 8", ( E '(UN) W ) ) D -e ( 8",uN, E"(uN)w 8", u N) D - e ( 8", uN, E' ( u N) 8",w) D $ - e ( 8",UN , E'(UN) 8",w) D cAvfc ll 8",w llD --+ O .

=

$

Thus by (4. 121 ) , U· satisfies (4.98) in the sense of distribution, and so it is the unique entropy solution. Theorem 4.20. Let (4 . 1 1 5) and (4. 1 1 7) hold. Then the solution of {4 . 1 12} tends to

a weak solution of the initial-boundary value problem of {4. 96} in Lfoc ( Q) strongly, 1 q < 00 . If in addition a < 1 , then it is the unique entropy solution.

$

We now present some numerical results. We consider the Hopf equation with Uo ( x ) = 1 + sin 1l'x and U ( I , t ) = 1. For time discretization, we use the second order Adams-Bashforth method with the step size T = 1 0 - 5 In all calculations, N = 128. The standard Legendre pseudospectral scheme corresponds to e = 0, while the Legendre vanishing viscosity scheme is implemented with e '" N-1 , M '" 5v'N and N) 2 q, = exp M< N.

-

!

.

•

( (1 - ) (1 - M) 2 '

1$

Figure 4.5 shows the numerical results of Legendre pseudospectral scheme before post processing, and Figure 4.6 shows the corresponding ones after post-processing. Figure 4.7 presents the results of Legendre pseudospectral vanishing viscosity scheme before

180

' Spectral Method� an d Their Applic� tions

post-processing. Figure 4 . 8 is for tile corresponding results after post-processing. In all figures, the real lines present the same exact solution of Hopf equation, and the dotted curves correspond to the numerical solutions. Obviously the Legendre pseudospectral vanishing viscosity method provides much better results. I.'

.

I.'

1.2

,..

.,

'.

, ,

:f.

." .,

·1

Figure 4.5.

..

., u

1.0

1.2

'

I

.

.

,.. ,.. ,..

·1

..,

,.,

Figure 4.6. I.'

1.0

I .'

I�

,.• o�.L , --. , -� "'.,,.-----::-----.,:7

Figure 4 . 7 .

Cha.pter

4.

Spectral Methods and Pseudospectral Methods

181

I,ci

Figure 4.8. Recently, Ma. (1996a.) developed a. Chebyshev-Legendre pseudospectra.! vanishing viscosity method for nonlinea.r conserva.tion la.ws which is a. combina.tion of Chebyshev Legendre pena.lty method in Section 4.4 and vanishing viscosity method in this section. He a.!so developed a. new method in which the viscosity term is ta.ken to be cPN ( ( l x2)- � D2 (q'U N ) ) , D = v'1 - :& 2 8", . It wa.s proved tha.t the solution of both schemes converge to the unique entropy solution provided tha.t certain rea.sona.ble conditions are fulfilled. Ma. (1996b) a.!so served a. Chebyshev-Legendre super viscosity method in which the viscosity term is of the form ( l ) + 1 c PN ( 1 - x2) - � D2''UN , s :::: 1 . Its solution a.!so converges to the unique entropy solution. The super viscosity technique for the Fourier spectra.! methods can be found in Ta.dmor (1993b). -

4.6.

,

(

)

S p ectral Approximat ions of Isolated S olut ions

In the previous sections, we studied various spectra.l methods for nonlinear revolu tiona.ry problems. Genera.lly spea.king, a.!l of those methods a.re a.pplica.ble to the corresponding stea.dy -problems. But for nonlinear elliptic equa.tions, the situa.tions are more complica.ted. The main feature is tha.t they ma.y possess severa.! solutions. In this section, we ta.ke the stea.dy Burgers' equa.tion as a.n example to describe the spectra.! a.pproxima.tions of isola.ted solutions. Let 11. = ( - 1 , 1) and II > O. J(x) is a. given function. Consider the problem

{

U8", U - 1-'8;, U = J, U = 0,

x E A, x = -I, 1.

( 4 . 122)

· Spectral. Method� and Their Applications

182

Set

a(v, w) = L o",v(x)o",w(x) dx, b(v, w, z) = ! L v(x)w(x)o",z dx,

v

v, w E H1 (A), V v, w E L4(A) , z E H1 (A). The solution of (4.122) is defined as a function U E HHA), satisfying ( 4.123) p,a(U, v) - b(U, U, v) = (f, v), V v E HJ (A). If (f, v) is a linear continuous functional in HJ (A), then by Fixed Point Theorem, . (4. 123) has at least one solution. Letting v = U in (4. 123) and f, v) l , sup 1( IlI f lll = 'EH�(A) Iv l1

we get that Wit :::; p,- l ll 1f lll . Assume that (4.123) has the solutions U and U· , and f) = U· - U . Then p,a (f), v) - b (f), u + f), v) - b ( U, f) , v) = 0, V v E HJ (A) . Putting v = f), we deduce that .,0.

We know that

II v I l L< :::; v'2 l v l 1 for any v E HHA) , and thus

Hence (4.123) has only one solution as long as 1 11 / 11 1 < If (4.123) has a unique solution, then it is easy to solve it by various spectral approximations as presented in Sections 2.4 and 2.5. Here we are more interested in some problems with multi-solutions. In this case, it is better to consider (4. 123) in an abstract framework. To do this, we need some preparations. For convenience, we shall adopt a unified notation as in Section 2.3, i.e. , == 1 for the Legendre · 1 approximation or (1 for the Chebyshev approximation.

p,2 .

w(x) = - x2) - '

Lemma 4.22. The imbedding

HJ,w CA)

w(x)

C

L�(A) is compact.

w(x) 1, the conclusion comes from the well-known imbedding theory. Now, let w(x) = (1 - x2 ) - ! . It suffices to verify that v E L�(A) vw! E L2(A) is Proof. For

==

-+

Chapter 4. Spectral Methods and Pseudospectral Methods

183

an isomorppism, and that E HJ,,,, (A) --+ W E HJ (A) , is a continuous mapping. The first fact is trivial. We now prove the second one. Let E HJ,,,, (A). We have • 1 1 1 = +

V�

v

v

a" (VW' ) a"vw' 2 xvw" Moreover by Lerp.ma 2.7, vw� E L2 (A).

a:rvw�

a" (vw� )

Clearly E L2 (A). E Hence l H (A). Next, the set V(A) is dense in H� (A) and so for any E HJ,,,, ( A), there exists a sequence E V(A), I = 1 , 2, . . , such that --+ in H�(A). Then is the limit in Hl (A) of which vanishes at x = - 1 , 1 . Therefore E HJ (A). It implies _ the continuity of the mapping E HJ,,,, ( A) --+ E HJ (A).

ZI

ZI v vw�

.

ZIW�

J.!

v

v

vw�

VW�

C

( 1 - x2 ) - ' . Firstly, let r = 1 , J.! = 0 , be a bounded sequence of H� (A). For each VI , there is E HJ,,,, ( A) and such that = + and

Lemma 4.23. For any 0 :$

< r, the imbedding H�(A)

Proof. It suffices to consider the case with W

and VI WI IP 1 E

=

H/:, (A) is compact.

1

v?

VI v ? W I

IP 1 ,

v?,m v?,m WI,m

By Lemma 4.22 and the equivalence of norms on we can extract subsequences and such that , --+ in L!(A) and --+ in LOO (A). Thus --+ + + in L!(A), and this completes the proof for r = , I and J.! = O. The general _ result follows from an inductive procedure and the space interpolation.

WI ,m VO W

WI ,m W

VI m v

Lemma 4.24. For any r > ! , H�(A)

c

LOO(A), and for any r

;:::

1 , H�(A) is an

Proof. We only have to verify the conclusions with = ( 1 - x2 ) -' . Since r 1 , H� (A) C H (A) C LOO (A) for r > ! . We next prove the second result. It suffices to verify that for r ;::: 1 , algebra.

w(x)

Indeed, for any integer r

lI a; (v w ) I ",

:$

r -l

;:::

1

w(x) ;:::

1,

I: C�IIVIl I'+ 1,w I l a; - l' w I L + l I a;v ll ", I l w l h.", :$ cllvll r,w I l W l r,w .

1' =0

Spectral MetHods' 'and Their Appli"cations

184

Then the space interpolation extends this result to any real numbers r

;,:::

1.

•

Now, let

aw(v, w) = L o",v(x)o",(w(x)w(x)) dx and define the linear mapping A (HJ.w ( A ) ) ' HJ. w (A) by aO/ ( Av , w) = (v, w), w E HJ.w ( A), ( 4.124) where (- , . ) denotes the duality pairing between (HJ.w ( A ) ) ' and HJ.O/ (A) . Lemma 4 . 2 5 . For any ;,::: 0, A is a linear, bounded and continuous mapping from H� (A) into H�+2 ( A ) n HJ.O/ ( A). For any - 1 :5 r < 0, A is continuous from (Ho.� ( A) ) ' into H�+ 2 ( A ) n HJ.O/ ( A ). :

-+

'if

r

Proof. If r is a non-negative integer, then the integration by parts in (4. 124) yields

- o�(Av) = v

I Av l r+ 2 .w cllvll r.O/ .

For real number r > 0, the same result follows from the and so :5 space interpolation. Finally for -1 :5 r < 0, the result comes as in Corollary 4.5.2 of Bergh and Lofstrom (1976) . •

A A

HJ.w ( A ) . Moreover, Lemmas L�(A) into HJ.O/ ( A ). Let HJ.w ( A) L!(A) G(A, v) = A (vo", v - J) . ( 4.125)

Now by Lemma 2.8, is a bounded mapping on 4.22 and 4:25 imply that is a compact mapping from -+ by A = .; and define the mapping G : IRI x

Clearly G is infinitely differentiable. Let DG be the Frechet derivative of G. Then D 2 G is bounded on any bounded subset of IRI x Moreover, since C we get

HJ.O/ (A).

LOO(A),

HJ.w ( A)

(4. 126)

HJ.w ( A ) HJ.O/ (A), by F(A, v) = v + AG(A, v).

Finally, define the mapping F : IRI

x

-+

Then problem (4. 123) can be written equivalently as follows: find that F A, U = 0.

( )

v E HJ.w (A ) such (4. 127)

Chapter 4. Spectral Methods and Pseudospectral Methods

185

Let > o}. Using the techniques in be any compact interval in = Lions (1969) , we can prove that there exists a branch of isolated solutions of (4. 123) , denoted by According to the description in Section 3.2, it means that there exists a positive constant such that

A*

IR! {A I A

(A, U(A)).

6, ( 4.128) 1 (1 + AD G(A, U(A)))v l kw � 6 1 v l h.w, V A A*, v H�,w ( A ). We begin to approximate the isolated solutions of (4. 127) . Let PN be the L! orthogonal projection and Pi/o be the HJ,w-orthogonal projection, i.e. , ( 4.129) aw (v - pi/ov , E HJ,w ( A ) . " Set AN = Pi/°A. Thanks to (4. 124) and (4. 129) , for any v E ( HJ,w ( A ) ) aw(ANv , 1P
E

=

=

=

Define

E

=

FN(A, v) v + ANG(A, v).

Maday and Quarteroni (1981) proposed a spectral approximation to (4. 127) . It is to find E 1P<;t. such that 4.130)

UN

(

We can use the Newton procedure or the modified Newton procedure to solve (4. 130). For instance, let u�) be the m-th iteration of and

UN,

We are going to analyze the convergence. To do this, we introduce the result of Brezzi, Rappaz and Raviart (1980) . Let and W be two Banach spaces. is a compact interval in is a C 1 mapping from X W into and is a linear continuous mapping from into W . Suppose that for all E and E W, the operator = and consider E C(W, W ) is compact. Let the problem = o. (4. 131)

IR1 . Q

ADQ (A, v)

B

B

A* B, A A A* v IF (A, v) V+AQ (A, v)

IF (A, U)

WN be the finite dimensional subspaces of W, N B, WN) and IFN(A, V) v + AN Q (A,V) for A E A*, v E WN .

Next let C(

=

=

Lemma 4 . 26. Assume that

0, 1 , . . . . Let

A*

AN E

Spectral Methods and Their AppHcations

186

em+1

Q

W

(i) is a mapping from A* x any bounded subset of A * x m � 1; (ii)

(iii)

(iv)

W,

E

nN C (W, WN ) E

E

B, and DmHQ

and

AN C ( W', WN) {(A, U(A)), A

into

Nlim -+oo I nNV - v ll w

=

0,

V

is bounded over

E

( 4.132)

OJ

(4.133)

v Wj

and

Nlim -+oo I A - AN l c(B '

W)

=

A* } is a branch of isolated solutions of (4 . 131) .

W,

Then there exists a neighborhood ® of the origin in and for N � No large enough, a unique mapping E A* -+ E such that

em H

A

UN(A) WN

Moreover there exists a positive constant any E A* ,

Cl independent of A and N, such that for

A I U(A) - UN( A) l I w $ Cl( I U(A) - nNU(A) ll w + I (A - AN) Q (A, U(A)) l w) .

(4. 134)

Following the line as in Corollary 1 . 1 of Maday and Quarteroni (1982a) , we have the following lemma. Lemma 4.27. Let

H be a Banach space and J( � W � H � B. Assume that

(i) the conditions of Lemma 4 . 26 hold; E

J(

E

DQ (A, V) C ( H, W') is continuous and OJ ( 4.135) J� I A - AN I (iii) if v H satisfies v + ADQ(A, U(A))V 0, then v W. (iv) U(A), UN(A) E J( and I U(A) - UN(A) I K tends to zero as N goes to infinity. (ii) the mapping

v

-+

E

C(W' ,H) =

=

E

Then for N � No large enough,

(4. 136)

Chapter 4. Spectral Methods and Pseudospectral Methods

187

Using the above two lemmas, we can state the following result.

Let {(A, U(A)), A E A* } be a branch of isolated solutions of (4.127). There exists a neighborhood ® of the origin in HJ,....( A ) and, for N � No, a unique C oo mapping: A E A* -+ UN(A) E lP� such that for any A E A* , UN(A) solves (..{.1 30). Furthermore, if the mapping A E A* -+ U(A) E H� (A) HJ,.... (A) is continuous for some r > I , then for any A E A*, there exists a constant b, depending on I U(A) l r,.... such that Theorem 4 . 2 1 .

n

(4.137)

= Take B = L!(A) , W = HJ, (A) , nN = = and in Lemma 4.26. By Theorems 2 . 1 1 and 2.18 and a density argument, (4.132) is valid. Since = (/ we have from Theorems 2 . 1 1 and 2.18, and Lemma 4.25 that

....

Proof.

(A - AN) V

g(A, V) G(A, V)

pt/, A A

- pt/ )Av, I (A - AN )v l h. .... ::; cN- 1 1 I AvI l 2 ,.... ::; cN- 1 ll v l w

and so (4. 133) holds. On use of Lemma 4.26, the first part of this theorem comes with

I U(A) -UN(A)l I l ,w ::; c ( I U(A) - P�o U(A) I h. .... + I (A - AN)G(A, U(A)) l h ,.... ) (4. 138) Since U(A) is a solution of (4. 127) , A G(A, U(A)) = -U(A). Thus Theorems 2 . 1 1 and 2.18, and (4.138) lead to I U(A) - UN(A) I h..... ::; cN1 -r Il U( ,X ) l I r,.... .

To obtain the L!-estimate, we check the hypotheses of Lemma 4.27. By (4.125) , Let H = B and J( = W . Then by Lemma 4.24, the first part of condition (ii) in Lemma 4.27 is fulfilled. Using Theorems 2. 1 1 and 2. 18, and Lemma 4.25 again, we deduce (4. 135) . Furthermore, condition (iii) of Lemma 4.27 is satisfied by Lemma 4.25 and condition (ii) of Lemma 4.27. Clearly condition (iv) is valid. Therefore (4. 136) leads to .

DG(A, V)W = 'x8.,(vw).

Since

. FN(A, U(A)) = (AN - A)G(A, U(A)) = U(A) - P�O U(A),

we have which completes the proof.

•

CHAPTER 5 SPECTRAL METHODS FOR MULTI- DIMENSIONAL AND HIGH ORDER PROBLEMS

In this chapter, we study the spectral methods in several spatial dimensions. We first list some results of orthogonal approximations in several dimensions, which are the generalizations of the results in Chapter 2. Next, we take the periodic problem of three-dimensional steady incompressible fluid flow and the stream function form of two-dimensional unsteady flow as two examples to describe the spectral methods for multi-dimensional problems and high order problems. The final part of this chapter is devoted to the spectral domain decomposition methods and the spectral multigrid methods. These techniques make the spectral methods more effective. In this chap ter, the Fourier approximations, the Legendre approximations and the Chebyshev approximations in several dimensions are used for elliptic, parabolic and hyperbolic equations respectively. These examples will help readers to handle the related tech niques.

5.1.

Orthogonal Approximations in Several Dimensions

This section is for orthogonal approximations in several dimensions. In principle, all results in Chapter 2 could be generalized to those in several dimensions. We first consider the Fourier approximations. Let x = (X l , " " x n ) and Q = (0, 21l't . Let lp be integers and 1 = ( h , 12 , . . . , ln ). Set III = lmax I lq l and t · X =
Chapter 5. Spectral Methods for Multidimensional and High Order Problems

189

Fourier tr!lJlsformation of a function v E L2 (0) is as

Sv(x) =

v e il .", L 11 1 =0 l 00

with the Fourier coefficients

VI =

( 2�) 10 v(x)e-il.", dx,

III = 0, 1 , . . . .

n

Let N be any positive integer, imd

VN denotes the subspace of all real-valued functions in VN. In this subspace, some inverse inequalities are valid. Firstly, for any 4> E VN and 1 :'5 p :'5 q :'5 00 , Next, let m be a non-negative integer. For any 4> E

(5. 1 )

VN

and I k l = m , (5.2)

and for any r � 0,

114>lI r :'5 cNr ll4>lI ·

(5.3)

The L 2 -orthogonal projection of a function v E L 2 (0)

PNv(x) =

1S

L vl e;I .",.

II I SN

Let H; ( O ) be the subspace of Hr (o) containing all functions with period variables. IT 0 :'5 I' :'5 r, then for any v E H; ( O ) ,

PNv is also the HT-orthogonal projection of v E H; (O) .

x Ci)

=

( x �M , . . . , x�n») and

{ I

O N = x U) x qUo ) The Fourier interpolant function in VN that

INv(x)

=

of a function

J'

v

(iI , h , . . . , jn ) ,

(5.4)

Now let j

=

21l'jq 0 < < 2N, 1 < < n } . - q- - -

2N + l '

21l' for all

q

E C(O) is such a trigonometric

190 If v

E

H; (fl) , 0

p.

:$

:$ r and r > 2 ' then

Spectral Methods' an d Their Applicat'i ops

n

l 1'

II v - IN v l :$ cNr -l' lI v ll r .

(5.5)

We next consider the Legendre approximations. Let fl = ( - 1 , l ) n . The Legendre polynomial of degree I is n

IT L/. (x q ) . L/(x) = q=l

The Legendre transformation of a function v

E

L 2 (fl) is as

00

Sv ( x) = :E v/L/(x)

1 / 1 =0

with the Legendre coefficients

v/ = q=l IT (lq

+

1 -2 r 10

)

v(x)L/(x) dx ,

III

=

.

0, 1 , . . .

Let IPN be the set of all polynomials of degree at most N in all variables. Several inverse inequalities exist in IPN. For any rP E IPN and 1 :$ p :$ q :$ 00, we have

On the other hand, for any

rP

E

IPN, 2

:$ p :$

00

(5.6) and I k l

= m,

(5.7) and for any r � 0,

(5.8)

The L 2 -orthogonal projection of a function v E L 2 (fl) is N PN V(X) = :E If r � 0 and

p.

:$

r,

then for any v

E

1 / 1 =0 H r (fl )

v/L/(x).

,

) is given in (2.53) . We introduce the inner product of Hm (fl) , i:e. ,

(5.9)

where

a(p.,

r

(v, W m =

)

:E (a! v , a! w) .

° :5 l k l :5m

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

The Hm -orthogonal projection Pfl : Hm (o) v E Hm ( o ) , If v E Hr ( o) and 0 :5

p,

(v - PNv, if» m :5 m :5 r, then

=

-+

191

IP N is such a mapping that for any

0, 'Vif> E IPN.

(5.10)

(5. 1 1 ) We also need another kind of orthogonal projection i n the numerical analysis of spectral methods. Let

{if> E IPN la:if>(x) = 0 on a�, I k l :5 m am ( v , w) = ( a: v , a: w) .

IP�,O =

L:

-

I}

,

l " l =m

The H.;"-orthogonal projection pJ;'o : H.;" (O) -+ IP�'o is such a mapping that for any v E H.;" (O), a m (v - pJ;'o v , if» = 0, 'V if> E IP�'o .

If v E H' (O) n H.;"(O) and 0 :5

p,

:5 m

:5

r,

then (5.12)

Let xU ) = ( xiii ) , . . . , 0 :5 jq :5 N, x�iq ) being the Legendre-Gauss type inter polation points. O N is the set of all xU ) . The Legendre interpolant INV(X ) E IP N of a function v E C(O) is defined by

, x !!n » )

If v E Hr (o), r > '2 and 0 :5 p, :5 r, then n

II v - IN VII ,. :5 cN2 ,.+� - r ll v llr .

(5.13)

In some special cases, we can improve the above result. For Legendre-Gauss interpo lation and r >

i,

For Legendre-Gauss-Lobatto interpolation and 0 :5 P, :5 min( 1 , 2r - n ) ,

Spectral Methods ' and Their Applications

192

For the Chebyshev approximations in several dimensions, let 1 n w(x) = II 1 - X�) - 2 , The Chebyshev polynomial of degree I is q =l

(

1j(x)

n = ( - 1 , l )n and

n

II TI • (x q ) .

=

q=l

v

The Chebyshev transformation of a function E L� (n) is as

Sv(x) =

vm(x) L: 1 =0 00

11

with the Chebyshev coefficients 2 n n 1 VI = - II - f v(xm(x)w(x) dx, I II = 0, 1 , . . . , 7r q =l CI . in where for 1 � q � n, C/ . = 2 for lq = 0 and CI . = 1 for lq � 1 . For any ifo E IP N and 1 � p � q � 00 . (5. 14) lI ifollL2; � cN � - Z; lI ifoIlL� ' Also for any ifo E IPN, 2 � P � 00 and I k l = m,

()

(5. 15) and for any r

�

0,

v vm(x). L: 1 =0

(5.16)

The L�-orthogonal projection of a function E L�(n) is If r � 0 and I-' �

PNv(x) =

r,

N

11 then for any E H� (n) ,

v

0'(1-', r ) is given in (2.53) . Furthermore, we define the inner product of H:: (n) by

(5. 1 7)

vw

( , ) m ,w

=

L: (a:v, a:wL ·

The H::-orthogonal projection P'N : H::(n) -+ IPN is such a mapping that for any E H:: (n) ,

v

O$ I " I $ m

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

193

v

If E H�(f!) and 0 $ P, $ 1 $ r, then

(5. 18)

Moreover, let

am,w (v, w) = kL:m (a;v, a;wL I l=

and

(5.19)

(5.20) am,w (v, w) = kL:m (a;v, a;(ww)) . I l= In particular, a w(v, w) = a l,w (v,W) and a w(v, w) = a l,w (v, W). The HJ,w-orthogonal projection PIt : HJ,w(f!) lP� is such a mapping that for any v E HJ,w ( n ), -+

If E H� (f!) n HJ,w(f!) and

v

r ;:::

1 , then

The other HJ,w-orthogonal projection for any E H , (f! ,

v Jw )

pit : HJ,w(f!)

-+

lP� is such a mapping that (5 . 21)

If E H�(f!) n HJ,w(f!) and 0 $

v

p,

$ 1 $ r, then (5.22)

x U) (x iM , . . . , x !tn» ) v

x �j.)

Let be the Chebyshev-Gauss type interpo = , 0 $ jq $ N, lation points. f!N is the set of all The Chebyshev interpolant of E a function E C(O) is defined by If E H�(f!) , r > "2 and 0 $

v

n

x U) .

p,

INv(x) lPN

$ r, then

(5.23)

194

Spectral Methods

n

w(x)

(-� xq) .

an d

Their Applications

We now turn to the orthogonal approximations on unbounded domains. First of all, let

= (O, oot

and

=

exp

The Laguerre polynomial of degree

LI(x) qII=l LI, (xq). The Laguerre transformation of a function v L� ( n ) is Sv (x) 1L:11=0 vILI (x) I is

n

=

E

=

00

with the Laguerre coefficients

VI in v (x)LI(x)w(x) dx, I I I =

For any

00 ,

=

0, 1 , . . . .

( 5.24 ) Also for 1 :::; P :::; n ,

v L� (n) is L: vI LI ( x). 1 1 1 =0

The L�-orthogonal projection of a function

Let

a

PNV(X) = >0

and

with the norm

For any

H� (n, ) { v I l v l r,w , a a

v H::, (n, ) and a

E

=

0 :::;

Jl

E

=

:::; r ,

of degree I is

n

= ( - 00 , oot

E

H�(n) I v tl xl H� (n) } Il v tJ + Xq ) t Ilr,w E

(1

where a is the largest integer such that Finally let

N

( 5.25 )

a

< r

( 5.26 ) + 1.

w(x) exp (- qt=l x;) . The Hermite polynomial HI(x) qII=l HI, (xq) .

and

=

=

n

Chapter

5.

Spectral Methods for Multidimensional and High Order Problems

The Hermite transformation of a function v

E

195

L! ({l) is as

v(x) = L v,H,(x) 1 1 1=0 00

with the Hermite coefficients

For any
00 ,

II
'

'

(5.27)

Also for 1 $ p $ n , The L!-orthogonal projection of v

For any v E H�({l) and 0 $

J.t

E

(5.28) L! ({l) is

N PNv(x) = L vIH,(x). 1 1 1=0 $

r,

we have (5.29)

The above results can be proven along the same lines as in Sections 2.2- 2.6. Some of them can be found in Canuto and Quarteroni (1982a) , and Bernardi and Maday (1992a) . The filterings and the techniques for recovering the spectriY accuracy in Section 2.7 can be generalized to the orthogonal approximations in several dimensions. For 1 example, let {l as

=

(0, 211")" and l i b

=

(� I:)

2 .

The filtering RN( 0:, (3) can be defined

( �!)P �,ei'."' ,

RN( o: , (3)
a,

(3 � 1 ,
Spectral. Methods and Their Applications

196

If in addition

p.

� 0, then (5.30)

The LOO-error estimates between RN(a, {3)PNv(X) and v(x) also exist. For instance, for any 4> E VN, we define the filtering RN(a, {3) by

Set

W(V i P ) = max Iv(x) - v(x') I , I"'-"" I�p

Cheng and Chen ( 9 56 ) proved that if a > for any v E C;' (fi) ,

1

and if a = m +

n -2-1 ] + 1 . (3 = [-

w

m+

RN(a, {3)PNv(X) - v(x) = 0

1,

m

w a!v i P) . m ( v i P ) = max m

being odd, then for any v

1 or

a

=m Ikl=

(

+ 1,

m

being even, then

(;m Wm (Vi �)) , E

C;'(fi) ,

For the sake of simplicity, we still use the notations LOO ( !l ) , LM !l ) , w�·r ( !l ) for the spaces of vector functions in the following discussions.

5.2.

Spectral Met hods for Multi-Dimensional Nonlinear Sys tems

The aim of this section is to present the spectral methods for multi-dimensional nonlinear systems. In particular, we are interested in spectral approximations of isolated solutions. For simplicity, we focus on the three-dimensional periodic problem of steady Navier-Stokes equation. As in Section 4.6, we shall rewrite it as an abstract operator equation in certain space. Then we use the Fourier spectral methods and Fourier pseudospectral method to solve it numerically. Finally the convergences of both schemes are given. N avier-Stokes equation is the fundamental equation describing incompressible fluid flows. Let !l = (0, 211")3 and L�( !l ) be the subspace of L 2 ( !l ) , containing all

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

1 97

functions , with the zero average over n. Denote by U(x), P(x) and p. > 0 the speed vector, the pressure and the kinetic viscosity, respectively. The components of U (x) are U (j) (x) , 1 :::; j :::; 3. Moreover f(x) stands for the body force, and f(x) = (J(1) ( x), f (2)(x) , J(3)(X)) . The steady Navier-Stokes equation is of the form

{ V(U· .UV)U= O,- p./lU

+ Vp =

f,

x E n, x E n.

(5.31)

In this section, we assume that all functions have period 27r for all variables. For fixing the values of P, we require that P E LMn). Let V = {v I v E Vp(n), V · v = O} and V be the closure of V in Hl (n). Set

a(v, w) = 10 Vv(x ) · Vw(x) dx , b( v, w, z) 10 v(x ) · (w(x) · V)z(x) dx. =

A weak solution of (5.31) is defined as a pair

(U, P ) E V

p.a( U, v ) - b(U, u, v ) = (f, v ) ,

'if

L � (n), satisfying

v E H: (n) . x

(5.32)

If (f, v ) is a linear continuous functional in V, then by the Fixed Point Theorem, (5.32) has at least one weak solution. If (5.32) has a unique solution, then it is not difficult to approximate it directly by the Fourier spectral method or the Fourier pseudospectral method. But in those cases, the trial functions
and for any

v E L2 ( n) ,

(�r eil.",

VI = (v,
Then it follows

10 (-p. /l U (x) + ( U(x) . V)U(x} - f(x)) . V
that

198

Spectral M/fthods and Their Applications U

V . = 0, we get - k tl U(x) . V (Pt(x ) dx = k (V . U(X))AtPt(x ) dx = O .

Integrating by parts and using

Therefore

10 ((U(x) . V)U(x) - f(x)) . V
I / I � = t I� . Since P E L� ( n ), Po = O . Thus for any v E H� (n) , 1o VP(x) . v (x) dx = - Jfo P(x)V . v(x) dx = i LI (/ . �I) (P,
(5.33)

where

q= l

from which and (5.32) , we obtain that -I-'

10 A U(x) . v (x) dx + 10 (U(x) . V)U(x) . v (x) dx f . -L ¥o 1 :1 2 (I �I) 10 [1 · ((U(x) · V )U(x) - f(x))] �I( X) dx = k f(x) . v (x) dx.

2 By (5.33) and P E LMO), (P,
{

Set

for =f.

I O.

(5.34)

b(v, w, z) = 10 [(w(x) · V)v( x) ] · z(x) dx I � �' 1 ( 1 . (w(x) . V)V( X ))�I(x) dx, (5.35) -L 1/0 I I 2 0 l . :&1 . ( I f,
Chapter 5. Spectral Methods for Multidimensional and High Order Problems

199

While P is determined by (5.34) . We can rewrite (5.32) in another way. To do this, let

bO (v, w, z) b1 (v, W, z )

i 'L#0 l (w(x) ' l) (v(x) . 'i,) ¢I (x) dx 10 - i 'L# 11 � 1�1 1 (w(x) . l)(v(x) . l) ¢, (x) dx, 10 2 l · � (\7 . w, (v(x) · l )¢l ) . = - ifr (\7 . w( x))(v(x) · z( x))dx + 'L 1#0 11 1 22 =

(J

(5.38)

(J

(J

(5 .39 )

By integrating by parts in (5.35), we obtain that (5.40) b(v, w, z) = bO (v, w, z) + b1 (v, w, z), 'V v, w E H; (O}. Moreover, if in addition w E V, then b1(v, ui, z) = O. Thus (5.32) can be presented as follows. It is to find U E H; (O), such that (5.41) We now verify the equivalence of the representations (5.3 1 ) , (5.37) with (5.34) and ( 5 . 4 1 with (5.34) .

)

Lemma 5 . 1 .

For any v, w E H; (O),

and for any w E L2(0),

I F(w) 1 ::; cl l f l l ll w l ·

Proof. The first result is clear. Next, we have f w (f, IF(w) l ::; II f ll l w ll + 'L 1#0 I w, j 1
Lemma 5 . 2 .

For any v, w, z E H;(O), I b(v, w, z) l ::; c l v h ll w lllll z I 1 3/4 ' I b( V, w, z)1 ::; cll v ll ll l w lll l z l h ·

Spectral M,ethods and Their Applications

200

Proof. We have from (5.35) that b(v, w , z) = 10 ( (w(x ) · V ) v (x ) ) z ( x) dx - � 10 w(;) (x)Oj v(m) (x)o-(m) (x) dx -

3

j'

where u (m) (x) =

1

L l ml 22 (I . i!) (/>t(x) , 1#0

�

Obviously I ud ::; lid which implies that for any

(5.42)

1 ::; m ::; 3.

r � 0,

Hence by imbedding theory, (5.42) reads

+

I b(v, w , z) 1 ::; cl v ll l 1 w llL' ( lI z I l L' lI u Il L' ) cl v h ll w l l � lI z ll � ::; cl v h ll w lh ll z ll � ·

<

The rest of the proof is clear. Lemma 5 . 3 .

•

Let v , w , z E Hi (o' ) . We have that for v E W1,4 (O,),

and for v E Ht (O,) , Proof. For any v E W1,4 (O, ) , we deduce from (5.42) that The second result follows from the fact that Ht(o' ) Lemma 5.4.

then

C

W1,4 (O, ) .

•

Let v , w , z E Hi (o' ) . If in addition v E W1,¥ (O,) and w E LI2 (O,) ,

If w E Ht (o' ) n V ,

then I b(v, w, z) l ::; e ll v II II w ll l• II z lh ·

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

201

Proof. By (5.42) ,

I b(v , w , z) 1 :::; cl l Y'v I l L¥ I W I L'2 ( l z l + I l u l ) :::; cl l v l w1, ¥ I W I l L'2 1 z 1 . If w E H t ( n ) V, then b y (5.38), I b(v, w, z)1 I bO(v, w, z) 1 :::; J,, tm=l I I: 1 1 1 2At,m (1 wU}(x )v (m}(x)(Mx ) dX) I n

1#0

=

fl

(5.43)

Thus (5.43) implies that

I bO (v , w , z) I :::; cl l v l l w l Loo l i z I I cl l v l l w i l t I z l h · :::;

(5.44) •

v , w, z E H: (n), I bO (v , w , z) l :::; cl l v l l l w l h l z i l i ' Let At, m be the same as before, and v , w , z E V ( n ). By (5.43), 1 2 m 2 :::; J,, t m=l (I: I I W I A{, I ) 2 (I: 1 1 1 �-2r (1 wU}(x) v(m}(X) ¢I (X )dX) ) 2 :::; cl l z l r i,I:m=l I wU} v (mt _r '

Lemma 5 . 5 . For

Proof.

any

I

1#0 3

1#0

fl

By putting r = 0, 1 , we obtain from imbedding theory that

I bO(v, w, z) I :::; c l z l l v l wl,< l w l wl,< :::; cl l z l l v i t Il w l t , I bO(v , w , z) 1 :::; cl l z l h Il v l u l w l u :::; cl l z l l l v l � I l w l i '

By the space interpolation, we assert that

•

Spectral Methods and Their Applications

202 Lemma 5 .6 .

Let v, w, z

and

E

We have that

H� (n) .

I bO (v, w, z) l :5 c l / v l /rl / w l/l / z lll>

On the other hand, if w

E

Ht (n),

for v

then

Proof. It is easy to derive the above estimates.

E

Ht (n) .

For instance, the last one comes

immediately from (5.44). Theorem 5 . 1 .

(5.41)

Proof.

•

The representation (5. 31) is equivalent to (5.37) with (5. 34), or to

with (5. 34) ·

It is clear that a solution of (5.31) is also a solution of (5.37) with (5.34) . Now, suppose that is a solution of (5.37) with (5.34) . We have from Lemma 5 . 1 , Lemma 5.2 and (5.37) that for any E V(n),

(U, P)

v

E H- � (n) and so E H� (n) n Ht (n) . By Moreover by Green ' s formula, imbedding theory, E L12 (n) n W1,¥ (n) . By virtue of (5.37) and Lemma 5.4,

U

/).U

U

I fLa ( u, v) l :5 c ( ll U llw1'lf ll U l / o 2 + Il f ll ) Il v ll· By Green ' s formula again, /).U L2(n) and so U H� (O) n H2 (O) . The integration by parts in (5.37) leads to (5.31). A standard argument tells us that the normal derivative of U is also periodic and so U H; (O). Furthermore by (5.34), L: 1 ((U . V)U(j) - f(i), ¢l) 1 2 . L: 1 1 I� I (P, ¢l W :5 L: j=l Since (U . V)U - f L2(f!), the above series converges and so P H; (f!) . Using E

E

E

1

E

(5.34) again, we have

1

3

E

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

203

and so P E, Finally it follows from (5.35) - (5.37) that On the other hand, by integrating by parts, we obtain that for 1 f 0,

l1-a (U, VI/JI ) = O.

H�(S1)nL�(S1).

a (U, V (I>t)

=

1 {)mU(j)(X){)m{)j �I(X) dx j,m=l 3

E

0

=-

3

E Im 1j

j,m=l

i l l l � E Ij Ulj l ,;" i l l l � (V · u, M . j=l (V . u, I/Jo) = 0 and so V . U = o . 3

1 {)mU(jl (X)�I(X) dx 0

(5.45)

In addition, The above statements lead to that P) is a solution of (5.3 1 ) . We now turn t o the equivalence of (5.37) with (5.34) and (5.41) with (5.34). If and so it U is a solution of (5.37) , then for any is also a solution of (5.41 ) . Conversely, if is a solution of (5.41) , then by taking VI/JI , 1 f 0 in (5.41) and arguing as in the previous paragraph, we can deduce that V · = o . Henc� V and for any E u, This means that is a solution of (5.37) . This completes the proof. _

(U,

v=

U

v E H�{S1 ), b(U, U, v) = b° (U, U, v) U v H�{S1), b(U, U, v) = b° (U, v).

UE

U

We are going to construct the Fourier-spectral scheme. For this purpose, we put (5.37 ) in an abstract framework. Let >. ..!:. and = Since

= E {S1 ) (HJ ( S1 ) ) ' . H! {S1 ) Hp! ( S1 ), E {S1 ) H- ! ( S1 ) and also E( S1 ) (H� ( S1 )) ' . We define an inner product on H� ( S1 ) by (5.46) (v, wh = a(v, w) + (v, w), V v , W E H;(S1), and a linear mapping A : (H� ( S1 ) ) ' -+ H� ( S1 ) by (5.47) (Av, wh = (v, w ) , V w E H;( S1 ). Moreover, by the theory of linear elliptic equations, A is a continuous mapping from W ( S1 ) with � - into Hr+2 ( S1) n H;( S1 ). Since E( S1 ) H-! ( S1 ), A is continuous from E( S1 ) into H�(S1)nH�(S1). But the latter space is comp&.ctly contained in H� ( S1 ), whence A is a compact mapping from E( S1 ) into H; ( S1 ) . Define G : rn.1 H; ( S1 ) -+ E( S1 ) by (5.48) (G( >' , v), w) = >'( b(v, v, w) - F(w)) - (v, w). From Lemmas 5.1 and 5.2, we know that for any >. E IR1 and v E H;( S1 ), I I G(>., v) II E :5 cl>'1 (II v ll � + Ilfll) . c

c

r

L

.

11-

C

C

x

Spectral Methods and Their Applications

204

Moreover G is a Coo mapping. Let D be the Frechet derivative of G. Then . for any positive integer m , DmG is bounded over any bounded subset of lR,l x According to (5 . 46) and (5.48) , (5.37) is equivalent to

(U, V ) l + ( G( .\, U) , v) = 0, \:I v E

Hi(f!).

H�(f!).

F(A, v) = v + AG(.\, v) . By (5.46)- (5.48) , we get that

an

(5.49)

equivalent form of (5 . 37) . It is to find U E

F('\, U) = O.

Hi(f!) such (5.50)

Let A* be a compact interval in IR\ and assume that (.\, U( .\ )) is a branch of isolated solutions of (5.50) . So there exists 8 > 0 such that for all .\ E A* and v E

11 (1 + ADG ( .\, U( .\ ))) vlll

H�(f!),

� 811vlll .

(5.51)

Now let VN be the same as in Section 5.1, and PN be the L 2 -orthogonal projection upon VN . It can be checked that for any v E

H�(n),

H;(f!) with

(5.4) asserts that PN is a continuous mapping in By (5.47) and (5.52) , AN E .c VN and

((H�(f!))' , )

(5.52) r

� o . Let AN = PNA. (5.53)

Maday and Quarteroni (19 82b) provided a Fourier-spectral approximation to (5.50) . It is to find UN E VN such that (5.54)

By (5.53) , (5.54) is equivalent to finding UN E VN such that By arguing as in the proof of Theorem 5.1 , it can be shown that \7 . UN = O.

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

205

E

Let { (A, U(A» , A A" } be a branch of isolated solutions of (5.50). There exists a neighborhood @ of the origin in H;(n), and for N sufficiently large, a unique Coo mapping A A" -+ UN( A) VN such that for any A E A", FN( A, UN( A» = o and U(A) - UN(A) E @. Moreover, if the mapping A E A* U(A) E H; (n) is continuous for some r 2:: 1, then for any A, there is a positive constant b1 depending on I U(A) l I r such that Theorem .5 . 2 .

E

E

-+

Proof. We first use Lemma 4.26, with B = E(n), W = H;(n), nN = PN, A = A and g (A,v) = G(A,v ) . Clearly condition (i ) of Lemma 4.26 is satisfied. Further, by ( 5.4) ( 5.55 )

and a density argument,

Since

A is a c?mpact mapping from E(n) into H;(n), we derive that Nlim �oo II (PN I)A llc(E,Hl ) = O. -+oo II AN - A llc('E H') Nlim

Therefore both ( 4.132 ) and (4.133 ) are also fulfilled. So there exists a unique branch E A*}. The combination of ( 4. 134 ) , ( 5.4 ) and ( 5.50 ) yields that '

{ (A, UN(A» , A II U(A) - UN(A) l h

� � �

P

=

-

(5.56)

p

c ( I U(A) - PNU(A) l h + I ( A - AN)G(A, U(A» 1 1) c ( I U(A) - PNU(A) l h + 1 (1 - PN)AG(A, U(A » lh ) ( 5.57 ) c Il U(A) - PNU( A) 1I1 � cN1 -r Il U(A) lI r .

For the L 2 -error estimate, we put Lemmas 5.3 and 5.4 ensure the first part of cQndition ( ii ) in Lemma 4.27. By C condition ( iii ) of Lemma 4.27 is also satisfied. Since ( 5.4 ) ensures condition ( 4.135 ) . Finally by ( 5.3 ) , ( 5.4 ) and ( 5.57 ) ,

((H;)' ,H;) ,

(I - PN)A, I U(A) uN(A) II K < I U(A) - PNU(A) I K + I PNU(A) - uN(A) I K < I U(A) - PNU(A) I K + Nl ( I U(A) - PNU(A) l h + I U(A) - UN(A) l d < cN- lII U(A)1 I 2 . �

E

A A - AN =

Spectral Methods B{ld Their .�pplications

206

Hence condition (iv ) of Lemma 4.27 is also fulfilled. Now, using (4. 136) and ( 5.4 ) , ' we obtain that

II U(,x) - uN(,x) 1 I .

� =

cII FN (,x , U(,x)) ll = II(A - AN)G(,x , U(,x» II II (PN - I)AG(,x , U(,x» II = II( PN - I)U(,x) II � cN-r I l U(,x ) ll r

which completes the proof.

•

In actual computations, we can use the Newton procedure to evaluate UN (,x) . For instance, let uWl(,x) be the m-th iterated value of UN (,x). Then the iteration is as follows

(I + ANDG ( ,x , uW) (,x»)) (uW+Il(,x) - uWl(,x») = - (u W ) (,x ) + ANG ( ,x , uW ) (,x»)) , m � O. On the other hand, the approximate pressure PN (,x) = PN( X, ,x ) could be determined

by

with

PN (,x) =

PN,O(,x ) = 0, PN,I(,x ) = I I� ( 1 . « UN (,x) . V) UN (,x) - f) , 1/>1) ,

:

1 � O.

E

( 5.58 )

n

Assume that the hypotheses of Theorem 5.2 hold, P(,x) H;(n) and r � 1 . Then there is a positive constant b2 depending on IIU(,x) ll r such

Theorem 5 . 3 .

L5(n),

that

L PN,I(,x ) Ip, ( ,x )

I I ISN

Proof. Let

IIP(,x) - PN (,x) I + N- 1 IIP(,x) - PN (,x) lh � b2N -r . PN(,x) = PNP(,x) and p(,x) = PN (,x) - PN(,x). By (5.34 )

(P (,x) , rpo ) = 0

and for I � 0,

�

and ( 5.58 ) ,

, · I l � (l · [(UN ( .\) V) UN(.\) (U(,x ) · V )U(,x) ] M . I l l � ( I . [« UN (,x) - U(,x» V) UN (,x ) + (U(,x) V) ( UN(,x) - U(,x»] , rpl ) . By integrating by parts and using V · U(,x) = V . UN (,x ) = 0, we obtain that for 1 � 0,

(P(,x), rpl)

=

-

�

.

I (P (,x ), rpl ) I

=

<

110 [(U(,x)

�J [

�J ax i

UN (,x» . I l (U(,x ) + UN (,x» . I l ¢l .L 1 ( ( UU) (,x) - uW(,x ») ( u (m ) (,x) + uWl(,x») , ¢I) I · J ,m=1 3

-

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

207

Consequently,

3

.

::; . � I ( UU) ( .\ ) - uW ( .\ ) ) (u (m ) ( .\ ) + u� ) ( .\ ) ) I J,m=l

2

3 . 2 3 m=1 3=1 2 < C IIU( .\ ) - UN ( .\) 1 I I 1 U( .\ ) +2 uN ( .\) II � ::; cIlU( .\ ) II � IIU( .\ ) - uN( .\ ) 1 I ,

::; � I UU l ( .\ ) - U W ( .\ ) I � I l u ( m) ( .\ ) + U � ) ( .\ ) I Loo

from which together with Theoerem 5.2 and ( 5.4 ) , we find that

•

-+

We now consider the Fourier-pseudospectral approximation. Define the mapping G : JR1 x H;(n) E(n) by ( 5.59 )

( G(.\, v ) , w) = .\ ( bO(v, v, w) - F( w ) ) - (v, w) .

Due to Lemmas 5.1 and 5.5, bO possesses all the continuity properties of b, except the first estimate in Lemma 5.4, which was used only in order to prove the regularity of U( .\ ) in Theorem 5 . 1 . So the mapping defined by ( 5.59 ) has the properties of G defined by ( 5.48 ) . In particular, ( 5.41 ) can be equivalently written in the form ( 5.50 ) where G is given by ( 5.59 ) . Now, let n N be the set of all Fourier interpolation points as in Section 5.1. For any v, w E C(O), the discrete inner product is given by

( V, W)N =

(� 2 7l' ) 3 � v(x ) w(x ) . 2 +1

"'HIN

Denote by INv and VI the Fourier interpolant of v E C(O) and the Fourier coefficients. Assume that f E C(O). Define b� (v, w , z) : VN C(O) and FN ( W) : VN C(O) by

-+

-+

Spectral Methods and Their . Applications

208

>.

Further, define

rn'!

Vk as (GN( A, V), W ) = A ( b� (v , v , w) - F(w » ) - ( V, W)N,

GN

:

x VN -+

(5.60)

and let

Let

AN

be the same as in (5.53) and set FN

:

]Rl VN x

-+

VN,

A Fourier-pseudospectral approximation of (5.41 ) is to find

UN E VN

such that (5.62)

By (5.53) , (5.60) and (5.61) , (5.62) is equivalent to finding

UN E VN

such that

By an argument as in the proof of Theorem 5 . 1 , it can be checked that We have the following result. E

'\7

.

UN = o.

Let { ( A, U( A», A A * } be a branch of isolated solutions of (5 ./.1), and rl > � , r2 > 2 . If f E H;' (O) and the mapping A E A* -+ U( A ) E H;2 ( O ) is continuous, then for N :?: No large enough, there exists a positive constant b3 depending on IIU( A ) II '2 and IIfll ." and a unique Coo mapping A E A* -+ UN ( A ) VN such that for any A E A *, FN ( A, UN( A» = 0 and

Theorem 5 .4.

.

E

N

In order to obtain an improved L 2 -error estimate, let M > be a suitably chosen integer. The corresponding set of Fourier interpolation points, the discrete inner product and the Fourier interpolation are denoted by OM, ( . , ·)M and 1M. If then M>

Nr':' ,

Define the mapping Giv( A,

v)

as

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

where :FN

: YN -+ C(a) is given by :F;' (v) = ( f, V)M - L 1 /11 2 ( I · VI ) (1 . f, (Pt)M · III$N,I#O 2

209

Let F;' (A, V ) = v + ANGiv(A, v) . A new Fourier pseudospectr al approximation is to find uiv E VN such that (5.63) F;' (A, Uiv(A)) = O. We have the following result.

Let { (A, U (A)), A E A*} be a branch of isolated solutions of (5. ..{.1) and iv be the solution of (5.63). If f E Hr -1 ( n) with r > 2 and the mapping A E A* -+ U (A) E H: ( n) H;(n) is continuous, then there exists a positive constant b4 depending on II U ( A) lIr and IIf llr - 1 such that for any A E A*,

Theorem 5 . 5 . u

n

In actual computations, we can use the Newton procedure to solve (5.62) or (5.63) . For example,

(I + ANDG'N (A, U�m) ( A) ) ) (U�m+1 ) (A) - U�m) ( A) ) - (U�m) (A) + ANGiv (A, U�m) (A))) , m � O. =

On the other hand, the approximate pressure iN(A) is given by

with Piv,O ( A) = 0 and for

1 "# 0,

�

Piv,I ( A) = �I « UiV (A) · l ) (UiV (A ) . 1) + if · 1 , (Pt) M · If M > N

.:1 , then

(5.64)

where bs is a positive constant depending on I I U(A) l I r and I I f IlT- 1 . Theorem 5.4, Theorem 5.5 and (5.64) were proved in Maday and Quarteroni (1982b ) .

Spectral Methods and Their Applications

210

For the Fourier approximations of unsteady N avier-Stokes equation, we refer to the early work in Orszag 1971a, 1971b ) , Fox and Orszag 1973a, 1973b ) , Fronberg 1977 ) , and Moin and Kim 1980 ) . Hald 1981 ) , Kuo 1983 ) , Guo ( 1985 ) , Guo and Ma 1987 ) , Ma and Guo 1987 ) , Cao and Guo 1991 ) , and Rushid, Cao and Guo 1994 ) provided various schemes with numerical analysis for compuational fluid dynamics. The applications of Fourier approximations to other multi-dimensional equations can be found in numerous articles, such as Wengle and Seifeld 1978 ) , Feit, Flack and Steiger 1982 ) , Guo, Cao and Tahira 1993 ) , and Cao and Guo 1993b ) .

( (

(

(

5.3.

( (

(

(

(

(

(

(

(

(

Spectral Methods for Nonlinear High Order Equations

Many nonlinear partial differential equations of high order occur in science and en gineering, such as the Korteweg-de Vries equation possessing solitons, the Laudau Ginzburg equation in the dynamics of solid-state phase transitions in shape mem ory alloys, etc. . In this section, we consider the spectral methods for such kind of equations. We shall take the initial-boundary value problem of two-dimensional Navier-Stokes equation in stream function representation as an example to describe the Legendre spectral method for solving initial-boundary value problems of nonlin ear revolutionary equations of fourth order. The corresponding numerical solution converges to the weak solution of original problem. The spectral accuracy is obtained for smooth solution. The prediction-correction method is also used for raising the total accuracy of numerical solution. The numerical experiments coincide with the theoretical analysis. 1 1 2 and 80 be a non-slip wall. The initial-boundary value problem Let 0 of two-dimensional Navier-Stokes equation is of the form

{

=( , ) -

E

8tU + (U . V)U - p.!:iU + 'VP = /, x 0, 0 < t ::; T, 'V . U = 0, x 0, 0 ::; t ::; T, ( 5.65 ) U = O, x 80 , 0 < t ::; T, x n. U(x, O) = Uo(x) , Let V = { v I v V(O), 'V . v = OJ. The closures of V in L2(0) and H l ( O ) are E E

E

E

denoted by H and V respectively. Let

a(v, w) = 10 'Vv(x) · 'Vw(x) dx, b(v, w, z) = 1o (w(x) . 'V)v(x) · z(x) dx.

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

211

The weak f�rm of (5.65) is as follows ·

(Ot U, v) + b(U, U, v) + p a(U, v) { U(O) Uo,

It is shown in Lions (1969) that if a unique solution in Tj V) n conservation =

L2 (O,

=

(f, v),

v

v E V, 0 < t :5 T, (5.66) x E n. and I E L 2 (O, Tj V'), then (5.66) has

Uo E H LOO(O, Tj H) . The solution of (5.66) possesses the

For numerical simulations, we can approximate (5.66) directly. But in this case, we should construct a trial function space whose elements satisfy the incompressibility. It is not convenient in actual computations. To avoid it, several alternative expressions are concerned. Firstly, we can take the divergences of both sides of (5.65) , and get

Ot eV' . U) + V' . ((U . V')U) - p l:l.(V' . U) + l:l.P = V' . f. By the incompressibility, we obtain

(U . V')U - pl:l.U {. Ol:l.Pt U ++ �(U) V' . I, =

+ 'V' =

P I, x E n, 0 < t :5 T, x n, 0 < t :5 T E

(5.68)

where

�(U) = 2 ( 02 U(1 )fhU(2) - fhU(1 )02 U(2 ») . Conversely if U and P satisfy (5.68) and . U(O) = 0, then for all t � 0, V' . U = 0. Thus (5.68) is equivalent to (5.65) provided that U and P are suitably smooth. The second formula of (5.68) is a Poisson equation for P at each time t. If we use it to 'V'

evaluate the pressure, then we need certain boundary conditions for the pressure. As pointed out in Gresho and Sani (1987) , we have that

O"P(x, t) = v . ( pl:l.U(x, t) + I(x, t)), x E On, 0 < t :5 T where O"P is the outward normal derivative of P . A simplified condition is O"P(x, t) = N(x, t), x E On, 0 < t :5 T. But the first condition is too complicated in actual calculations, while the simplified one is not physical.

Spectral Methods and Their Applications

212

The second way is to introduce the vorticity H(x, the stream function w(x, and J(v, w) = 82 w iAv - 81 w82 v, V V, w E Hl (n) .

t) ,

Then

{

(5. 6 5) i s equivalent to

atH + J(H, w) - p,/:).H = '\l -/:).w = H, H(x, O) = Ho (x) ,

X

x E n, x E n, x E n.

J,

0 < t ::5 T, 0 ::5 t ::5 T,

t)

(5. 69)

Since the incompressibility is already included, we do not need to construct the trial function space with free-divergence. Clearly w(x, = 0 on the boundary. But it is not easy to deal with the boundary values of the vorticity. In some literature, it is assumed approximately that H (x, = on an. But it is not physical. The third representation of Navier-Stokes equation is the stream function form. Let

t)

t) 0

{

Then the stream function form is as follows

8t /:).w + G(w, w) - p,/:).2 W = g , aw w = - = o, an w(x, O) = w o (x) ,

x E n, x E n, xEn

0 < t ::5 T, 0 < t T, ::5

(5. 70)

where 9 = - '\l x J. The main merits of this expression are remedying the problems mentioned in the above and keeping the physical boundary values naturally. But the appearance of the terms G(w, w) and /:). 2 W brings some difficulties in theoretical analysis. We shall derive the weak form of For any v, z E w 1 •4 (n) and w E H 2 (n), let

(5. 70).

Obviously

J(v, w, z) + J(z, w, v) = O.

(5. 7 1)

J(v, w, z) = -(G(v, w), z).

(5.72)

It can be verified that We can estimate IJ(v, w, z} 1 in several ways.

Chapter 5. Spectral Methods for Multidimensional and High Order Problems Lemma 5 . 7. For

' an y

213

v E HJ (fl),

Proof. By a density argument, it suffices to prove the conclusion for such a function, we observe that for x = (xt , X 2 ) E fl,

1:1 :

(8lv(X» 2 :5 2 18lv (Xl, x 2 ) 1 1 8�v (Xl, X 2 ) 1 dXl , (�v(X» 2 :5 2 1 ' 18lV (Xl! x 2 ) 1 1 �82 V (Xl! x 2 ) 1 dX 2 , Thus

v E CO" (fl) .

For

etc.

Similarly This completes the proof. Lemma 5 . 8 . For

any

•

v, w E HJ (fl), I J(v, v, w ) l :5 2 11'\7 v ll ll � v llll � w ll ·

Proof. By integrating by parts, we obtain that

J (v, v, w )

=

- 10 (8lV82 V8�w - (81 v? 8l82 w + (82 v) 2 �82 W - 8lV82 V8�w) dx + 10 (81 V82 W8�V - 81 v8l w8l 82 v + 82 v82 w81 82 v - 82 v8l w8�v) dx.

Integrating the last integral by parts again, we find that it vanishes. So by Lemma

5.7, I J (v, v, w ) 1 :5 118l v l b 1 182 v ll L4 " 8� w" + 1 1 8lV I l � . 11 8l82 W ll + 11 82 V Il�. 11 8l82 w ll + 118lV Ib 11 82 VI lL' " 8�w " :5 (2 11 8lV I l�. 1 I 82 V I l�. + 1 1 8lV lli. + 11 82 V lli. ) . 2 I 81 82 w II 2 + " 8�W ,, 2 + " 8�wl :5 2 11'\7 v ll ll� v llll� w ll ·

(

n

1

1

2"

2"

•

Spectral Metnod� and Their Applications

2 14

If v, z E HJ ( f! ) and w E H2 ( f! ) , then

Lemma 5 . 9 .

I J (v , w , z) l :5 2 11� w ll (IIVvll ll� v Il IlVzlI lI� zll) ! .

Proof. By Lemma 5.7, I J (v , w , z) 1

v'211� w Il IlV v IIL' IIV zII L' :5 2 11� w ll ( IIV v ll ll� v Il IIV z ll lI�zll) ! . :5

•

Lemma 5 . 1 0 .

For any v , w , z E HJ ( f! ) ,

Proof. By integrating by parts, we have that J (v , w , z) = 10 (OlW02Z0:V ..... �W�ZOl02V + 02W02Z01 02V - 02WOIZO�V) dx + 10 (OlVOIW��Z - 02VOIWO:Z + OlV02WO�Z - 02V02WOI02Z) dx. (5.73) Then the desired result follows from an argument as before. Lemma 5 . 1 1 .

Let v , w , z E HJ ( f! ) . If in addition w E Hr ( f! ) with r > 2, then

I J (v, w, z)1 :5 cll� vlll l w ll r IlV zlI · Proof· We first let 2 r 25 ' Set r = I-' + 1 , 2 _2 I-' Hl ( f! ) '-+ LP ( f! ) and Hr - 2 ( f! ) '-+ La ( f! ) Thus

< <

I J (v , w , z) 1

< :5

Next, let

•

.

a: ==

and f3

=

I-'

_2

l'

Then

II�WIlLa ( IIOIVllo' II02Z 11 + Il o2V l lv' II OI Z II ) cll� v ll ll w ll r IlV z lI ·

r � �. Since HI'- l (f! ) '-+ L4 ( f! ) , we see that

•

Chapter 5. Spectral Methods for Multidimensional and High Order Problems Lemma 5 . 1 2 .

Le t v, w , z·E Ha ( !1 ) .

215

If in addition v E Hr ( !1 ) with r > 2, then

I J(v, w , z) 1 � c ll v llrll.6. wI I IIV z ll , I J(v, w , z) 1 � c ll v llr IIV w ll ll.6. z ll .

Proof. Since W- 1 (!1)

'-+

I J(v, w , z) 1

L<Xl ( !1), we get that �

:S

IlolV IlLoo II.6. w ll llo2Z I1 + II 02V llLoo II.6. w ll llolZ II c ll v llrll.6. w I I IIVzll·

On the other hand, we have that

\In 02VOlwoiz dX \ � I I 02V IILoo IIVw ll l l.6. z ll ·

Moreover for

and for

2 < r < '25 '

r 2: % ,

The remaining terms i n (5.73) can b e estimated similarly. Finally, the second con clusion follows from imbedding theory as in the proof of the previous lemma. _ The weak solution of ( 5 . 7 0) is a function that

{

111

E L 2 ( 0, Tj Ha) n L<Xl ( 0, Tj H1 )

such

(OtV1l1 (t), V v) + J (1l1(t), 1l1(t), v ) +p( .6. 1l1(t), .6. v) + (g(t ) , V )C(H-2(n),H�(n» = 0, v v E Ha ( !1), ° < t � T, x E n. 1l1 (X, 0) = 1l1o(x),

(5.74) It is shown in Guo, He and Mao (1997) that if 1l1o E HJ ( !1) and 9 E L 2 ( 0, Tj H- 2 ), then (5.74) has a unique solution. The solution satisfies the conservation

Spectral MetJ; ogs .and Their Applications

216

We can adopt Legendre approxi � ations or Chebyshev approximations to solve (5.74) numerically. For the sake of simplicity, we focus on the Legendre spectral method. Let lP�f = lPN n H� (O) and

(v, wh = (Vv, Vw), (v, wh = ( b. v, b. w),

v

v, W E HJ (O) , V v, w E H� (O) .

It can be checked that in the space H� (O), the inner product (v , wh here is equal to the inner product a 2 (v, w) in Section 5 . 1 , and I v l 2 = lI b.v l l . The orthogonal projections P;:;'o : HO' (O) --+ lP�o , m = are two such mappings that for any v

E

1 , 2,

HO' (O),

(v - p;:;,o v , ¢>L = 0,

V ¢> E lP�o , m = 1 , 2. It is not difficult to sh�w that for any v E HJ (O) , I l v - Pk o v l 1 1

--+ ° as N 00 . Clearly the projection p 2 ,0 is the same as the projection p;/ in Section 5 . 1 . Now let tPN be the approximation to W. A Legendre spectral scheme for (5.74) is to find tPN(X, t ) E lP�o for all t E f4 (T) such that

{

--+

(Dr VtPN(t), V¢» + J (tPN(t) + orDr tPN ( t ) , tPN(t) , ¢» +/1- (b. (tPN(t) + O' rDr tPN(t)) , b.¢» + (g(t), ¢» = 0,

tP(O ) = tPN,O(X),

where 0' , (j are parameters, ° ::; 0', 0 ::;

(5.76)

1 and

I I W o - tPN, o l l l -+ 0, as N --+ 00 . (5.77) 0 1 For W o E HMO), we take tPN,O = PJII WO. While for W o E H� (O) , we take tPN, O = Pi/W o o Both of them satisfy (5.77). Thus the approximations on the initial level are well defined. Now, assume that tPN(t - r ) is known. Let r be suitably small and A(v, w) = (Vv, Vw) + /1-O'r ( b. v, b. w) + or J (v, tPN(t - r) , w) , V v, w E lP�o . By Lemma 5.9, A(v, w) is a continuous bilinear elliptic form on lP�o x lP�o . Hence by Lemma the numerical solution at the time t is determined uniquely. So (5. 76) has a unique solution as long as W o E HJ (O) and 9 E L2 (0, T; L2(0) ) . Especially, if 0' = 0 = then we deduce from ( 5 . 7 ) and ( 5 7 ) that

2.1 ,

�,

�

1

.6

IIVtPN(t ) 11 2 + /1-r L I Ib.V'N(s) + b.tPN(S + r ) 1 I 2 +r

seRr(t - r )

L (g(S), tPN(S) + tPN(S + r) ) ds = IIVtPN,0 11 2

seRr(t - r )

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

217

which is a reasonable simulation of (5.75). In practical computations, a good choice of basis functions is essentially important. Shen proposed a nice set of basis functions. Let Lm (Y) be the Legendre polynomial of degree m. Set

(1994)

Then

lP�P = {(Pi(x) 1 0 :S It , 12 :S N - 4}.

B y the orthogonality of Legendre polynomials, both the matrix with the elements ((Pz,
which simplifies the calculation and improves the condition number of the related matrix essentially. The numerical solution ¢N( x, given by (5.7 ) is a discrete approximation to w ( x, But we can creat a continuous version by the interpolation in time Define two step functions w� ) (x, : [0, TJ -t lP�o , m = such that for E R.,.(T) ,

t)

t).

t)

6 1, 2,

t.

t

( 1996) proved the following results. Theorem 5 . 6 . Let W o E HJ (fI) , g E C(0, Tj L2) and ¢N,O = Pj/ w , :S �:,C. being a certain positive constant independent of N and Then w�) , m 1, 2, converge to the weak solution W in the following sense: (i) w�) W in L 2 (0; Tj H6 (fI)) weakly, as N 00, 0; (ii) w�) W in LOO (O, Tj H6 (fI)) weak-star, as N 00 , 0; Guo and H e

o T

T.

-t

-t

-t

=

T -t

-t

T -t

21

8

Spectral Methods �d Their Applications

(iii) I}i�) --+ I}i in L2(0, Tj HJ ( n )) strongly, as N

If in addition 7

=

--+

00,

7 --+ O .

O (N-4), then

(iv) I}i�) --+ I}i in L2(0, Tj HJ(n)) strongly, as N --+ 00, 7 O . I t i s noted that i f 0' , 6 > !, then the conclusions (i)- (iii) hold for any 7 = (N-4). If in addition l}io E HJ(n) and ¢N,O p;/l}i o , then the conclusions (i) -(iii) are valid for any 7 O . Furthermore, if 7 = (N-4) or l}i o E HJ ( n ), ¢N,O = p;/ l}i o , 7 0, --+

CJ

CJ

then the conclusion (iv) holds for 0' = 6 = 1. If I}i is smooth, then provides the numerical solution with high accuracy. Let qo 0 and e be a positive constant chosen below. Set =

--+

--+

(5. 76)

�

E (v, t) = IIV'v(t) 1I 2 + J.! 7

(0' + �) lI�v(t) 1I 2+ � J.!7

�

aeR.(t-T)

( lI�v( s ) 1I 2 + q07 I1DT V'v( s ) 1I 2 ) .

(5.78)

Theorem 5 . 7. Assume that . (i) I}i

E

L2 (0, Tj WO n HJ) n HI (0, Tj Wl ) n H2 (0, Tj Hl) , wo and ¢N,O = p;/l}io ;

E

HJ( n ) , g

E

(ii) ro � rl 2 ; (iii) one of the following conditions is fulfilled (a) 0' > � ' 7 = (N-'\ ) , A � max (�, 7 - 2ro ) , (b) 0' = � ' 7 (N-4 ) , 2 I ) 0' < "21 ' 7 N � ' being a positive constant such that for any J.! c�(1 _ 2 0' ) C(O, Tj L2)

=

0

=

( C

4

CJ

Cp

v E IP�o , II� v ll � cpN2 11V' v ll .

J.!,

Then there exists a positive constant bI depending only on n and the norms of and g in the mentioned spaces, such that for all t E RT (T), If 0' > � and 6 is suitably large, then the above result is vaild for any 7 --+ O .

I}i

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

2 19

Proof. LeOliN = P't;° ifl . We derive from (5.74) that

{

(D,. V'WN(t), V'ifJ) + J(WN(t) + OrD,. W N(t), WN(t), ifJ) + j.£(�(WN(t) + urD,.wN(t)), �ifJ) o = -(g (t), v) + B(w(t), WN(t), ifJ), V ifJ E IP� , 0 < t ::; T, W N( O ) = p;/ Wo, x E fl

(5.79 )

where

(V (D,.iflN - 8tifl ) , V ¢1) + J(iflN + 8r D,. iflN, iflN, ¢1) - J(iJ!, ifl, ¢1) + par (D,. � ifl, � ¢1) . Let W = .,pN - iJ!N. We obtain from (5.76) and ( 5 7 9 ) that (D,. VW (t) , V ¢1) + J (W (t) + 8r D,. W (t) , iflN(t) + W et), ¢1) + J (iflN (t) + 8r D,.iflN (t ), W et), ¢1) +o p (� (W (t) + ar D,. W(t») , � ¢1) = - B (ifl(t) , ifl N(t), ¢1) , 'V ¢1 E IP� . By taking ¢1 = 2 W (t) + e r D,. W (t ) and using an argument as before, we obtain from (5.71) that D,.I I VW (t)1 1 2 + r (e - 1) li D,. VW( t) 1 2 + 2pl l �W(t)1 I 2 +pr (a + n D,.II �W (t)1 I 2 (5.80 ) +pr2 (ea - a - �) I D,. � W(t)1 I 2 = � Gj (t) .

where

G1 (t) = -r(e - 28)J (W(t), W(t),D,.W(t») , G2 (t ) = -r(e - 28)J (W (t), iJ!N (t ), D,. W (t») , G3(t ) = -2J (iflN(t) + 8r D,. iflN(t), W et), W (t») , G4 (t ) = -fod (iflN (t) + 8rD,.iflN(t), W(t) ,D,. W(t») , Gs(t ) = 2 ( D,.iflN(t) - 8tifl (t), �W (t») - for (V ( iflN (t) - 8tifl (t» ,D,. VW (t») , G6 (t) = -2par (D,.� ifl(t ) , � W(t») - fopur 2 (D,.� ifl (t) , D,. � W(t») , G7 (t) = -2J (iflN (t) + 8rD,.iJ!N (t) , iflN(t), W et») + 2J (ifl(t), ifl(t ), W(t») , Gs (t) = -erJ (ifl N (t) + 8r D,. ifl N(t ) , iflN(t), D,. W (t») + er J (ifl (t), ifl(t), D,.W(t») .

220

Spectral Methods aDd Their Applications

I Gj (t) l .

c

D

Let > ° and be an undetermined positive We are going to bound Lemmas and 5.8-5.10, we have that constant. By (5.71)

I G1(t)1 S 2TI � - 281 \ VW (t)\J I �W(t) I I DT� W(t ) 1I S coT I DTVW(t)1I 2 + CITN4 I1 VW(t)1I 2 1 � W(t) 1I 2 , I G2 (t)1 S 2TI � - 2ol l � wN(t)1 1 ( I VW (t) I I �W (t) I I DT VW (t) I I DT�W(t)\1 ) � 2 (Vrl � - 20II l � WN (t)I I I \ � W(t)l \ t I DT VW (t) l I t) (T I VW(t)I I I D T�W (t) l I ) t S T I V W (t) I I DT�W(t) 1I + T(� - 20)21 1 � WN (t)1 1 2 1 � W(t) I I DT VW (t) 1I S cO IlT2 I1 DT�W(t)r + 4coll1 I V�(t)1I 2 + cO T I DT VW(t)\r + C2 T I �W(t)\r , IG3(t )1 S 2 \J (W (t) , W (t) , WN(t) + 8TDT WN(t)) \ S 2D I V W (t) I I � W (t)ll l I � (WN(t) + OTDT WN(t ) )11 S D I � W(t)11 2 + c3 1I VW (t ) 1I 2 , 1

2

_

I G4 (t)1 S 2�T I VW (t) l I ! I � W(t)lI ! (I I DTVW(t) lI ! I DT�W(t) lI t ' I � ( WN(t) + OTDT WN (t))1 1 + I DT�W(t)II I V (WN(t) + OTDTWN(t))l l t I � (WN(t) + OTDT WN(t)) I ! ) S 2�TI I WI I C(o,T;H2) (II DT VW (t) lI t I DT�W(t) lI t I V � (t) I ! I � � (t) lI ! + I DT�W (t) I I VW(t) lI ! I �W (t) l I t) S 2�TI I Wl l c(o,T;H2) I DT VW (t) 1I 2 I 1 DT�W (t) 1 I 2 I1 VW (t) 1 I 2 1 � W(t) 1I 2 + cOIlT2 I1 DT�W(t) 1I 2 + c:: I l wl l �(O.T;H2) I VW (t) I I �W (t) 1I S 2�TI I WI I C(o.T;H2) I DT V � (t) l I ! I DT � � (t) l I ! I V � (t) l I t 1I � � (t) lI ! + cOIlT2 I1 DT�W(t) 1I 2 + D I �W(t) 1 I 2 + c4 1I VW (t) 1 I 2 S 2coIlT2 I1 DT�� (t) 1I 2 + 4C�Il Il VW (t) 1I 2 + cO T I DT V � (t) 1I 2 + D I � W(t)1I 2 + c4 I1 VW (t) 1I 2 + C4T I �W(t ) 1I 2 1

1

1

1

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

where

221

Cl = c1o ; ( e - 2D) 2 , C3 = IIwll � (o,T ;If2 ) ' -c

�

I Fs (t) 1 and I F6(t) l . It can be verified as before that IID1'WN(t) - otw(t) 1I 2 ::; IID1'WN ( t) - D1'W (t) 11 2 + IID1'w (t) - otw(t) 1I 2 +1' 21' r+1' w ( s) 2 ds . (5.81) ::; -;;:2 itr 11 0. (w (s) - wN(s))1I 2 ds + "3 it I o� 11 A similar estimate is valid for 11 '\7 (D1'wN(t) - Otw (t» 1I 2 . Thus 1' I Gs ( t) 1 ::; D 11L).� ( t) 11 2 + cOT IID1' '\7� ( t) 11 2 + � + 110. ( W (S) - wN(s» 11 2 ds + CS * 110. (W(S) - wN ( s» l I� ds + CsT * II O�W ( s) 11 2 ds 1' + CST 2 + II O� W ( s) l l : ds

We next estimate

[

1

where

Cs =

[ max ( e, � ) . Clearly 2�0

1

1

t I G6( t) 1 ::; D 1 1 L).� ( t) 11 2 + CO/1- T 2 1ID1'L).� ( t) 1 1 2 + l1;T +1' lIo.w(s) lI� ds where Ca = /1- (1' 2 of Lemma 5.10,

(� + 4�0 e T 2 ) . Finally, we estimate I G7(t) 1 and I G8 (t) l . By virtue

2 1 J (WN(t) + DTD1'WN(t) , W (t) - WN (t), � ( t») 1 ::; 8cllwl l c(0,T ;H2 ) IIW (t) - WN(t) lI f IIW (t) - WN(t) lIl llL).� (t) 11 ::; D 11L).�(t) 11 2 + c7 ,1 Ilw (t) - wN(t)ll l ll w(t) - wN (t) 1I 2 16 ' . t Lemma where C7, 1 = c 11 W 11 2C T 2 . Accordmg 5.9, D

(0, ;H )

( 5 . 82 )

�

2 \ J ( wN(t) + DT D1' W N ( t) - W (t), W (t), � (t») I ::; 4c llwll c(0,T ;H2) 11L).� (t) 11 IIW (t) - WN(t) ll f IIW (t) - WN(t) ll l + DT IID1'WN (t) 1I 2 ::; D 1IL).� ( t) 1 1 2 + C7,1 1IW (t) - WN (t) III IIW (t) - wN(t) 11 2 t + C7,2 T itf +1' II 0.W ( s) 11 22 ds (5.83 )

(

)

Spectral Methods ahd Their Applications

222

� 52cI I 1l1 11�(o,T;H2)' Let C7

where C7, = 2 to (5.83) leads

=

2C7,1 + C7, . 2

The combination of (5.82) and

t

r IG7 (t) 1 ::; 2D I tJ.W (t) 11 2 + c7 11 1l1(t) - 1l1N(t)1I1 1l1l1(t) - 1l1N(t) 11 + C7T + 118.w(s)ll� ds. 2

I Gs (t) 1 <

Similarly,

2CO J1,T 2 1I DrtJ.W(t) 1 1 2 + cs Il W (t) - WN(t ) 111 1l1l1(t ) - WN(t) 11 2 r + CS T itr + 11 8.w(s)11 22 ds

: (2 5 2 )

where Cs = cC + II w ll�(O,T;H2)' We now take c J1, previous inequalities into (5.80 ) . Then

D=

:2

and substitute the

Dr I I V7W(t)l r + T (e - 1 - 4co ) II Dr V7W (t) 1 2 + � II tJ.W(t) 11 2 + J1,T 0- + Dr I tJ.W(t)1 1 2 + J1,T 2 - 0- - - 6co I DrtJ.W(t) 11 2 ::; A1 1IV7W (t) 11 2 + A2 (t) IItJ.W(t) 1 2 + A3 (t)

( �)

(eo-

�

)

(5.84)

where

e as follows. For 0- > ! , we take 2 12 max ( 1 + 4co + qo , 0- + co ) . 20- - 1

We next choose the value of

Then

e=

6 =

eo- - 0- - !e - 6co � 0 and so

T (e - 1 - 4co ) I I Dr V7W(t)11 2 +J1,T 2 - 0- - - 6 co IIDrtJ.W(t)11 2 � qoT IIDr V7W (t) 1 1 2 .

(eo-

�e

)

( 5.8.5 )

Cha.pter 5. Spectral Methods for Multidimensional and High Order Problems

0" �2 and 7 � 0 ( N-4 ), then we take

223

If =

� { = 1 + 4eo + qo + JLC� 7N4 G + 6eo) 2 and so (5.85) also holds. Finally for < � and 7 N4 < 2 JL cr> ( 1 {

2

0"

20"

)

' we take

{ � 6 = ( 1 + 4eo + qo + JLc� 7N4 (0" + 6eo ) ) (1 + I'c� 7N4 (0" _ �)) -

1

a.nd so (5.85) is still true. Let

p (t) = 7 By summing (5.85) for t E

A3 (S).

R.,.(T), we obtain that

E ( � , t) � p (t) + 7 If

L

seR. (t-.,.)

L

.eR..{t-.,.)

(A 1E ( � , s) + A2 (S) I ��(s ) r) ·

(5.86)

7 is suitably small, then

By virtue of Lemma 3.2, if for certain

t 1 � T, (5.87)

then for all

t � t1 ,

In order to get the convergence rate, we only need to estimate (5.87) . Clearly by (5. 12),

p (t)

and to check

p(t) � b1 (7 2 + N3- 2ro + N-4 + 7N- 2 ) • Furthermore if 0" � � and 7 = 0 (N- 4 ), then (5.87) holds for t 1 = T and N large . 1 enough. If 0" > 2 ' then the values of { and all do not depend on 7 N4 • By Lemma 3.2, we only require A 2 (t) � 0, i.e., Cj

(5.88)

Spectral Methoids and Their Applications

224

being a positive constant independent of N and T. Since � 2, a sufficient is that T = o(N->' ) , ).. � max 7 - 2 . Finally, the above statements together with. the triangle inequality lead to the first part of this theorem. then we can take e = 28 and so C = = O. If in addition 0' and 8 � Therefore the condition ( a) can be weakened to any T -> O. •

b

2 condition for (5. 8 8)

(�,

�,

>�

ro)

ro

l C2

Theorem does not provide an optimal estimate. But if \[f is a little smoother, then the optimal estimate exists, stated as follows.

5.7

Theorem 5 . 8 . Assume that (i) condition (i) of Theorem 5.7 holds; (ii) ro � rl > 2, or ro > rl = 2 and E LOO (O, T; Hr) for some r E ( 2 , ro); (iii) one of the following conditions is fulfilled (a) 0' > �, T = (N->') , ).. � max (�, 6 - 2ro) , (b) 0' � 'T 0 (N-4 ) , (c) 0' "21 ' TN4 :::; Jl C� (12 20' ) Then for all t E k,.(T), \[f

0

=

=

<

_

\[f

where b3 is a positive constant depending only on Jl , n and the norms of and g 1 in the mentioned spaces . If in addition 0' > "2 and 8 is suitably large, then the above result holds for any T O . Proof. We rewrite (5.80) as Dr II\7W (t) 1I 2 + T ( e - 1) IIDr \7W (t) 1 I 2 + 2 Jl II � W (t) 1I 2 + Jl T ( + £) Dr I W (t) 11 2 + Jl T 2 (e O' - 0' �) I Dr�W(t) 1 1 2 = Rl ( t ) + R2 ( t ) (5.89) ->

0'

-

with

6

L Gj ( t ), R2 (t) = G7 (t) + GS (t). R1 (t ) = j=l

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

I R1 (�)1

Since has been handled in Theorem 5.7, we need only to estimate precisely. If > then E LOO(O, Tj for E We have

r1 2,

w

Hr ) r (2, r1 ) ' 4 G7 (t) =. jL=l Gr,j (t)

225

I R2 (t)1 more

where

G7,1 (t) = 2J (WN(t) - W (t), W(t) - WN(t), q,(t)) , G7,2 (t) = 2J (w(t) + t5-r DTw(t), W(t) - WN(t), q,(t)) , Gr,3 (t) = 2J (w(t) - WN (t) - t5-r DT WN(t ), W(t), q, (t)) , G7At) = 2t5-rJ (DT WN(t) - DTW(t) , W(t) - WN(t), q,(t)) . By Lemmas 5.8 and 5.12, we obtain that

I G7,1 (t)1 ::; 4 1 � q,(t) l l I V' (W(t) - wN(t)) l l � (W(t) - WN(t))1 I ::; D 1 � q, (t) 1 2 + C7,3 1 I W (t) - WN(t)l I � and

I G7,2 (t)1 ::; 2c Il w (t) + t5-rDT W(t)l l r I W (t) - WN(t)1 1 1 1 � q,(t) 1 ::; D 1 � q,(t) r + C7,4 I 1 W (t) - WN (t)l I � where C7, 3 = � I W llioo (O , T ; H2 ) and C7,4 = � I W l i oo (O , T ; H r) ' Moreover by ( 5.71 ) and Lemma 5 . 1 1 ,

Since

I DT WN(t)1 I 1 ::; -r- � (1* I O.WN(S)I I � dS) � ::; -r- � ([+T I O.WN(S)I I � dS) � and I l otWN(t)1 I ::; I l otw( t)1 I ' we see that 2 2 I G7,3 (t) 1 ::; D I � W- (t) 1 2 + C'i,4 I 1 W (t) - WN (t)1 I 21 + Cr,s-r itr+T l I o.w(s)1 1 22 ds

226 where

Spectral Methods

and'

Their Applications

C7,S = 282C7,4 ' Using Lemma 5.9, we deduce that

. l i D,. (llT(t) - IlTN(t)) ll i � D 1 � q, (t)1 1 2 + C7, S T 2 1 1 � (llT(t) - IlTN(t))1 I 2 1 I D,. (llT(t) - IlTN(t))l l l . l i D,. (llT (t) - IlTN(t))1 I 2 +,. � D I � q, (t) r + C7, S T I � (llT(t) - IlT N(t)) 1 1 2 ([ 118. (llT(s) - IlT N(S )) I � dS ) � . (1* 118. (llT(s) - IlTN(s)) I � dS) < D I � W (t) 1 2 + C7,S [+,. 11 8. (llT(s) - IlTN(s)) ll ; ds + 41 c7,sT 2 1 � (llT(t) - IlTN(t)) 1 I 4 i(t +,. 118. (llT(t) - IlTN(t)) ll � ds 82 where C7 ,6 = c ' The above statements lead to D I G7 (t)1 � 4D 1 � q, (t) 11 2 + c7 111lT(t) - IlTN(t) l I � + C7 It t+,. 118. (llT(s) - IlTN(s)) I � ds + C7T It t+,. 1 8. IlT(s) 1 I 22 ds + C7T2 11� (llT(t) - IlTN(t))1 1 4 1t t+,. 118. (llT(s) - IlTN(s))1 1 22 ds where C7 = max (C7 ,3 + 2 C7 , 4 , C7, S , C7 ,S ) ' Similarly, 1

1

2

where

Cs

=

(C7, 7 + 2C7,S , C7,9 , C7,10) and 4e ce 2 T',Hr) , C7,7 = -l ' coil l llT llioo(O'T,'H2) , C7,S = -4coil-I l IlT I Loo(o

max

Now, by taking = obtain from ( 5.89 ) that

D i6

C7,1 0 = c482oe . ll

and using an argument as in the derivation of ( 5.84 ) , we c

D,. 1 \7q, (t) 1 2 + T(e - 1 - 4co) I D,. \7q, (t) 1 2 + � 1 � q, (t) 1 2 + IlT (0- + �) D,. I � q, (t) I 2 + Il T 2 (e o- - 0- - � - 8co) I D,. tlq, (t) I 2 � Al I \7q, (t) I 2 + A2 (t) I � W (t) I 2 + A3 (t), ( 5.90 )

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

227

where

Let

qo ;::: 0

and choose

�

as follows. For

(1' > 21 ' we take

)

� ;::: 6 = max 1 + 4co + qo , 2(1'2(1'+-16co 1 . Then

�(1' - (1' - 21 � - 8co ;::: 0

(

and so

(

�

7"(1-�-4co ) li D.,. V � ( t)1 1 2 +1'7" 2 �(1' - (1' - � - 8co 7" = 0 ( N-4 ) , then we take

If (1' =

�

and so

(5.91) also holds.

and

) I D.,.6�(t)11 2 ;::: qo7" li D.,. V �(t)11 2

.

(5.91)

Finally, for (1' < � and

2

2 4 . 7" N < Jt cr>2 ( 1 - 2(1' ) ' we put

and then (5.91) also follows. Now, by the same procedure as in the proof of Theorem 5.7, we obtain from (5.90) an estimate similar to (5.86). In particular, we have

p (t ) = 7"

L: A3(S) sERr(t-.,.)

=

0

(7" 2 + N2 - 2ro + N- 2rl + 7" N2 - 2rl )

•

and

Spectral Methods Their Applications

228

T = ( N-4 ) , then for all t E R... (T) , p(t) ::; � 2�TN . On the other hand, for > �, we require that A2(t) ::; 0 , i.e. , ) ::; b4 , T ::; b4 , T N4 (T + +T b4 being a positive constant independent of and T. The proof is complete. We now turn to numerical experiments. For describing the errors, let x �jq ) and w{jq) be the Legendre-Gauss-Lobatto interpolation points and weights. The error If

0"

::;

�2

and

0

N 2 - 2 To + N- 2 T,

2

0"

N2 - 2 T,

N

_

E( tPN, t) is defined as

1 eN (1 + cos ) (1 + cos 2) (5. 92) 10 The numerical results of scheme (5.76) are presented in Tables 5.1-5.3. The first table shows the high accuracy of (5. 7 6) even for small and the good stability in long time calculation even for large T. Table 5. 2 indicates that when -+ and 0 proportionally, the numerical solution converges to the exact solution. Table 5. 3 shows that the computations are also stable for very small even if = O. Table 5.1. The errors E(tPN, t) , fl = 0. 5 , 0" = 0. 5 ,b = O , = 10. T = 0. 5 T = 0.1 T = 0. 0 5 T = 0. 0 1 T = 0.005 T = 0. 0 01 t = 10 2. 497E - 2 4. 9 95E - 3 2. 4 97E - 3 4. 9 95E - 4 2. 4 97E - 4 5.000E - 5 t = 20 2 . 4 85E - 2 4.969E - 3 2. 4 84E - 3 4. 9 70E - 4 2 . 4 85E - 4 4. 9 83E - 5 t = 30 2. 438E - 2 4.880E - 3 2. 440E - 3 4. 8 79E - 4 2. 4 39E - 4 4. 8 76E - 5 In the previous part, we considered the two-level spectral scheme (5.76). Since we I

We take the test function

W(x, t)

t

=

7r X l

7rX

•

N,

T

N

-+

fl,

00

fl

N

adopt partially implicit approximation for the nonlinear term, the convergence rate in time t is of first order. It limits the advantages of spectral approximations in the space. A modified version is the following

{

(D ... \11/;N(t) , \1
�J ( 1/;N(t) + 1/;N(t + T ) , 1/;N(t) 1/;N(t T ) , E IP�/ , t E il... ( T) , ) L\
+

+L\ tPN( t + T 2 tPN(O) = P,;0 Wo,

,

-

x E n.

+

+

+

Chapter

5.

Spectral Methods for MuItidimensional and High Order Problems

229

Table 5.2. The errors B ( 't/JN, 30), p, = 0.5, 0" = 0.5, 8 = O. N=8 N=6 N = 10 7 0.1 6. 151E - 3 4.920E - 3 4.880E - 3 7 = 0.01 3.179E - 3 5.367E 4 4.879E - 4 7 = 0.001 3.091E - ·3 1.301E - 4 4.486E - 5 =

-

able 5.3. The errors B ( 't/JN' 20) , 0" = 0.5, 8 = O , N = 1 O. p, � 0 p, = 10 - 3 P, = 10- 4 7 = 0.01 4.325E - 4 4.418E - 4 4.452E - 4 2.153E - 4 2.224E - 4 2.296E - 4 7 = 0.005 4.214E - 5 4.248E - 5 4.272E - 5 7 = 0.001 This scheme is of second order in time. But we have to solve a nonlinear equation at each time step. It costs a lot of computational time and brings some difficulties in theoretical analysis. It is well known that if we use certain reasonable prediction correction schemes for nonlinear problems, we could get the accuracy of second or der in time and evaluate the values of unknown functions explicitly, e.g. , see Guo (1988a) . Following this idea, we construct a Legendre prediction-correction procedure for (5.74) . Let 't/Jr;.(x, t) be the predicted value of 't/JN(X , t). The prediction-correction scheme is as follows �

( V ('t/J�) (t + ) - 't/JN(t)) , V ¢;) + J ( 't/JN(t), 't/JN(t), ¢;) + � (� ('t/JN(t) + 't/J�) (t + ) ) , � ¢;) + (g(t) , ¢;) = 0, V ¢; .E IP�o , � (V('t/JN( t + ) - 1/JN(t)),. VI/» + � J(1/JN(t), 1/JN(t), 1/» + � J(1/JW)(t + ) 1/JW)(t + ) 1/» + "2p, (� ( 't/JN(t) + 't/JN(t-+ )) � ¢;) + 2"1 (g(t) + g et + ) ¢;) = 0, V ¢; E IP�o , 't/JN( O ) = PN2 0 Ilio, x E !l. 7

7

r

r ,

7

,

r ,

7 ,

(5.93) 4 Let E( v , t) be the same as in (5.78), and r = CJ ( N- ) . It can be proved that if Ili E L2 (0, Tj H ro n HJ) n H I (0, Tj W l ) n H 2 (O, Tj H 2 ) n H3 (0, Tj H I ) , Ilio E HJ (!l) and g E H2 (0, Tj L2), then there exists a positive constant bs depending only on p,, !l and the norms of Ili and g in the mentioned spaces, such that for all t E i4(T) , E (Ili - 't/JN , t) :$ bs

( 4 + N2-2ro + N- 2rl + r N2 - 2rl ) . 7

We now present some numerical results of scheme (5.93). The definition of error

Spectral Methods and Their Applications

230

E (t/JN, t) is the same as in the previous paragraph. We first take the test function (5.92) and list .the errors in Tables 5.4-5.6. The first table shows the high accuracy even for small N, and the good stability in long time calculation even for large Table 5.5 indicates the convergence of sch�me (5.93). By comparing Table 5.2 and Table 5.5, we find that (5.93) gives more accurate numerical solutions than (5.76). Table 5.6 indicates that (5.93) is stable for small 1-', even if I-' = O. r

T.

Table 5.4. The errors E(t/JN, t}, I-' = 0.5, N = 14. t=4 t=2 t=5 t=1 t=3 3.919E - 6 4.331E - 6 4.782E - 6 5.279E - 6 5.825E - 6 = 0.1 5.905E - 8 6.524E - 8 7.208E - 8 7.964E - 8 8.798E - 8 T = 0.01 = 0.001 6.576E - 10 7.189E - 10 7.873E - 10 8.635E - 10 9.482E - 10 T

T

Table 5.5. The errors

T T T

E(t/JN, 30), I-' = 0.5. N=8 N = lO N = 12 = 0.5 3.612E - 4 3.933E - 4 4.070E - 4 = 0.1 9.155E - 5 3.675E - 5 3.712E - 5 = 0.01 9.672E - 5 1.749E - 6 8.469E - 7 Table 5.6. The errors

E(t/JN, 20), N 10. 1-' = 0 = Hi-3 I-' = 10-4 = 0.04 3.487E - 6 2.103E - 5 2.114E - 5 = 0.025 2.861E - 6 1.658E - 6 1.469E - 6 =

I-'

T T

We next take the test function

We use (5.76) and (5.93) to solve (5.74) with I-' = 0.05, a = 1, b = 0.01 and N = 12, and list the errors in Table 5.7. It is shown again that scheme (5.93) gives much better numerical results. In particular, scheme (5.93) is very stable in long time com putation, even for small N and large T .

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

231

Table 5.7 The errors E('l/JN , t). scheme (5.76) scheme (5.93) T = 0.1 T = 0.4 T = 0.01 t = 20 2.694E - 6 3.309E - 5 3.853E - 4 t = 60 1.625E - 7 2.588E - 6 1 .613E - 4 t = 100 3.559E - 8 5.712E - 7 9.883E - 5 Bernardi, Coppoletta and Maday (1992a, 1992b). studied the Legendre spectral methods for a two-dimensional steady fourth order problem. Shen (1994) developed an efficient direct solver for one-dimensional steady fourth order problems.

5.4.

Spectral Domain Decomposition Metho ds

In this section, we consider spectral domain decomposition methods. The spatial computational domain is divided into several adjoining, nonintersecting subdomains, within each of them we look for a polynomial solution. Some fulfillments are required on the common boundaries of adjoining subdomains. This approach stems from its ca pability of covering problems in complex geometry. Orszag (1980) , and McCrory and Orszag (1980) studied the patchings of complex domains. Moreover, this approach also allows local refinement to resolve internal layers or discontinuities, maintaining however the spectral accuracy enjoyed by the standard spectral methods. In the previous sections, we already used the Fourier spectral and Fourier pseudospectral methods for an elliptic system, and the Legendre spectral method for a high order parabolic equation. In this section, we take a quasi-linear hyperbolic system as an example to describe the Chebyshev pseudospectral domain decomposition method, and prove the convergence for a special case. The numerical results coincide with the theoretical analysis. Let n be an open bounded domain in lRn . U and f are two vector functions with the components U (j ) and f U) , 1 � j � p. A q (U) are p x p matrices, 1 � q � n . Consider the problem

{

�

Ot U + Aq (U)oq U = f, U(x, O ) = Uo(x),

x E n, 0 < t � T,

(5.94)

Spectral Methods and Their Applications

232

with suitable boundary conditions on an x (0, T] . For any e E define the characteristic matrix for the direction e as

JR'"

such that l e l 2 = 1 ,

"

A W = L A q (U)eq . q= 1

(5.94) is assumed to be hyperbolic in time. It means that for any such e, A(e) has P real eigenvalues and moreover it is diagonalizable. Let ).,. be the eigenvalues of A(O and V" be the corresponding left eigenvectors, 1 $ k $ p . Then Suppose that ). " > 0 for 1 $ k $ Po and ),,. inner product of Vk with (5.94), we obtain

(5.95) <

0 for Po + 1 $ k $ p. By taking the

(5.96) Denote by (T1 , . . . , T,,_1 ) the system of Cartesian coordinates of the hyperplane or thogonal to the direction e. Then for each = 1 , . . . , n - 1 , Th = (Th" " " Thn ) E JR" , Th . e = 0 and I Th I = 1 . Owing to the identity

h

,,-1

Oq U = eqOe U + L Th .O'Th U, h=1 it follows from (5.96) that

We refer to (5.97) as the compatibility equations for (5.94). Now, let be an ( n - l ) -dimensional manifold, and v be the unit normal direction to Taking e = v, (5.97) restricted to becomes

r.

r

r

r,

The right-hand side of (5.98) depends on the tangential derivatives of U on while the left-hand side yields a combination of transport equations along a direction that is normal to If is the boundary an and v is oriented outward n ( see Figure

r. r

Chapter

5.

Spectral Methods for Multidimensional and High Order Problems

233

5 . 1 ) , then, for 1 � ::; k ::; Po , ( 5 . 98) yields Po transport equations according to which information are propagated from the inside to the outside of O. These Po equations are called the compatibility equations for O. On the other hand, if r is the common boundary of the adj oining sub domains 01 and O2 , taking as II the normal direction to r, oriented from 01 to O2 (see Figure 5 . 2 ) , then the first Po equations of ( 5 . 98) are the compatibility equations for 01 , Obviously, for Po + 1 ::; k ::; p, (5 . 98) provides the compatibility equations for O2 ,

v

Figure 5.2.

Figure 5 . 1 .

The use of compatibility equations for single domain pseudospectral methods was � advocated in Gottlieb, Gunzburger and Turkel ( 1 982) , and Canuto and Quarteroni ( 1 987) . Gottlieb , Lustman and Tadmor ( 1 987a, 1 987b) developed the analysis of the stability and convergence for the case of dissipative boundary conditions . The issue of the compatibility equations at sub domain interfaces was addressed in Cambier , Ghazzi , Veuillot and Viviand ( 1 982) .

Now, let 0 be an n-dimensional hypercube, .0 = .01 U .o2 , .o1 n .o2 = r, 01 n 02 = iP where r is orthogonal to one Cartesian direction. We also assume for the moment that the solution U is continuous across r. Let be any positive integer , and 'EN be the set of Chebyshev- Gauss-Lobatto interpolation points of the reference hypercube [- l , l ] n . Next , let .om,N be the sets of the corresponding points in .om , and Om,N = .om,N n Om , m = 1 , 2 . Moreover, rN = .o1 , N n r = .o2,N n r, since we use the same number of interpolation points in .01 and .02 ,

N

Quarteroni ( 1 990) served a Chebyshev pseudospectral domain decomposition scheme for (5.94) . It is to look for U 1 ,N (X , t) E IPN for x E 01 an d u 2 N ( x t ) E IPN for x E O2 , satisfying a set of equations. Firstly, they fulfill that ,

8tU1, N + :L: Aq (Ul,N) 8qU1 ,N = j, q=l

,

n

x

E 01 , N ,

( 5 . 99)

Spectral Methodll ana Their Applications

234 "

8tU2,N + L: Aq (U2,N) 8qU2,N = q=l

I,

x

(5. 1 00)

E 0 2,N,

(5. 101)

In addition, we require that Vi.

+ Ak8vU1,N) = Vi. . - t (Aq (U1,N) I: Thq8ThU1,N ) q=l h=l

. (8tU1,N

(I (

Vk · (8tU2,N + A k8vU2,N) = Vk .

I-

I: Thq8ThU2,N t (Aq (U2,N) h=l

q=l

) ))

,

1 :5 k :5 Po ,

, Po

(5. 102)

+ 1 :5 k :5 p,(5.103)

where v is the unit outward normal direction to 0 1 on r, A( k) and V( k) are the eigenvalues and the left eigenvectors of the matrix A( v ) , and T is the tangential direction on r. Indeed, (5. 102) are the Po compatibility equations for 0 1 on r, while (5. 103) are the P - Po ones for O 2 on r . Furthermore at each point of fi1,N belonging to a "face" W of 80dr, we set Vi.

.

(8tU1,N + A k8vU1,N) = Vi. .

(I

-

� (Aq (U1,N) E Th. 8ThU1,N) ) ,

1 :5 k :5 Po ,

(5. 104) where v is now the outward normal direction to 0 1 on W, T is the tangential direction on W, while A k and Vk are the eigenvalues and the left eigenvectors of the matrix A( v ) . But here Po is not necessarily the same as in (5.102). The remaining P - Po equations that we need at each interpolation point of W must be provided by the physical boundary conditions that supplement (5.94) . Similarly, at each point of fi 2,N belonging to a "face" W of 80 2 /r, we enforce that

(5.105) where v is the outward normal direction to O 2 on W, T is the tangential direction on W , while A k and Vi. are the eigenvalues and the left eigenvectors of the matrix A( v ) . But here Po is not necessarily the same as before. The remaining p Po equations are prescribed as boundary conditions. -

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

235

It is n9ted that if the matrix A q is not constant, then the eigenvalues as well as the eigenvectors in (5.102) - (5. 105) can be generally expressed in close form in terms of v and the components of the physical variables. We now turn to the case in which r is no longer an arbitrary surface, rather the front of a shock propagating throughout n. Let n l and n 2 be the adjoining sub domains separated by r, and w be the speed with which r propagates in the direction v, v being the outward normal vector to n l on r as before. Assume that there exists a vector function F such that Aq (z) = oz Fq (z). Then (5.94) can be written in conservation form as .

OtU + L oqF (U) = /, q=l n

x

E

n, 0

� T.

(5. 106)

The weak solutions to (5. 106) are allowed to be discontinuous across the interface r, satisfying the Rankine-Hugoniot condition n

w [ U) - L Vq [Fq(U) ) = 0 q=l [z) denotes the jump of the values of z on the two sides of r .

where In the context of the above Chebyshev pseudospectral approximation, the above equation reads as n

W (U 2 ,N - U I , N )

- L Vq (Fq (U 2 ,N) - Fq (U I ,N) ) = o . q=l

(5.107)

At each interpolation point Xr E rN , (5.107) yields p equations for the 2p + 1 unknowns which are u(i) p and w . Thus we need p + 1 further in1, N ' u(i) 2 ,N ' 1 dependent conditions provided by the compatibility, which, in the current situations, can be determined as follows. Let us define the characteristic matrix for n l at the point on r, A (V( l » ) = f: Vq Aq (U I ,N)

< J. _<

q=l

and denote by A �l ) and Vj.( l ) the eigenvalues and the left eigenvectors respectively. Similarly denote by A �2) and Vj.( 2) the eigenvalues and the left eigenvectors of the matrix n A (v ( 2 » ) = - :E vq A (U 2 ,N ) .

q= l

· In general, UI , N =f. U 2 ,N on r and so A �l ) =f. _A �2) , Vj.( l ) =f. VP) . Assume that the eigenvalues are ordered as 1 �k�p-1

Spectral Metho.9s an d Their Applications

236

and that r is the surface of propagation of a k1-shock . 1

�

k

l�

P such that ( see Smoller ( 1 983 ) )

So there exists an integer

( 5 . 1 08 ) Figure 5 . 3 illustrates an example of a kl -shock . For fixed time straight line with slope

w,

t,

we draw in bold the

l 2Ai ), � Ai�) p, . �

the speed of the shock front . On its left , we label with k

1

the strai ght line whose slope equals the characteristic speed we label on its right the straight lines with slope

Figure 5 . 3 .

p=

3, kl

1

k

� p.

S�milarly,

= 1.

The compatibility equations for f h are given by the following

V,,(l )

k

atUl ,N + Ail) all(1) U1,N) Vio(l ) . (f - f (Aq (UI,N ) I: ThqaT.Ul,N) ) ,

p- k

l+

1 equat ions

(

.

=

q= l

h= l

� p,

kl

�

kl

+ 1 � k � p.

k

while those for fh are given by the k l equations

Vio(2) . (atU2,N + Ai2) all(2) U2,N) VP) ( f - f (Aq (U2,N ) 'f Thq aT.U2,N ) ) , h=l q=l {Th } =

As usual ,

.

p

-

is the set of vectors t angent i al to r at the p oint

tion. Note that ( 5 . 1 08 ) can be rewritten in the form \ (1) " ", - I

<

w

<

( 2)

\ " ", +1 '

Xr

under consi dera

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

237

We now consider the iteration by sub domain algorithm for the solution of the domain decomposition problem. For simplicity, assume that the solutions are contin uous across the interface r. Let D(v) and E( v ) denote the matrix of eigenvalues of A(v) and the matrix of the corresponding left eigenvectors, so that

E( v ) A(v)

=

D(v)E(v) .

(5.109)

Let U1,N,m and U2 , N,m be the m-th iterated values of U1,N and U2,N respectively. At each point x E rN, set Moreover, we denote by Epo (v) the Po x P matrix given by the first Po rows of E( v ) , and by Ep _ po ( v ) the (p - Po ) x P matrix given by the remaining p - po rows of E(v) . Then U1 , N,m+1 satisfies the interior equations (5.99), the interface equation (5. 102) together with

Xl,m = U l,N,m

Ep -PO ( V )U1 , N,m+ 1

=

Ep-po ( V ) x2,m ,

X

E rN ,

and finally the boundary equations given by (5. 104). Similarly, U2,N,m+ 1 satisfies the interior equations (5.100), the interface equation (5.103) together with and finally the boundary equations given by (5. 105) . It is noted that the limit solu tions U1 N = m--.. limoo U1 ' N m and U2 'N = mlim .... U2 N m fulfill the conditions I

I

oo

'

,

EpO ( V )U2 ,N = Epo ( V )U1 ,N ' Ep-po ( V )U l,N = Ep -po ( V )U2 ,N,

which in turn ensure the fulfillment of the continuity requirement (5. 101). We next consider a special case in which 0 = ( - 1 , 1 ) , 0 1 = (-1, a ) , O 2 = ( a , 1 ) , l a l 1 . Clearly r = { a}. The system (5.94) is reduced to

OtU + A(U)o",U = /,

x

E 0, 0 < t � T,

(5. 1 10)

while the compatibility equations (5.97) become (5. 1 1 1 )

Furthermore by (5. 109) ,

EotU + DEo",U = Ef.

<

Spectral Meth6ds and Their Applications

238

Obviously, D and E depend on U , if A does so. Suppose that at the interface point x = a , the first Po eigenvalues of A are positive, and the others are negative. Let Dpo and Epo denote the Po X P matrices obtained suppressing the first Po rows aID and E respectively. Then the following Po equations (5. 1 12) provide the Po compatibility equations for 0 1 . Similarly, denoting by Dp_ po and Ep_po the lower part of the matrices D and E, the equations (5.1 13)

p

provide the - Po compatibility equations for O 2 . The compatibility equations for O 2 at x = 1 are derived in a similar way, and take the form (5. 112). At x = - 1 , the compatibility equations for 0 1 take the form (5. 1 13). The system (5. 110) is completed by the initial condition

U(x, O)

=

Uo(x) , x E

0,

(5.1 14)

and by a set of boundary conditions at x = ±1. For instance, B1 is a Po x P matrix and B2 is a - Po) x P matrix, 91 and 92 are two given vector functions, and

(p

for x for x

=

=

-1, 1.

(5. 1 15)

We now apply the Chebyshev pseudospectral domain decomposition method to the Chebyshev-Gauss-Lobatto interpolation points in the (5. 1 10). Let yU) = cos reference interval [- 1 , 1] ' and

�,

At each time t, we look for U 1,N E JI>N for X E 0 1 and U 2 ,N E IPN for x E O 2 . They

Cbapter 5. Spectral Metbods for Multidimensional and Higb Order Problems

satisfy a set of equations as follows

_

OtUI,N + A (UI ,N ) Ox UI ,N = f, OtU2,N + A (U2,N) Ox U2 , N = f, Epo OtUI , N + Dpo Eox UI , N = Epo f, Ep - po OtU2,N + Dp _po EoxU2,N = Ep - po f, U l ,N = U2,N, Ep- po OtU1 ,N + Dp_po Eox U1,N = Ep - po f, Epo OtU2,N + DpoBox u2 ,N = Epo f, B1U1,N = gl , B2U2, N = g2 ,

239

x = x�j), 1 ::; j ::; N - 1 , x = x �j) , 1 ::; j ::; N - 1 , x= x= x= x = -1, x = 1, x = -1, x = l. 0,

0,

0,

Obviously, the first two equations in the above set are the internal equations. The next three equations are the interface equations. The remaining ones are the boundary equations. We now write the iteration. Let X 1,m = U1,N,m and X 2,m = U2,N,m at x = 0. Then we look for U1 ,N,m+1 E IP N in 11 1 , such that

x = x�j), 1 ::; j ::; N - 1 , OtU1 ,N,m+ 1 + AO",U l,N,m+ 1 = f, x = 0, Epo OtU1,N,m+ 1 + Dpo Eox U1,N,m+1 = Epo f, x = 0, Ep_po UI , N,m+1 = Ep -po X 2,m, Ep -po OtU1,N,m+1 + Dp_po Eo",U1,N,m+ l = Ep -po f, x = - 1 , x = -1, B1U 1 , N,m+1 = gl , where A , D and E depend on U1,N,m+ 1 ' Similarly, we look for U 2 , N,m+ 1 E IP N i n 112, such that

OtU2 ,N,m+ 1 + AOx U2,N,m+1 = f, Ep -po OtU2 ,N,m+ 1 + Dp_po Eox U2 , N,m+1 = Ep - po f, Epo U2 ,N,m+ 1 = Epo X 1,m , Epo OtU2,N,m+ l + Dpo Eox U2 ,N,m+ l = Epo f, B2U2 , N,m+1 = g2 , where A , D and E depend on U2,N,m+1 '

x = xU2 ) , 1 -< x= x= x = 1, x = 1,

J'

<

-

N - 1,

0, 0,

We shall consider the convergence of the above iteration for problem (5.110), (5. 1 14) and (5. 1 15). For simplicity, let p = 2, a be a constant, l a l < 1 and

240

Spectral Methods

and

,!heir Applications

). 1 = a + 1 > 0, ).2 ' = a - I < O. Let D = Diag {>. l ' ). 2 } and 1 1 1 E -- .J2 1 - 1 ' Clearly A = EDE . We require the initial value Uo ( x ) to be a continuous vector function with a first component U� l) (X ) vanishing at x = ±l. The boundary condition is that U ( l )( -1, t) = U ( 1 )(I, t) = 0 for 0 < t ::; T. Let T be the mesh size in time t, P = -T1 , ). = A 1 and p. = -A 2 > o. We use the Euler method for the temporal discretization. When we advance from the time t to the time t + T , we denote the values U1,N (X, t + T ) , U2,N (X, t + T ) , (3 U1,N (X, t) + f(x, t) and (3u 2,N (X, t) + f(x , t) by U1,N, U2,N, h ,p and 12,1' for simplicity. Then we obtain that x = x 1U) , 1 -< J. -< - N - 1 , (3U1,N + A8",U1,N = h,p , x = x 2(i) , 1 <- J. <- N - 1 , (3U2,N + A8",U2,N = 12,1" (5.116) (3E1U1,N + AE18.,U1,N = Ed1,p , x = a, (3E2U2,N - p.E28.,U2 ,N = E2f2,p , x = a, Then the eigenvalues

(

and

{

)

{ E2U1, N = E2U2,N , E1U2,N = E1U1,N,

x = a, x = a.

(5 . 117)

x = -1 , x = 1, (5.118) x = -1, x = l. At the ( m + l ) th step of the iteration, the solutions U1,N, m +1 and U 2,N, m + 1 satisfy (5.1 16) and (5.1 18). The matching interface condition (5.117) is dealt with as follows

N, m , { EE12U1,U2,NN,,mm+1+1 == E2U2, E1 U1,N,m ,

x = a, x = a.

(5.119)

The proof of the convergence of the above iteration is rather involved. It is con veniellt to carry it out using the characteristic form of the equations. To this end, define the error functions e"), , m = E ( U"), ,N,m - U"), ,N ) , I = 1, 2. Clearly,

Cha.pter 5. Spectral Methods for Multidimensional and High Order Problems

Since DE

= EA, (5. 116), (5.118) and (5.1 19) imply that x = x P) , 1 $. j $. N - 1 , x = -1, x = -1, x = o, x = o.

241

(5. 120)

Similarly, we have that

x = xU) x = 1, x = 1, x = o, x = o. 2

,

1

<

-

J. - N - 1 , <

(5.121)

We start by proving two important lemmas. Lemma 5 . 13. Let TJ > 0 and c be two given real numbers, and y (i) be the Chebyshev

Gauss-Lobatto interpolation points in the reference interval [-1 , 1] . The function

v E lP N satisfies

{V

- TJ8yv

v = c,

= 0,

y y

= y {i) , 1 $. j $. N, = y eO) = 1 .

(5 . 122)

Then there exists a constant UN( TJ) such that IUN( TJ) I < 1 and v ( - 1 )

= UN(TJ)c. Proof. Let 1/(y) b e the Chebyshev polynomial of degree I, 4> N -l ( y ) = 8yTN(Y) and 4> N ( y ) = y8yTN (y) . Since y (i) , 1 $. j $. N - 1 are the roots of 8yTN(y) , we deduce

from the first formula of (5. 122) that

(5.123) where e is a constant that can be determined by the inflow boundary condition, i.e. , the secong formula of (5. 122) . Let �-Y.I be the Chebyshev coefficients of 4>-y, N - 1 , N. Following Canuto (1988), we know that �-Y.I � 0 for 0 $. 1 $. N - 1 , N. Next, let
i, i =

i=

(5.124)

Spectral Methods

242

and Their Applications

and

(5.125) , Owing to

(5.123), we state that (5.126)

Taking into account

(2.72), we obtain from (5.124 ) and (5.125) that 0 5: 1 5: N

and 0 5: / 5: N. Clearly for T/ > 0, Moreover, since T,(l)

=

�f/" � 0, t/Jf/,' � 0, 1

for any I,

(5.126)

(5.127)

0 5: I 5: N.

leads to

and so sign e d) = sign e e ) . Furthermore, the solution of Chebyshev coefficients

(5.123)

has the following

(5.128) whence sign Put UN (T/ )

=

(vI ) = sign e e ) ,

0 5: 1 5: N.

�(-1)' (�f/" t/Jf/,') (� (�f/'P t/Jf/,p) )

Since 1/(-1) = ( _ I )', we obtain from ensures that !UN (77 ) ! < 1 for 77 > o.

+

-l

+

(5.128) that v(- l) = UN (T/ ) e.

Finally,

(5.127) •

Chapter

5.

Spectral Methods for Multidimensional and High Order Problems

243

Lemma 5. 14. · Let 1/, c, y U) and O'N.( 1/) be the same as in Lemma 5.13. The function

v E lP N

Then

satisfies

{ VV += 1/c,01lV = 0,

y = y U), 0 $ j $ N - 1, y = y e N) = - 1 .

(5.129)

v(1) = O'N (1/)c.

Proof. We first note that

(5.130) where the constant p has been determined by the second formula of (5.129) and the fact that OyTI(1) = 12, OyT( -1) = ( _ 1 )1+1/2• Consider now the following auxiliary problem

{ Ww -= 1/0yW = 0, c,

We want to show that A s a matter of fact,

w( y )

y = y U), 1 $ j $ N, y = y (O) = 1.

v( - y ) = w( y ), Iyl $ 1.

(5.131)

satisfies

p* =

c - 1/oyw(1) 2N2 '

and thus

w( - y) - 1/ 0Il W( - y) = p* (1 - y )OyTN ( - y) , Iy l $ 1 . Clearly Oy1i(-Y) = (-1)1+ 10111i ( y ). Also by (5.131), oll w(- y ) = - ol/v(y ).

Thus we

get that

v(y) + 1/ 01l (y) = ( _ 1) N - lp· (1 _ y)OIlTN (Y). Since ( _ 1) N - lp• = p and (1 - y )oIlTN (Y) = �N - l (Y) - �N(Y), we conclude that v satisfies (5.130), whence v satisfies the first formula of (5.129). In view of Lemma 5.13 and (5.131), we know that v(1) = w(-1) = O'N(1/)c. Furthermore, (5.128) and (5.131) lead to

•

Spectral Methods and Their Applications

244

We are now in a position to state some results. Let ).. ' = ).. " =

2)" .8(1 + a) ' 2)"

.8(1 - a) '

(5.120) satisfies

Lemma 5 . 1 5 . The solution of

Proof. Let y ( x)

21' , I' = .8(1 + a) ' 21' " I' = .8(1 - a) "

e�� + I (a)

=

- O"N ( ).. ' ) O"N (p, ' )e��(a) .

be the affine transformation from fi 1 to the reference interval [- 1 , 1]. Let us define v E lP N such that

= 2( x + 1 ) ( a + 1 ) - 1 - 1

) (x), v ( y (x)) = e 1,( 2m+I

-

x E !1 1 .

It is readily seen from the first formula, the third formula and the fifth formula of (5.120) that v ( y ) is the solution of (5. 122) , provided that Tf = 1" and c = e�� (a) . Thus by Lemma 5.13, Next, define w E lP N such that

(5.132)

By virtue of the first formula, the second formula and the fourth formula of (5. 120) , w ( y ) is the solution of (5. 129), provided that Tf = )..' and c = -e�� + I ( - I ) . So by Lemma 5.14 and (5.132) ,

e�� + I (a) = w ( l )

=

O"N ( ).. ' )c = - O"N ( ).. ' ) O"N ( I" )e�� (a) . •

Lemma 5 . 16. The solution of

(5.121) satisfies

e 2( 2, m) +1 ( a )

- O"N ( ).." ) O"N ( " ) e 1(1), m ( a ) .

Proof. Let y (x) 2(x - a) (I - a) - 1 - 1 be the affine mapping from fi 2 to the interval [- 1 , 1], and define v E lP N such that =

I'

=

v(y (x))

=

e�� +I (x), x E fi 2 •

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

Owing to ( 5.121 ) , by Lemma 5. 14,

v(y )

is the solution of ( 5. 129 ) with

TJ

= )," and e =

e��+I (l ) = uN (A") e��(o:). On the .other hand, let w E IP N such that w(y(x )) = e �� +I (x ) , x E fi 2 • According to ( 5.121 ) , w(y ) is the solution of ( 5.122 ) with TJ = p," Thus b y Lemma 5 . 1 3 , (2 ) +I ( 0:) = -UN ( I'") e (2,1m) +1 ( 1 ) e 2,m

e ��( o:) .

and e =

from which and ( 5. 133 ) , the desired result comes.

245 Thus

( 5.133 )

-e��+I(l). •

Now define the interface error

m�

1,

and set

u'N( A, p" 0:) = max (u� ( A' )U� ( p.') , uM A" )U� ( I' " )) . Obviously, uN(A, I' , O: ) < 1 .

Lemma 5 . 17. For

m�

2,

Proof. By the last formula of ( 5.120 ) , the last formula of ( 5.121 , Lemmas 5.15 and

)

5.16, we find that

e��+I( O: ) = ei��(o:) = -UN( A' )uN ( p.')e�� _l ( O: ) = -UN (A ' )uN ( p.')ei��( o:)

and

e1,(2m) +I ( 0: ) = e 2,(2m) ( 0: ) = -UN ( ) UN ( I'" ) eI,( 1m) - 1 ( 0:) = -UN ( UN ( I'") e (l) 2,m ( 0:) . \ 11 1\

Therefore

em+I

\ 11 1\

(

) ((ee�(o:)r + (e��(o:)r)

U� ( A')U� ( p.') (e�� ( 0: ) ) + U�(A") U�(I'") ::; u'N( A, p" a

)em•

)

2

+

(e�� (0:) )

2

•

Spectral Methods and Their Applications

246 Lem ma 5 . 1 8 . For m 2: 1,

N

�� + 1 (x) L: vtTl(y( x)) 1=0 N e e� + l ( x ) = L: wtTl(y(x)) 1=0 e

=

e

�� ( a )

(� (�

' - O'N ( IL e N

x E [-1, a] ,

) 1 � (�I'I ,I + ¢I'I,I) 1/ ( Y (x)), 1 ) �� ( ) (� (� I + ¢>'I,I ) )

Proof. From the proof of Lemma 5.15, we deduce that for I"

'

I + ¢I'I,I)

a

>."

-

. L:( _ 1) 1 (�>'I , I + ¢ >'I, I) TI ( Y (x)) . 1=0

Since IT1 (y) 1 ::; 1 for IY I ::; 1, we know from the above two equalities and the last formula of (5.120) that for x E [-1 , a] ,

In a similar way, we see from the proof of Lemma 5.16 that for

x E [a, 1),

The above estimations lead to the conclusion.

•

Theorem 5 . 9 . The iteration applied to (5. 1 1 6}- (5. 1 1 8) converges, as m

---t 00 .

Proof. The result follows from Lemmas 5 . 1 7 and 5.18. Actually, since E i s nonsingular, the convergence of ej, m implies that of U,,/, N, m - U,,/,N , 'Y = 1 , 2. •

In the final part of this section, we present some numerical results. The time discretization considered is the second-order Crank-Nicolson scheme with the mesh size T = 0.01. The test function is

U ( 1 ) (X, t)

=

et arctg

(ax + i) ,

U ( 2 ) (X, t)

=

e t arctg

(ax i) . -

We use the Chebyshev pseudospectral domain decomposition method to solve this problem. For comparison, we also use single domain method for this problem. In Tables 5.8 and 5.9, we list the maximum norms of the errors between the exact solution and the corresponding numerical ones at t = 1 , -with different values of a and

.

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

247

N. It can be seen that t �e domain decomposition procedure provides better numerical results, in particular, for the solutions possessing oscillations. Furthermore, if the matrices associated with the pseudospectral method are full, then the use of several sub domains allows the reduction of the overall complexity. It is noted that similar results are also valid for multi-sub domain method and the Legendre pseudospectral domain decomposition method.

Table 5.8. The errors of single domain method. a = 10 a=1 a=4 N = 4 4.767E - 4 1 . 288E - 2 2.812E - 2 N = 8 1 . 094E - 4 5.079E - 3 2.345E - 2 N = 20 2.572E - 7 2.860E - 4 1 . 005E - 2

Table 5.9. The errors a=1 N = 4 1.937E - 4 N = 8 7.377E - 7 N = 20 2.577E - 7

of two domains method. a = 10 a=4 6.1 19E - 3 3.668E - 2 1 .024E - 3 1.893E - 2 1.943E - 4 9.231E - 3

Another early work of spectral domain decomposition methods for hyperbolic sys tems was due to Kopriva (1986, 1987). We also refer to Quarteroni (1987b), Macaraeg and Streett (1988) , and Macaraeg, Streett and Hussaini (1989) for this technique and its applications. Carlenzoli, Quarteroni and Valli (1991) also used it for com pressible fluid flows. Recently Quarteroni, Pasquarelli and Valli ( 1991) developed heterogeneous domain decomposition method arising in the approximation of certain physical phenomena governed by different kinds of equations in several disjoint subre gions. Various applications of domain decomposition methods can be found in Chan, GlowinBki, Periaux and Widlund ( 1988) , and Canuto, HUBsaini, Quarteroni and Zang (1988). For the applications of Chebyshev pseudospectral methods to other nonlinear problems, we also refer to Hitidvogel, Robinson and Schulman (1980) , and Maday and Metivet ( 1986).

248 5.5.

Spectral Methods and Their Applications Spectral Multigrid Methods

Multigrid methods have been used for spectral approximations since the beginning of the last decade. Zang, Wong and Hussaini (1982, 1984 ) described this approach and its applications to periodic problems and non-periodic problems. It makes the spectral methods more efficient. In this section, we present some basic results in this topic. We first let A = (0, 211') and consider the model problem

{

x < 00 , - 00 < x < 00 . 2 and /j = J (xU» ) . Let N be an even positive integer. Set xU ) = approximation to the value U (xU» ) , 0 :5 i :5 N - 1 . Then -o�U = J, U (x) = U (x + 2 11') ,

Uj =

if - I

E

/ l12

-00 <

;}

u/ exp

(5.134) Let Uj be the

(2'7:"Z) '

u/ being the discrete Fourier coefficients, 2 ii l , 1 N�- l Uj exp � u/ = N =0 3 A sensible approximation to the left-hand side of

(5. 135)

(- 11' )

(5.136)

211' ii Z) . � 12 u- / exp (�

(5.134 ) at xU )

is

�

.

/ = -if

.

Let V = ( uo , . . , UN - I ) and F = (10 , . . , IN -I ) . Moreover, we denote by D a diagonal matrix with the elements Djk = PSj,k , and by C a matrix with the elements 1 211' ii Cjk = exp � . N Clearly

( - k)

(C - 1 ) jk = exp Cll'� k ) .

Set L = C- 1 D C. The pseudospectral approximation may be represented by

LV = F.

(5.137)

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

249

The original problem becomes the calculation for unknown vectors, as in McKerrel, Phillips and Delves Orszag and Haidvogel and Zang proposed various iterations to solve Let W be a relaxation parameter and Vm be the m-th iterated value of V. is The Richardson iteration for

(1980) ,

(5.137).

(1981).

(1979)

(5.137)

It can be checked that L has the eigenvalues Al 1 2 and the corresponding eigen1 vectors WI with the components w/ exp � . Let Vm Vm - V. It can be resolved into an expansion in the eigenvectors of L . Each iteration reduces the I-th error component by a factor A / ) - WAf. Clearly the iteration is convergent, if for alI I, - WAd < In the c�rrent context, Amin 1 and Amax = � N2 . So is convergent for < W < -ffi . The best choice of W results from minimizing I A I I for A E [Amin ' Amaxl . The optimal relaxation parameter for this single-grid procedure

11

0

= (") = (27rij ) V( = 1 =

1.

IS

WSG

It produces the spectral radius

= Amax +2 Amin

.

= AmAmaxax +- AAmmiinn ·

(5.138)

=

(5.138) V( )

(5. 1 39)

Unfortunately, P SG :::::: - J. which imples that 0 (N 2 ) iterations are required to achieve the convergence. This slow convergence is the outcome of balancing the damping of the lowest frequency eigenfunctions with that of the highest frequency one in the minimax problem described in the above. The multigrid approach takes advantage of the fact that the low-frequency modes ( I l l < can be represented just as well on a coarser grid. So it settles for bal ancing the middle-frequency eigenfunction ( I l l with the highest frequency one ( I I I Jt) . Namely, it damps effectively only those modes which cannot be resolved by Amid , and obtain that on coarser grids. To do this, we replace Amin in PS G

1

=

It)

(5.140)

= It) (5.139)

WMG

The multigrid smoothing factor .

= Amax +2 Amid

.

(5. 1 41) (5.142)

Spectral Methods and Their Applications

250

measures the damping rate of the high frequency modes. In this example, Amid and so I-'MG 0.6, independent of N. We present the multigrid process by considering the interplay between two grids. The fine grid problem with NI interpolation points is written in the form

= �:

=

The decision to switch to the coarser grid is made after the fine grid approximation WI to VI has been sufficiently smoothed by the relaxation process, i.e. , after the high frequency content of the error VI - WI has been sufficiently reduced. Let Lc be a coarse grid operator and v., be the correction of WI on the coarse grid with Nc interpolation points. Furthermore denote by R a restriction operator interpolating functions from the fine grid to the coarse grid, and

The auxiliary coarse grid problem is

After an adequate approximation We to imation is updated by

v.,

has been obtained, the fine grid approx

where the prolongation operator P interpolates functions from the coarse grid to the fine grid. We can repeat the above procedure with a nest of grids and a control structure to reach a desired result. We now construct the restriction operator R and the prolongation operator P. Consider the problem

8", (a(x)8",U(x)) = f(x) , x E A

=

(5. 143)

where A = (0, 211") in the periodic case or A ( - 1 , 1 ) in the Dirichlet case. We first consider the periodic case. On the coarse grid, the discrete Fourier coeffi cients of the corrections uj , at the coarse grid interpolation points xU) are computed using (5. 136) with N = Nc• The fine grid approximation is then updated by

Uj

+--

Uj +

� -1

L u / exp /= - �

(i lx (j» ) ,

0 5 j 5 NI - 1 ,

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

251

where xU) are the fine grid interpolation points. The restriction operator is con structed in a similar fashion. It turns out that except for a factor of P and R are adjoint. An explicit representation of the prolongation operator is Pj k =

( ( � - k )) . .

1 �- l exp 27ril Nc / = L: +1 -�

'

i,

Nc

It sums to yield B (r) =

where

{

. Nc - 1 , sin (7rr Nc) cot ( 7rr) - cos (7rr Nc) ,

The corresponding restriction operator is Rjk =

1 N, B

(

.

�c

-

r is an integer, otherwise.

)

k ' . N,

We next consider the Dirichlet case, and use Chebyshev approximation. The prolongation operator is Pj k =

where Ck

=

2 for k

=

-

7rlj 7rlk 2 L:N. c,l cos cos :--

Ck Nc

0, Nc and Ck

=

/ =0

Q

�

1

Nc

1 for 1 :$ k :$ Nc - 1 . This sums to

where (r)

N,

One kind of restriction operator is' defined as

r is an integer,

otherwise.

252

Spectral Methods and Their Applications

{

Ck 2 for k = 0, Nf and Ck = 1 otherwise, and I Nc r is an integer, :4 + 2 ' Q(r) = � + � cos ( ; (Nc + 1)) sin (1rr;c) csc ( ; ) , otherwise.

where

=

1r

1r

The other is defined by the adjoint requirement, i.e. ,

For the problem

(5.143), the discrete operator on the fine grid is (5.144)

Ajk (x U» ) bj,k , DN, is the "+k cot (- (J - k )) , j :f. k, -(-1)' 2 Nf (5.145)

where is the diagonal matrix with the elements = matrix related to the differentiation. In the periodic case,

A

(DNI )

"10 = J

{I

1r

a

"

0,

j

=

k.

In the Dirichlet case,

Cj( _j 1)i+kk , j :f. k, Ck (x_x( ) -(k)X ( ») 2 ( 1 - (x (k») 2) ' 1 $ j k $ Nf - 1, 2NJ + 1 ' j = k 1, 6 ( ) - 2NJ + 1 j = k = Nf. 6 =

(5.146)

=

Many multigrid investigators have advocated choosing the coarse grid operator so that

(5.147) 4 Both the Fourier and the Chebyshev first-derivative operators defined by (5.1 5) and (5.146), satisfy (5.148)

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

253

However, (5.147) itself is not satisfied, if the coarse grid analog of (5.144) is used to define Lc , except the case in which a (x) is a constant. On the other hand, much of the _efficiency of the pseudospectral method is lost if (5 .147) is used to define Le . Some compromises were suggested by Zang, Wong and Hussaini (1984). The most satisfying one seems to be using (5.144), but with the restricted values of a(x) in place of the pointwise values. The Chebyshev restrictions should be performed with R.

We now turn to two-dimensional problems. Let n = self-adjoint elliptic equation

(-1, 1)2

and consider the

(5.149)

(N

with the Dirichlet boundary condition. We use pseudospectral method with + 1)2 Chebyshev-Gauss-Lobatto interpolation points. It leads to a discrete set of equations like (5.137). The convergence rate of Richardson iteration on a single grid is governed by the ratio of the largest eigenvalue to the smallest eigenvalue of L , referred to as the condition number. The multigrid condition number is the ratio of the largest eigenvalue to the smallest high frequency eigenvalue. In our current context, Amax = o , Amid = 0 ( 2) and Amin � f . An effective preconditioning is essential for multigrid as well as for single-grid iterative schemes. The preconditioned Richardson iteration can be expressed as

(N4 )

N

v

+-

V + wH-1(F - LV )

where H is a preconditioning matrix. An obvious choice of H is a finite differencel approximation HFD to the differential operator in (5.149). But for multi-dimensional problems, these finite difference approximations are costly to invert. An attractive alternative is to use an approximate LU-decomposition. In this case, H is taken as the product of a lower triangular matrix C and an upper triangular matrix U. Buleev (1960) and Oliphant (1961) proposed one such type of preconditioning by taking C as the lower triangular portion of HFD and choosing U such that the two super diagonals of CU agree with those of HFD , see Zang, Wong and Hussaini ( 1984). Zang, Wong and Hussaini (1982) also gave a similar decomposition denoted by HRS , in which the diagonal elements of C are altered from those of HFD to ensure that the row sums of HRS and HFD are identical. Both types of preconditioning can be computed by a simple recursion. Let Uj, k be the values of U at the interior grid point x p l , x�kl .

(

)

Spectral Methods and Their Applications

254

Suppose that a five-point finite difference approximation to (5. 149) at this point is given by

and that a five-diagonal incomplete L U factorization is given by

( £V ) j,k = Vj,kUj,k + (3j,k Uj-l,k + /j,k Uj,k-l , (UV ) j,k = Uj,k + Aj,k Uj+l,k + O'j,k Uj,k+ 1 ' Set

(3j,k = dj,k, /j,k = bj,k , I/j,k €j,k - (3j,kO'j,k-l - /j,kAj-l,k - a ( (3j,kAj,k-l gj,k \ = -, Ai,k I/j,k hj,k O'j,k = - . I/j ,k

=

+ /j,kO'j-l,k ) "

Then a = 0 gives HLU5 and HRS 5 uses a = 1. Straightforward modifications are made near the boundaries. A more accurate factorization can be achieved by including one extra nonzero diagonal in £ and U . Suppose that

( £V ) j,k = Vj,kUj,k + (3j,k Uj-l,k + /j,k Uj,k-l + {}j,k U i+l,j-l , (U V ) j,k = Uj,k + Aj,k Uj+l,k + O'j,k Uj,k+ 1 + Xj,k U i -l,j+1 '

Put

/j,k = bj,k, {}j ,k = -/j,kAj,k-l , (3j,k = dj,k - /j,kXj,k-l , Vj ,k = €j,k - /j,kO'j,k-l - {}j , kXj+l,k-l - (3j, kAj-l,k - a ( >.j + l,k-l {}j , k - Xj-l,k(3j, k ) , 1 (g . , 0" +1 {) ' k ) A3', k l/j,k J k - 3 k-l 3 , ' -(3j,kO'j-l,k Xj,k = , I/j,k hj,k O'j,k = - . I/j,k

=

I

5.

Chapter Spectral Methods for Multidimensional and High Order Problems 255 Once again, a 0 gives the HLU7 version and a 1 gives the HRS7 version. Tables 5.10 and 5.11 list the multigrid condition numbers and the multigrid smoothing rates for the preconditioned procedure related to (5.149) with al(x) a 2 (x) 1. Table 5 10 Multigrid condition numbers N HF� L Hr;JsL Hr;J 7 L HRJsL HRJ7 L 8 1.733 1. 863 1.756 1.984 1.796 16 1. 874 2.134 1.932 2.925 2.172 32 1. 941 2. 324 2. 054 4. 425 2.934 64 1.974 2.525 2.191 8.675 5.607 128 1. 990 2.763 2.307 =

=

==

==

Illg

' ' d smoot h ' rates. 5.11. MuItlgn HF�L Hr;Js L Hr;J 7 L 8 0. 268 0. 3 01 0. 2 74 16 0.304 0. 362 0.318 32 0. 320 0.398 0. 345 64 0.328 0. 433 0;373 128 0. 3 31 0. 469 0.395

Table N

Finally, we present some numerical results. Let take the test function

U(x)

=

sin

al(x) a2 (x) ==

(a1rx 1 + �) sin (a1rx 2 + �) .

==

1 in (5.149) and

We use spectral multigrid method (SMG) and finite difference multigrid method (FDMG) to solve it. The errors and the CPU times in seconds ( on a CDC Cyber for these methods are shown in Tables and Clearly the former method provides more accurate numerical solution for the same CPU time.

175)

5.12 5.13.

256

Spectral Methods and Their Applications Table 5.12. The errors and CPU times, 1 O! =

Errors

N

FDMG

SMG

CPU Times FDMG SMG

2. 37E - 2 4. 42E - 5 0. 08 5.73E - 3 8. 53E - 13 0. 24 1.42E - 3 1. 25E - 14 0.76 3. 55E - 4 2. 59 8. 88E - 5 9. 7 1 Table 5 13 The errors and CPU times ,

8 16 32 64 128

Errors

N

8 16 32 64 128

FDMG +

SMG +

0. 3 5 2. 3 2 5. 6 2 O! =

5

CPU Times FDMG SMG

1. 68E 0 1.73E 1 0. 0 7 0.33 1. 92E - 1 2. 56E - 2 0.19 1.19 4.11E - 2 5. 25E - 10 0. 53 7. 84 9.87E - 3 2. 26 2. 44E - 3 9.52

Spectral multigrid methods have been used for nonlinear problems, such as in Streett, Zang and Hussaini and Zang and Hussaini

(1985) ,

(1986).

CHAPTER 6 MIXED SPECTRAL METHOD S

I n studying boundary layer, channel flow, flow past a suddenly heated vertical plate and some other problems, we encounter semi-periodic problems. It is reasonable to deal with them using Fourier approximations in the periodic directions. Sometimes, the orthogonality of Fourier system reduces the resulting schemes to a set of separate equations for the Fourier coefficients of unknown functions at each time step, and thus it is easier to solve. We can use finite difference approximations or finite element approximations in the non-periodic directions. But in these cases, the accuracy in space is still limited due to these approximations, even if the genuine solution of an original problem is infinitely smooth. This deficiency can be removed by mixed Fourier-Legendre approximations or mixed Fourier-Chebyshev approximations. The purpose of this chapter is to present these two mixed spectral methods with their applications to semi-periodic unsteady incompressible fluid flow.

6.1.

M ixed Fourier- Lege ndre Approximat ions

In this section, we introduce the mixed Fourier-Legendre approximations. Let = = and 0 = To X < < ,A = < < describe the approximation results using different orthogonal systems in different directions, we define some non-isotropic spaces as follows,

(XI, X 2 ), AI {xI ! - 1 Xl 1} 2 {X 2 ! 0 X 2 211"}

2 Hr , s(o ) = L 2 (A 2 ' H' n H8 (A 2 ' L (Ad) I I Mr , 8 ( O ) = Hr , s(o) n H (A 2 , Hr- (A I » n H,- I (A 2 , H I (A I » , Xr , 8 ( O) = H S (A 2 ' Hr +1 (A I » n H8 +1 H' (A I » ,

,

(AI»

(A2 , 257

Al A2.

r, s r, s r, s

� 0, � 1, �°

x

258

Spectral Methods and Their Applications

equipped with the norms

Il vllw. ' (o) = (1 I v ll l' (A w (At} ) + I v 1 1-' (A .£2 (At} » ) i , i l I VIl Mr (o) = ( I v ll 1-r + I l v l 1-' (A W-' (At} ) + I v 1 1- .-1(A H ) , (At} » i I v ll xr = (IIv ll 1-'(A2 .W+1 (At}) + I v ll 1-'+ l (A W(At} » ) . •.

(o)

.•

.•

.•

•

(o)

•.

•.

•.

Denote by Vp(n) the set of all infinitely differentiable functions with compact sup ports in Al and the period 211" for the variable X 2 ' The closures of Vp (n) in w·'cn) , Mr.s cn) and Xr . BCn) are denoted by H�;; cn) , M�;;cn) and X�;;cn) , respec tively. For the sake of simplicity, we denote the norms (o) and , respectively. Furthermore, also by and stand for the semi-norms, associated with the norm and , and , s Hr.scn) of the set involving all respectively. Besides, H; Cn) denotes the closure in infinitely differentiable functions with the period 211" for the variable X 2 , etc. Let M and N be any positive integers. Denote by the set of all polynomials of degree at most M on the interval Al and lP� l = l n HJ CA l ) ' Denote by V 2 . the set of all trigonometric polynomials of degree at most N on the interval A 2 , and by the subset of V 2 containing all real-valued functions. Set ® several inverse inequalities are valid. and ® 2 In the space

I . I x r (o) 1 · I Mr

I . I w." I . I Mr. .

.•

I . I xr

I . IIw. ' (o) , I . I Mr I . Iw 1 I · lIw. ' 1 · I Mr .•

.•

.•

lP M.l lPM.

VN.2

N. V,,'}.N = lP�.1 VN. '

VM.N ,

Theorem 6.1. For any 4> E VM.N and 2 � Proof. Set

4>1 CX I )

Then 4>1 E

114>II Loo

.•

.•

<

l14>dI L oo(At} cMN i ll 4> ll . �

��

�

00,

= -2111" lA. 4> (Xl , X2 ) e-i1"2 dX2 .

lPM•1 and by Theorem 2.7, �

p

N. VM.N = lPM. I VN.2

f

� cM �

��

C6 . 1)

II4>dI L2 (At} � cMN i ( � 1 4>l I i'(At} )

i

��

Consequently, which leads to the conclusion.

•

Chapter 6. Mixed Spectral Methods

259

As a consequence of Theorem 6. 1, we have that for any <1>, 'IjJ E

VM.N, (6.2)

Theorem 6.2. For any E VM•N , Proof. Let

that

, (X I)

1101 <1>11 2

be the same as in

(6.1). We know from the proof of Theorem 2.8

= 27l' IILI $N II Ol <1>d/;" (Al) � 1 !7l' M4 IILI$N lI d/i,'(A,) = � M4 I1<1>1I 2 .

On the other hand, 1102 <1>11 2

=

27l' IILI $N 12 II <1>11 i,. (A,) � N2 11 <1>1I 2 .

Then the desired result follows.

•

v v(x) mL= L vm .I Lm (xd ei1x, O 1 ' 1 =0

The mixed Fourier-Legendre expansion of a function E L 2 (!1) 00

IS

00

=

where

Vm. 1 is the Fourier-Legendre coefficient, Vm.l = 217l' (m + �) 10 v(x)Lm (X I) e -i1x• dx.

The L 2 (!1)-orthogonal projection PM.N : L; (!1) VM.N is such a mapping that for any v E L; (!1), - PM.N V, <1» 0, "1 <1> E VM .N ,

(v

or equivalently,

PM.NV ( X) =

M

=

-+

Lm(� I )eilx• . mL=O I 'L'Vm. I $N I

Theorem 6.3. For any v E H;·8(!1) and r, s 2:: 0,

26 0

Spectral Methods and Their-Applications

Proof. Let

anq. PM : L 2 ( At}

1 , X ) e i l.,. dX 2 , { VI(X ) = l 2 7r lA2 V(X l 2 -

-+

(6.3)

IPM, be the L 2 (A l )-orthogonal projection. Then l PM,NV (X) = 2::: PMv l (xI )eil"2 . 111$ N

By virtue of Lemma 2.2, I I V - PM,NvII 2

+

2::: I I vl - PMvdli2 (Al ) 27r 2::: I I vdli2 (Al ) 111$ N 111 > N :::; cM- 2r 2::: II V d11r(At l cN- 2 • 2::: 1W' l I v d l i'(Al ) 111> N 111$ N :::; cM- 2r l l v l l i'(A2 ,w( Al » CN- 2 B l v I 1' (A2 P (At l ) · 27r

+ +

It is noted that for any v

E

�

H�:;(!l) and r, 8

•

0, (6.4)

When we use the mixed Fourier-Legendre approximations for semi-periodic prob lems of partial differential equations, we need different kinds of orthogonal projections to obtain the optimal error estimates. For instance, the H l (!l)-orthogonal projection Pk,N : H� (!l) -+ VM, N is such a mapping that for any v E H� (!l), (V ( v - PiI,N V) , V¢»

+ ( v, ¢»

=

While the HJ (!l)-orthogonal projection plf?N (!l) any v E HJ,p (!l) ,

Theorem 6.4. If v E M;'B (!l) with r, I I v - piI,N v l l " :::; c (Ml - r

8

0, V ¢> E VM, N .

-+ V,,'},N is such a mapping that for

� 1 , then for J.L = 0, 1 ,

+ Nl-B) (M,,-l + N,,-l ) II V I l Mr,

• .

If, in addition, for certain positive constants Cl and C2 ,

(6.5) then

Chapter 6. Mixed Spectral Methods

261

Proof. Let VI ( Xl ) be as before, and Pk be the Hl (l\.t )-orthogonal projection. Set V. (X) 1 L1 :5N (PiIVI (Xl )) ei1x, . =

By Theorem 2. 10,

I l v -w l � :::; I v - v. l I � < 211' I!LI $N ( l i vi - PiIvdl k1 (A') + [2 1 vl - PiIvdl i' (A, » ) + 211' L ( I vdl k1 (A' ) + 12 I 1 vdl i2 (A, » ) :::; cM2-2IlrI>lLN N ( I vdl kr (A' ) + 12 I 1 vdl kr-1 (A1» ) I :5 + cN2 - 28 L 128 - 2 ( I vdl k1 (A, ) + 12 1 I vdl i2 (A' » ) :::; c (M1 -r +I!NI>Nl -S) 2 1 I v l �r" . inf

wE VM.N

Then the first conclusion with J.L = 1 follows. By means of the duality, we deduce that

I v - PiI,NVI l :::; c (M1 -r + N1 -S) (M-1 + N-1) I V I l Mr" .

The last result comes from the above inequality and (6.5) .

v

Theorem 6 . 5 . If E HJ,p(n)

n

M;,8 (n)

with r, s 2: 1, then for J.L = 0, 1,

If, i n addition, (6. 5) holds, then

pj,l

Proof. Let Vl (Xl ) be as before, and be the HJ (Al )-orthogonal projection. Put v. (X ) 1!LI:5N (pltvl(xl )) ei1x2• =

By virtue of Theorem 2 . 1 1 ,

•

262

Spectral Methods and Their Applications �

+

+

�

�

+

c 1LN (i vi - pl/ vd1-'(A') [2 1 v - pli�NvdI 12 (A, ) ) 1 I$ c I ILI >N (I vd1-'(A' ) [2 1 I vl lli2(A, ) ) cM2-2r ILN ( I vdl 1-r(A' ) [2 I1 vdl 1-r-l (A, ) ) I I$ CN28-2 ILN [28-2 (Ivd1-'(A, ) 12 1 I vdl i'(Al) ) I I> c (M-r N-8 ) 2 I 1 VIl �r, +

+

+

+

• .

The remaining results come from an argument as i n the last part of the proof of Theorem 6.4. • In numerial analysis of mixed Fourier-Legendre spectral schemes for nonlinear problems, we also need to bound the norm One of them is stated in the following theorem.

I Pli�NVllwr,p .

Theorem 6.6. Let (6. 5) hold and 8

n

> !. IIv E HJ.p ( O ) H; ( A2 , H1 ( A1 )) , then

I Pli�NVI I Loo � c ll v I H'(A2 .H1 (AtllIf, i n addition, v E X;· 8 ( O ) with r � 1 , then Proof. We first let VI ( Xl) be as before, and let Obviously

VI E HJ(A1 ) and vi E lP�. l ' Define

We assert from the orthogonality of the Fourier system and the property of that Obviously

PlioN

Chapter

6. Mixed Spectral Methods

263

By Schwarz inequality and Poincare inequality, we deduce that

I VI -v71 I iT' (A, ) + [2 1 I vl - v7I1 i' (A, ) :::; c ( l i vi -
Theorem 2 . 1 1 , we derive from the above inequality and (6.5) that

I vI - vi 1 I h-, (A, ) + [2 I 1 vI - villi' (A, ) :::; cM2-2r ll vdl h-r(A, ) '

By means of duality, (6.6)

I PJJ�NVllw�. oo , JL = 0, 1 . Clearly, I PJJ�NVl l w�, oo :::; I ILI ::;N (1 I 8i viIlLOO(A, ) + (1 + 111") I v7I1LOO(A, ) ) ' JL = 0, 1 . (6.7) We can bound the terms on the right-hand side of (6.7). Let 1M be the Legendre Gauss-Lobatto interpolation on the interval A 1 . Then 1 81 V7I 1 L OO(A, ) :::; 1 81 Vi - IM81 VII I LOO(A, ) + 1 81 VI -1M81 VI I Loo (A, ) + 1181 VdILOO(A, )' (6.8) We now estimate

According to Theorem 2.7 and (2.63) , we know that for r > ! ,

1181vi - IM81 vII I LOO(A, ) < cMI I 81 vi - IM81 Vdl L2 (A, ) :::; cM ( i vi - viI Hl (A, ) + I 81 VI - IM81 VdI L' (A, » ) (6.9) :::; cMl vl - viI Hl (A, ) + cM1 -r ll vdl w+1(A, ) ' Let 0 6 :::; !. By imbedding theory and (2.63) , we have that for r > ! , 1 81 VI - IM81 VdILOO(Al) :::; cll81 vI - IM81 VI lI Ht + 6 (Al) :::; cM1 -r ll vdlw+ 1 (Al) ' (6.10) <

Moreover, the imbedding theory implies that for r > !,

181 VdLOO(A, ) :::; c ll vdl w+1(A, ) .

The combination of (6.6) and (6.8)-(6 . 1 1 ) leads to that for

r

2:

(6. 1 1 ) 1 and

8

>

!,

l L V LOO(A, I I I ::;N 1181 iI l ) :::; cM I ILI ::;N I VI - v7I H (Al) + c IlLi ::;N I vdl w+1(A, ) < (M1 -r + 1 ) II2:N I vdl w+1(A, ) I ::; < c (II2:N (1 + [2 fS ) ( I2:N (1 + [2) " I VdI W+1(A, ») I I ::; I ::; C

1

"2

Spectral Methods and Their Applications

264 Similarly, we have that

L I l v7 1 I LOO (A, ) s: cli v Ii H'(A2 ,H' (A, ) ,

III� N

L I l i li vi I i L OO (A, ) s: cli v I IH'+'(A2 ,W(A, ) '

III� N

The proof is complete.

•

Finally we consider the mixed Fourier-Legendre interpolation. Let 1M be the same as in the previous paragraphs and IN be the Fourier interpolation on the interval respectively. Set IM,N = 1MIN = IN IM.

A2

Theorem 6.7. If v E H: (A2 , W(A 1 )) n H; (A2 , H"'(A1 )) n H; (A2 , H>'(Ad) , O < (

( )

O

a s: min r, A ) , S: j3 S: mi n s , , and r, s , A, , > � , then I I v - IM,Nv I I H,B(A2 ,H"(A, )

s: cM2 ",-r + t Ii V I i H,B(A2 ,W(A, ) + cN,B-s l l v I I H'(A2 ,H" (AI » + cq ( j3) M2 01 - >. + t N,B --y Il v Il m ( A2 ,H� (AI » )

where q (j3) = ° for j3 > � and q(j3) = 1 for j3 s: � . Proof.

Let I be the identity operator. We have that

where

Dl

=

I I v - IM v II H,B(A2 ,H"(Ad )

+ I l v - INv Il H,B(A2 ,H"(A, ) ,

D2 = II (IM - I) ( IN - I) v II H,B(A2 ,H"(A, )

.

From Theorems 2.4 and 2.12,

If j3 > � , then by Theorems 2.4 and 2. 12,

If j3 s: � , then

The proof is complete.

•

Chapter

6. Mixed Spectral Methods

265

It is noted that if 0 a :::; 1 and a < min(2r - 1 , 2>' - 1 ) , then we can improve Theorem 6.7 by using (2.63) . In this case, we have that

Il v -

:::;

IM,NVl l wJ(A2 ,HQ(A, )) :::; cM"- r l v I l H�(A2 ,W(A, » + cNfj- sll vI I H'(A2 ,HQ(Al » + cq ( j3 )M"->'Nfj-,,! I l v II H� ( A2 , H A (A, ») '

q( j3 ) being the same as in Theorem 6.7. Theorems 6.3 and 6.5 can be found in Bernardi, Maday and Metivet (1987) .

6.2.

Mixed Fourier- Chebyshev Approximations

This section is devoted to the mixed Fourier-Chebyshev approximations. Let A I , A 2 1 and n be the same as in the previous section. Set w(x) = w(x I ) = (1 - XD - 2 , and

(v, w) w = Define

10

1

v(x )w(x )w(x) dx, I l v ll w = (v, v) � .

{v I v is measurable, I l v ll w < We define the spaces L�(n), H� (n), W� ,p (n) and their norms in a similar way. We L� (n)

oo}.

=

also introduce some non-isotropic spaces as follows

H�, S(n) = L 2 (A 2 ' H� (A I )) n HS (A 2 ' L�(A I ) ) ' r, S 2: 0, l �, 8(n) = H�, 8(n) n H I ( A 2 , H:- I (A I ) ) n W - ( A 2 , H�(Ad ) , r, s X�, S(n) = H8 ( A 2 ' H:+ I (Ad ) n W+ 1 (A 2 , H:(A I ) ) , r , S 2: 0

M

2:

1,

equipped with the n9rms

The meaning of Vp(n) is the same as in the previous section. The closures of Vp (S1) in H�, s(n), M�, S (n), X�, 8(n) are denoted by H�:; ,w(n) , M�:; ,w(n), X�:; , w(n) respectively. Their norms are denoted also by I . IIH:" , I . for simplicity. The corre I . The space H;:�(n) is the sponding semi-norms are denoted by I . I ':; and I .

1 M:' " Il x.:;' · 1 M':;' " H "

Spectral Metbods and Tbeir Applications

266

closure in H:;;" ( n) of the set containing all infinitely differentiable functions with the period 211" in the x 2 -direction, etc. Let M and N be any positive integers. The subspaces lP M, ! , lP1t, l ' have the same meanings as in Section 6 . 1 . We first give some inverse in and equalities.

VN, 2 , VN,2 , VM,N

VA},N

Theorem 6.8. For any 1> E

VM,N and 2 ::; p ::;

00 ,

Proof. Let 1>1 ( Xl) be the same as in (6. 1 ) . According to Theorem 2.13, we deduce

that

1 1> I Loo whence

::; IILI �N I 1>dI LOO(A, ) (IILI�N ::; c ( MN) !

::; cMt

1 I ILI �N l 1>dI L�(A, ) l 1 1>d l h(A ) = c (MN ) ! II 1> ilw ')

1

2'

As a consequence of Theorem 6.8, we have that for any 1> , 'IjJ E

This completes the proof.

Theorem 6.9. For any 1> E

11 1> 'ljJ I I � ::; cMNII 1> II � II'ljJII � ·

•

VM,N ,

(6.12)

VM,N ,

Proof. Let 1>1 be the same as in (6. 1 ) . We see from the proof of Theorem 2.14 that

The mixed Fourier-Chebyshev expansion of a function v E L�(n) is

The rest of the proof is clear.

v(x)

= mL=O ILII=O vm,ITm( Xl)eilx2 00

00

•

Chapter

6. Mixed Spectral Methods

267

with the Fourier-Chebyshev coefficients

(

1 v(X)Tm(x l )e -ilx2 dx . Vm , l = 1r 2 Cl in The £ 2 (O)-orthogonal projection PM, N : L; ,,,, ( O) -+ VM, N is such a mapping that for any v E L; ,,,, ( O ) ,

or equivalently,

PM,NV ( X )

=

M E E vm ,l Tm ( Xl ) eilx• . m =O I I I:5 N

Theorem 6 . 1 0 . For any v E H;:�(O) and r, II v

�

s

;:::

0,

PM, NV Il", � C (M-r -+- N-' ) II v I l H':;'· .

Proof. Let vl (xd b e the same as in (6.3) , and PM : L�(A l )

orthogonal projection. Then

PM,N V (X)

=

-+ IPM,l b e the L�(Al )

E PMvl (x l )ei lx2 . Il l:5 N

Using Lemma 2.5, we find that

IIv - PM,NV I I

=

2 1r E I I Vl - PMvdl ii!,(At l

Il l :5 N

+ 21r EN II vdlii!, (A, ) l' I >

+ cN-2• EN W· ll vdli� (A, ) I l l> I l l:5 N cM-2r l l v l l p(A. ,H':;(A,)) + C N- 2 0 l l v llh'(A2 ,Li!,(A, ) ) .

� cM - 2 r E II vdlh':; (A' ) �

It is noted that for any v E H�:; ,,,, ( O) and r, s

;:::

•

0,

(6 . 1 3) When we apply the mixed Fourier-Chebyshev approximations to different semi periodic problems of partial differential equations, we adopt different orthogonal pro jections to obtain the optimal error estimations. For instance, the H� (O)-orthogonal projection Plt,N : H!,,,, ( O) VM,N is such a mapping that for any v E H!,,,, ( O) ,

-+

Spectral Methods and Their Applications

268

While the HJ.w e!1)-orthogonal projection PJ.,;.�N that for any v E HJ.p .j!1),

Theorem 6 . 1 1 . If v E M;:� (n) with r,

8

:

HJ.p .we!1)

--+

Vi} . N is such a mapping

� 1 , then for f1- = 0 , 1 ,

If in addition (6. 5) holds, then

Proof. Let VI (X l ) be as before, and Pit be the H!(A l )-orthogonal projection. Put

v. (X) = L: (PIt vl (X l ) ) ei lx2 . I I I� N

By Theorem 2.16,

Il v - pk. NVII i.w � <

<

inf Il v - w l l i w � Il v - v .lli w '

W€VM,N

1

2 11' L: ( l ivi - Pk vdliI';'(A, ) + { 2 11 vl - Pk vdlh(A, ) ) I I I� N 2 11' L: ( 1I v l l J.';'(At) + { 2 I1 vdl i�(At } ) I I I> N

+

cM2 - 2 r L: (I I vdlJ.':;(A1 ) + { 2 I1 v ll��-1 (Al ) ) I I I� N

+ cN 2 - 2 • L: 12.- 2 ( 1 I vdliIJ, (A, ) + { 2 1I vdl bA, ) ) I I I> N

<

r

c ( Ml- r + N l- · l lv l l ��.' .

S o the first result i s true for

f1-

= 1.

Theorem 6 . 1 2 . If v E HJ.p.w(n) If i n addition (6. 5) holds, then

n

The rest of the proof i s clear.

M;:�(n ) with r, s � 1 , then for f1- = 0, 1 ,

•

Chapter

6. Mixed Spectral Methods

Proof. Let

269

VI (Xl) be as before, and pJ/ be the HJ,w (A l )-orthogonal projection. Put V.( X) = ILN (pif�l VI (Xl)) eilx2 • I I$

By Theorem 2.18, :s:

:s:

:s:

in! I l v - wl kw :s: cllv - v.l kw c wEVM, N c I ILI $N ( i vi - pifovd��(A, ) + [2 1 vl - pitvdlh(A, » ) + c L ( I vd��(AJ ) + 1 2 I1 vdll�(A '» ) II I >N cM2-2T. ILN ( I vdl �':;(A, ) + 12 1 I vd l�':;-1 (A, » ) I I$ + cN2-2 8 L: 12 8 -.2 ( I vd��(A' ) + 12 1 I vdl h(AJ ) ) II I >N l (M -T + Nl - ·r I v l t-.:;' · · c

The rest of the proof i s clear.

•

In numerical analysis of mixed Fourier-Chebyshev spectral schemes for nonlinear One of them is stated in the problems, we have to estimate the norm following theorem.

IIpif�NVllw.:;'p .

Theorem 6 . 1 3 . Let

If in addition

S

>

!.

v E x;:�(n), then

Proof. Firstly let

Clearly

(6.5) hold and r,

VI ( X l) be as before, and set

VI E HJ,w (A1 ) and vi E lP�,l . Define

v

If E HJ,p,w (n) n H; (A 2 ' H:(A l ) ) , then

270

Spectral Methods and Xheir Applications

w E HJ,w (A t }, then by Lemma 2.8, al,w (w, w) ;::: � l w l ��(A, ) + [2 I1 w ll h (A, ) . Moreover, for any w, z E HJ,w(A 1 ), If

al ,w (vi - vi, ¢i) ° for all ¢i E lP�,I ' we see that � l I v, - vill ��(A, ) + 12 11 vI - vil l i�(A, ) ::; al,w (VI - V; , VI - vi ) al,w (lil - vi , VI - ¢i) ,

Since

=

=

V

¢i E lP�,I '

Consequently,

� l I v, - vill ��(A, ) + [2 1 vI - vil l h (A, ) ::; ..j� G ll v, - ¢i 1 l ��(A, ) + [2 1 vI - ¢i ll h (A' » ) .

Now, let pJ/ be the same as in the proof of Theorem 6.12. By taking and using Theorem 2. 18, we deduce that If'

M,l

¢i pllv, =

� l I v, - vill ��(A, ) + 12 1 vI - vil l i�(A, ) ::; cM2-2r ll vdl �':;(A, ) .

(6.14) II vl - viIlH�(A' ) ::; cM!'-r llvdIH.:;(A') , J.I 0, 1 . We now estimate IIpli�NV I w.. . oo , J.I = 0, 1 . Obviously, I l Pli�NVllw.., oo ::; IILI $N ( 11 8r viIlLOO (A, ) + ( 1 + I J.l I !' ) I vi i Loo (A' » ) , J.I = 0, 1 . (6.15) We can bound the terms on the right-hand side of (6.15) . Let 1M be the Chebyshev

By means of the duality,

=

Gauss-Lobatto interpolation on the interval A I . Then

Using Theorems 2.13 and 2.19, we obtain that for r >

I � vi - IM8I VdI LOO(A, )

�,

cMtl l 8lvi - IM8I VdI L�(A, ) ::; cMt ( l vi - VdHMA, ) + 1 8lVI - IM8I VdIL�(A' » ) < cMt I vi - vdH�(A, ) + cMt -r l l vdI H':;+ ' (A') ' (6. 1 7) <

Chapter

6. Mixed Spectral Methods

271

w( y) = W(X I ) ' Let iMw . Let 0 < 8 < ! .

!.

Now, let W E Hr (A I ) and r > Set Xl = cos y , 1\. = (0, 1l') and be the corresponding Fourier interpolation. Then � = We have

iM

t 1. 6 (A ) l I iMw - wil H2t 1. + 6 (A) -< cM � -r llwIl W (A) ' I l iMw - wI I LOO(A) -< cl l iMw - wil H2By the continuity of the mapping W w , II IMw-w II L OO(A, ) ::; cM t -r ll wIIH':;(A, ) . Thus (6.18) II OIVI - IMolvd I LOO(A, ) ::; cMt -r ll vdI H':;+ ' (A, ) . _

_

-t

Clearly,

II OI VIII LOO(A, ) ::; cll vdlw+' (A, ) ::; cll vdIH,:;+' (A,) '

The combination of (6. 14) and (6. 16) - (6. 19) yields that for r, s >

(6.19)

!,

< cM � l:N I VI - vil dJ,(A,) + c LN II vdI H':;+ ' (A, ) O 1 1L1 �N II IViIl L OO(A,) 111 � 111 � ::; c (M t -r + 1 ) 11L1 �N Il vdIH,:;+ '(A, ) , , < c ( 1LN (1 + lZrs) ( 1LN (1 + 12) " IIVdl �':;+ ' (A, ) ) 1 1� 1 1� 2"

Similarly

The proof is complete.

2"

, , 11E1 �N IlvdILOO(A, ) ::; cllvI I H'(A. H':; ( A,» 11L1 �N 1111IvdI LOO(A, ) ::; c ll v I l H'+'(A.,H':; ( A, )) . 1M

•

Finally, we consider the mixed Fourier-Chebyshev interpolation. Let be the same as in the previous paragraphs and be the Fourier interpolation on the interval = = A 2 . Set

IN IM,N IMIN INIM · Theorem 6.14. If v E H: (A2 ' H: ( AI ) ) H; (A2 ' H� (AI )) HJ (Az, H; (Ad ) , 0 ::; a ::; min( r, A ) , 0 ::; j3 ::; min ( s, ) and r, s , A" > !, then Il v - IM,NV IlHl'(A2 ,H�(A, )) ::; cM2or -r ll v Il Hf3(A. ,H':;(A, )) + cNi3- s llvI I H '(A2 ,H�(A, )) + cq( j3 )Mzor - .\Ni3 -'Y llvI I H� ( A2 ,H; (A, )) where q(j3) = 0 for j3 > ! and q(j3) = 1 for j3 ::; ! ,

n

n

.

Spectral Methods and Their Applications

272

Proof. We have with

D1 = D2 =

I l v - IMV I H!l(A2 ,H�(Al)) + I l v - INV I Hil(A2 ,H�(Al)) ' II (IM - I)(IN - I)V Il H!l(A2 ,H�(Al)) '

From Theorems 2.4 and 2.19,

For (3 >

!, we have

while for (3

:::;

!,

The proof is complete.

•

Theorems 6.10, 6.12 and 6 . 14 can be found in Guo, Ma, Cao and Huang (1992) , and Guo and Li (1995a) . Quarteroni (1987a) also considered this kind of mixed approximation. The filterings are also available for the mixed spectral methods. For example, let

¢(x )

=

M N

L: L: �m" Tm (X1)ei'x2 . m=O 1 ' 1=0

Following the same line as in the proofs of Theorems 2.26 and 2.27, we can prove that if 0 :::; r :::; 01 and 0 :::; S :::; 02, then In order to keep the spectral accuracy, we can take OJ = oj ( N ) and oj ( N ) N � oo, j = 1 , 2, see Guo and Li (1995b).

� 00

as

Chapter

6.3.

6. Mixed Spectral Methods

273

Applications

In the final section of this chapter, we take two-dimensional unsteady Navier-Stokes equation as an example to describe the mixed spectral methods. For simplicity, we focus on the mixed Fourier-Legendre spectral schemes. Let and be the same as in the previous sections, and - 1 , 1 , ° :s; :s; 21l'} . I and > ° are the speed, the pressure and the kinetic viscosity, respectively, and and t) are given functions with the period 21l' for the variable We look for the solution with the period 21l' for the variable such that

At , A 2' X2

00 = {x X l = Il U = ( U( l ), U(2 ) ) . Uo(x), Po(x) f(x, X2 . (U, P) X2 , Ot U (U . V)U - Ill::. U V P = f, E 0, ° t :s; T, E 0, ° :s; t :s; T, V · U = O, E 00, ° t :s; T, U = O, x E n, U(x, O) = Uo(x), X E n. P(x, O) = Po(x), +

+

X

X

X

0 U(x, t), P(x, t)

<

<

(6.20)

Let

A(v) = k v(x) dx and Lg ,p (O) = {v I v E L;(O), A(v) = OJ. For fixing the values of P, we require that P(x, t) E L� ,p (O) for all O :S; t :S; T. Let Vp = {v I v E 1\(0), V · v = OJ. Denote by Hp and Yp the closures of Vp in L 2 (0) and HI (O) . Set (6.21) a(v, w) = (Vv, Vw), b(v, w) = (V · v, w). v E Yp, ° < t :s; T, x E n. (6.22) By an argument as in Lions ( 1969), it can be shown that if Uo E Hp and f E L2 (0, Tj V;), then (6.22) possesses a unique solution in L2 (0, Tj Yp) n Loo(O, Tj Hp) .

The weak form of (6.20) is

(Ot U(t), v) + « U(t) . V)U(t ) , v) { U(O) = Uo,

+

ll a(U(t), v) = (f, v),

'if

The solution satisfies the conservation which is exactly the same as (5.67) . We can approximate (6.22) directly with trial function space belonging to the space Yp. To avoid this, we recall some artificial compression methods. The first was due to Chorin (1967) . It means that instead of the continuity equation, we consider the approximate one as (6.23)

274

Spectral Methods

and Their Applications

The solution of this auxiliary problem tends to the genuine solution of (6.22) as {3 -+ 0, under some conditions, see Lions (1970). Guo (1981b) modified this approach, in which the continuity equation is approximated by {30tP + V . U - a{3!:::.. P = 0,

{3 > 0,

a being ce)."tain positive constant. It was used for finite difference schemes of N avier Stokes equation with strict error estimates. Recently Shen (1992) proposed a new artificial compression as V · U - {3!:::.. P = 0, (3 > O. We shall use (6.23) in this section. On the other hand, the convective term U · V U can be expressed as d( U, U ) where (6.24) Clearly

(d(v, w), z) + (d(z, w) , v) = 0, V v, Z E H 1 (n) , w E H� (n) .

Let M and N be any positive integers, and

(6.25)

r be the mesh size in time. Set

VM,N = IP�,l ® VN,2 , WM,N = {w E IPM, l ® VN,2 I A(v) = O} . The approximations of U and P are denoted by UM,N and PM,N, respectively. A mixed Fourier-Legendre spectral scheme with the artificial comp)."ession for (6.20) is to find UM,N( X, t) E VM,N and PM,N( X , t) E WM,N for all t E R,. ( T) such that (D.,.UM,N (t) + d(UM,N (t) + orD.,.uM,N (t) , UM,N (t» , V) + j.ta(uM,N (t) + urD.,.UM,N (t) , v )

-b (V, PM,N(t) + (JrD.,.PM,N (t» = (J(t), v) , V v E VM,N, t E R,. (T), (3 (D.,.PM,N(t), W) + b (UM,N(t) + (JrD.,.uM,N(t), W) = 0, V w E WM,N, t E R,. (T), UM,N( O ) = Pk,N UO , PM,N( O ) = PM,N PO ,

(6.26) where 0, (J' and (J are parameters and 0 :5 0, (J', (J :5 1. If ° = (J' = (J = 0, then (6.26) is an explicit scheme. Otherwise, we need an iteration to evaluate the values of UM,N and PM,N at each time step. In particular, if 0 = (J' = (J = then by taking v = UM,N(t) + UM,N(t + r) and w = PM,N(t) + PM,N(t + r) in (6.26) , we derive from

�,

Chapter

6. Mixed Spectral Methods

275

( 6.25 ) that I I UM,N(t) 1I 2 + f3 I1PM,N(t) 1I 2 + � JLr =

:L .

.e�(t - .,.)

I IUM,N( O ) 11 2 + f3 I1PM,N (0) 1 I 2 + r

IUM,N(S) + UM,N (S + r) l �

:L

(f( s ) , U M, N ( S ) + U M,N ( S + r )) . ( 6. 2 7 )

This simulates the conservation ( 5.67 ) suitably. We now analyse the generalized stability of ( 6.26 ) . For simplicity, we only consider the case with D = u = 0, 0 > Assume that UM,N( O ), PM,N ( O ) , f and the right-hand side of the second formula of ( 6.26 ) have the errors u,p, j and 9 respectively. They induce the errors of UM,N and PM,N, denoted by u and p respectively. Then we get from ( 6.26 ) that

!.

(D.,.u(t) + d (u(t), UM,N(t) + u(t)) + d (UM,N (t), u(t)) , v) + JLa (u(t), v ) -b (v, p(t) + 9rD.,.p(t)) ( l, v) , 'V v E VM,N, t E R.,. (T) , f3 (D.,.p(t) , w) + b (u(t ) + OrD.,.u(t), w) (g, w) , 'V w E WM,N, t E R.,. ( T) , U ( O ) = uo , E n , p( O ) = Po , E n . ( 6.28 ) Let e > O. Taking v = u(t) + 9 rD"u(t) in the first formula of ( 6.28 ), we obtain from ( 1.22 ) and ( 6.25 ) that e) IID"u(t) 11 2 + 2JLlu(t) l� + 8JLrD.. lu(t) l� 3 D" lIu(t) 1I 2 + r( 2 0 - OJLr2 ID"u(t)l� - 2 b (u(t) + Or D"u(t), p(t) + Or D"p(t)) + :L Fj(t) j=1 ( 6.29 ) =

=

x

x

-1-

where

F1(t) = 2 (d (UM,N (t), u(t)) , u(t)) , F2 (t) = 2 0r (d (u(t), UM,N(t) + u(t)) , D"u(t)) , F3(t) = 20r (d (UM,N(t), u(t)) , D.,.u(t)) . Next taking w = p(t ) + OrD"p(t) in the second formula of ( 6.28 ), we get that e) IID"p(t ) 11 2 + 2 b (it(t) + OrD"u(t), p(t) + OrD.,.p(t) ) f3 D"lIp(t) 11 2 + f3r( 20 :5 f3l1 p(t) 1 12 + (� + 0;; ) II g(t) 1I2 . ( 6.30 )

-1-

276

Spectral Methods and Their Applications

Putting

( 6.29) and ( 6.3 0) together, it follows that + 2 1lIu(t) l� + BllrDT lu(t) l�

We now estimate easy to see that

3 - Bllr2 IDTu(t) l� + L Fj(t) j =I

I Fj (t) l . To do this, we need some bounds for II d(v, w ) lI . It is II d(v , w) 1I � cllvllwl , oo ll w lh , II d(v, w) 1I � cllvl h ll w llwl,OO .

By virtue of Theorem 6.1, we can also see that for any v , w E

VM,N ,

Using the above three estimates, we obtain

i

IFI(t) 1 � lllu(t) l � + c

(1 + � IIUM'N(t) II�T1'OO ) lIu(t) 1 I2 ,

cB2r cB2r M 2 N lIu(t) 2Iu(t) I�, I F2 (t) 1 � c rIlDTu(t) 1I2 + 1I c-IIUM,N(t) II�T1,oo llu(t) lI � + c cfP r I IUM,N(t) 1 IW1, 2 oo llu(t) lI 2l' 1 F3 1 � c r II DTu(t ) I 12 + c Substituting the above estimates into ( 6.3 1 ) yields that DT ( lIu(t) 1I2 + ,Bllp(t) 11 2 ) + r( 2 B - 1 3c ) ( IIDTu(t) 1I 2 + ,BIIDTP(t) 112 ) + Illu(t) l� + BllrDT lu(t) l � - Bllr2IDTu(t) l� � Al (lIu(t)1I2 + ,B llp(t) 112 ) + A2 (t) lu(t) l� + A3(t) ( 6.32 ) -

where

Chapter

6. Mixed Spectral Methods

277

Assume that c and qo are suitably small positive constants such that (6.33)

Then by Theorem 6.2,

Let E( v , w , t) = I l v (t) 1 I 2 + ,8 l l w(t)1I 2 +r

2:

seRr(t - T)

(qor I l DT v (s) 11 2 + qo,8r I I DT w( s) 11 2 + I v (s) l n (6.35)

and p(t) = lI uo ll 2 + ,8 l1po ll 2 + (} Ilr luo l � + r

By summing (6.32) for t

E

2:

seRr( t - T)

A3 (s) .

(6.36)

R.,. (t - r), we derive from (6.34) that

E (u, p, t) � p(t) + r

2:

seRr( t -T)

(At E ( u , p, s) + A2 (s) lu(s) I O .

By applying Lemma 3.2 to the above inequality, we obtain the following result. and 0, {} > !, r � bI ll in for some tt E R.,. (T) , then for all t E RT ( tI ) ,

Theorem 6 . 1 b35 . Let 0 p(tI) e b2 t, �

2 rM N

(6.26),

= a =

(6.33) hold.

If

where bI , b2 and b3 are some positive constants depending on Il and II UM,N II C(0,T;W1 , OO )

•

We next deal with the convergence of (6.26) with 0 = a = 0 and {} > !. Let PM1 0,N U, U = UM,N - UM,N and P = PM,N - PM,N P , We get from (6.20) and (6.26) that

[

UM,N =

( DT U(t) +_ d(U(t) , UM,N (t) + U(t)) + d( UM,N(t), U(t)) , v) + J1,a(U(t), v) - b(v, P(t) +(hDT P(t))

=

( Gt (t) + G2 (t) + G3(t), v) - b(v, G4 (t)),

(3(DT P(t) , w ) + b(U(t) + (hDT U(t), w ) U( O )

=

p( O )

=

0

=

't v E VM,N , t E RT(T) ,

- ,8 ( DT P(t) , w) + (Gs(t) , w ) , 't

w

E WM ,N , t E RT (T) ,

Spectral Methods and Their Applications

278

where

Gt(t) = 8tU(t) - DTUM,N (t), G2 (t) = (U(t) . V) U(t) - (UM,N (t) . V) UM,N (t), G3 (t) = "21 (V · UM,N(t)) UM,N(t), G4 (t) = p et) - PM, N P(t) - ()rDTPM,N P(t), Gs(t) = V (U(t) - UM,N (t) - ()rDTUM,N (t)) . -

•

By Theorem 6.5 and (4.24),

I Gt (t) 1 I 2 � cr l U ll k' (t,t+T;L' ) + � (M-2r + N- 28 ) I U l k1 (t,t+T;Mr, , ) .

By Theorems 6.5 and 6.6, we deduce that for

I G2 (t) 1 I 2

r, s > 1 ,

1 (U(t) . V) U(t) - (UM,N (t) . V) U(t) 1 I 2 + I (UM, N (t) . V) U(t) - (UM, N (t) . V) UM,N (t) 1 I 2 c ( II U(t) lI � l,oo Il U(t) - UM,N(t) 11 2 + I UM,N (t) 1I100I l U(t) - UM,N (t)II� ) ( M - 2r + N-2. ) ( I U(t) lI �r+ l" + l + I U(t) I tvl ,00 )

� 2

� �

C

U = 0, we derive in a similar manner that I G3 (t) 11 2 = I G3 (t) - � (V . U)U f � (M-2r + N-2. ) ( l 1 U(t) ll �r+ l" + l + I U(t) l I tv" OO ) .

Since V .

Obviously,

•

C

C

I G4 (t) 1 I 2 � (M-2r + N -2B ) I P(t) lI kr" + c()2 r ll P l k1(t,t+T;£2 ) , I Gs(t) 11 2 � (M -2r + N -2S ) II U(t) lI�r+ l , ' +l + c()2 r l U l k'(t,t+T;M" ' ) . C

Finally, by an argument as in the proof of Theorem 6.15, we get the following result.

Let 8 = q = O, () > �, r � b4J.t in (6.26), and (6.33) hold. If for

Theorem 6.16.

r, s � 1 ,

n

n

n

n

U E C (0 " T · H0t,1' Mr+1,s+ 1 Wt,oo) Ht (0 T · Mr,s) H2 (0 T · L2 ) P E e (0 T · Hr," ) n Ht (0 T· L2 ) then for all t E Rr (T) , E ( U M,N UM,N , PM,N - PM,NP, t) � bs (� ( r 2 + M-2r + N- 28 ) +(3) where b4 and bs are some positive constants depending on J.t and the norms of U and ,

,

p

P

-

P

in the spaces mentioned above.

,

,

p ,

"

p

"

p

'

Chapter

6.

279

Mixed Spectral Methods

We find from Theorem 6.16 that the best choice is f3 = OCr) = 0 (M- 2 r + N-28 ) , and the corresponding accuracy is of the order f3! . It seems that the artificial com pression brings the convenience in calculations, but lowers the accuracy. Indeed, if we compare the numerical solution to other projection of the exact solution, called the Stokes projection, we can remove this problem, and get a better error estimation for any f3 2': O. We shall introduce this technique in the next chapter. Obviously, since we use the approximation (6.24) for the nonlinear term ( U · V)U, . a reasonable simulation of conservation (6.27) follows, even the trial functions do not satisfy the incompressibility. Also because of this reason, the effects of the main nonlinear error terms are cancelled, i.e. ,

(d (u(t), u(t)) , u(t)) = (d ([r(t), [r (t) ) , [r(t)) = O. Therefore the nice error estimates follow. · Conversely, if the trial function space VM, N fulfills the incompressibility, then it is better to approximate the convective term by

d( UM,N, UM,N) ,

d( v , w)

=

81 (w(I)V) + 82 (W( 2 )V) .

In this case, the conservation can also be simulated properly. For the error analysis, we have

�

(J (u(t), u(t)) , u(t)) = (lu(tW, V · u(t)) .

As V . u(t) =I- 0 usually, this term cannot be cancelled. This fact shows that a good approximation to the nonlinear convective term plays an important role. Bernardi, Maday and Metivet (1987) applied the mixed Fourier-Legendre spectral methods first to semi-periodic problems of steady Navier-Stokes equation. The mixed Fourier-Chebyshev spectral methods have also been used for semi periodic problems of various nonlinear differential equations, such as the vorticity equation in Guo, Ma, Cao and Huang (1992) , and the Navier-Stokes equation in Guo and Li ( 1996). Recently, the Fourier-Chebyshev pseudospectral methods have also been developed, e.g. , see Guo and Li (1993, 1995a) .

CHAPTER 7 COMBINE D SP E CTRAL METHOD S

In many practical problems, the domains are not rectangular. We might use some mappings to change them to rectangular and then use pure spectral methods to solve the resulting problems numerically. But it is not easy to do so. On the other hand, such procedures usually bring in the complexity and errors. Thus it is better to adopt the finite element methods in those cases. However, sometimes the intersections of the domains may be rectangular, such as a cylindrical container. So it is reasonable to use the combined spectral-finite element methods or the combined pseudospectral-finite element methods. Since finite element methods are based on Galerkin method with interpolations of functions, the latter seem more attractive. In this chapter, we present the combined Fourier spectral-finite element methods for semi-periodic problems, and the combined Legendre spectral-finite element methods and the combined Chebyshev spectral-finite element methods for non-periodic problems. Finally, we take Navier Stoke� equation as an example to show the applications of these combined methods.

7. 1 .

Some Basic Results in Finite Element Methods

In this section, we recall some basic results of finite element methods in one dimen sion, as a preparation of the forthcoming discussions. Let A = ( - 1 , 1 ) and x E A.. Decompose A into 2 M subintervals D..j , -M $ j $ M - 1 . hj represents the length of D..j . Let h = -M<max j <M-l hj . Assume that there exists a positive constant p inde- -

pendent of the partition such that IIl,ax h $ p. Let k be non-negative integer -M$J$M-l j and nh , k be the Lagrange interpolation of degree k over each D..j . Define h

Sh , k = {v I v is a polynomial of degree $ k on D..j , -M $ j $

280

M - I}.

7.

281

Chapter Combined Spectral Methods

n

Moreover Sh,k = th,k n Hl (A) and S�,k inverse inequalities exist.

= Sh,k

Lemma ·7. 1 . For any E Sh,k and 1 ::5

p ::5 q ::5

HJ (A) . In the space Sh,k , several

00,

l1j

Proof. is a polynomial of degree at most k on each with the left extreme point = xU) . Let y x - xU») - 1, A = {y I l y l < I } and �(y) = (x). By virtue of

:( .

Theorem 2.7,

J

On the other hand,

Consequently, 1 I I I Lq(llj) ::5 c

from which the desired result follows.

Lemma 7.2. For any E Sh,k and 0

(��)

::5 r ::5

1

1

. - r;

1 I I I LP(llj ) •

1' ,

( h ) r -/I-

1 1 <1> 1 1 /1- ::5 c k2

I I l I r .

Proof. By Theorem 2.8 and an argument as in the proof of the previous theorem, we

deduce that Hence

1 1 <1> 1 1 /1- ::5

( h ) r-/I-

C k2

1 1 <1> 1 1 . ,

I'

=

0, 1.

The general result comes from induction and space interpolation. The L 2 (A)-orthogonal projection Ph ,k : L 2 (A) any v E L 2 (A),

--+

•

Sh,k is such a mapping that for

(7. 1 )

282

Spectral Methods and Their Applications

For simplicity, let c(k) be a generic positive constant depending on k. We have from Ciarlet ( 1 978 ) that for any v E � < r :5 k + 1 and 0 :5 I-' :5 min(r, 1 ) ,

W(A),

IJ v - n h,k V I I :5 c (k)hr-I-' Iv l r , I-' Il v - nh,kv lJwl',oo :5 c( k) hr-I-' Iv l wr,oo . Lemma 7.3. For

any v E Hr ( A ) , �

<

r

:5

k

+

1

and 0 :5

I-'

(7.2) (7.3)

:5 min(r, 1 ) ,

Proof. We first assume that r � 1 . Since n h,k V E Sh,k , it follows from Lemma 7.2 and

(7.2) that

I l v - Ph,k v l J l-' :5 Il v - n h,k l i l-' + I J n h,k v - Ph,kV 11 1-' < c(k ) hr - I-' I v l + ch-I-' I J n h,k V - Ph,k V I I . r

Since

we obtain from (7.2) that

Next let 0 :5 r

<

(7.4) 1 . By (7.4) and projection theory, I l v - Ph,k V I J :5 c( k) h l l v l l b I J v - Ph,k V I I :5 c(k) l J v l J ·

Thanks to the space interpolation, we assert that for any 0 :5 f3 :5 1 ,

O n the other hand, by (7.4) ,

Using the space interpolation again, we obtain that for 0

:5 f3 :5 1 ,

Chapter

7.

Letting fJ

=

Combined Spectral Methods

283

I=;' we complete the proof.

•

-+ Sh,k is such a mapping that for (a", (v - Pl,/c v) , a", ¢) + (v , ¢) 0, ¢ Sh,k . (7. 5 ) The HJ {A)-orthogonal projection: HJ (A) -+ S� , k is such a mapping that for any v HJ (A), (7.6) Lemma 7.4. If v Hr (A) , 1 $ $ k + 1 and 1 - k $ $ I , then v

E

any

The H I (A)-orthogonal projection: HI (A) H I (A),

=

v

E

E

E

Lemma 7 . 5 . If v

E

Jl.

r

H' (A) n HJ (A) , 1 $ r $

k + � and 1 - k $

Jl.

$ I,

then

Lemma 7.5 comes from Douglas and Dupont (1974) . Lemma 7.4 can be proved in the same manner. Guo and Ma (1991) gave a result similar to Lemma 7.3. Finally, we give a special Poincare inequality which will be used in the forthcoming sections. Let A = {y I a $ y $ b } , v E H I (}.) and I v ( Y * ) 1 = in! I v (Y) I .. Then ilEA

v 2 (y)

2

f I v ( z ) l l az v (z ) 1 dz + I v (Y* ) 1 2

$ 2 I 1 v l b (A) l v I H1 (A)

+ b � a II v ll �2 (A) '

and so (7.7)

If, in addition, the mean of v(y ) over A equals zero, then by the continuity, inf I v (y ) I = yEA O. Thus

284 7.2 .

Spectral Methods and Their Applications C o mbined Fo urier- Finit e Element Approximat io ns

We discuss the combined Fourier-finite element approximations in this section. Let = and n = l X We use the l = notations in Section such as H;,S (n) , H�:; (n) , M;,S(n) and X;,S (n) . and > O. Sh,k,l and Sh,k,l are the sets of functions on as are integers, .� VN, 2 is the set of trigonometric defined in the above section. S� ,k,l = Sh,k,l n Set v,.,k, N = Sh,k,l I8l VN, Vh,k, N = Sh,k,l I8l VN, 2 and on functions of degree V� k,N = S�,k,l I8l VN, 2 ' Several inverse inequalities are valid in the space Vh,k, N '

A

A A2 . {X l I I XII < 1}, A2 {X 2 1 0 < X 2 < 211"} k N 6.1, k 0, N A I, HJ(AI). :$ N , A 2 • 2,

Theorem 7. 1 .

00 ,

For any rfJ E Vh,k,N and 2 :$ p :$ IlrfJI I LP :$ C (k:N ) ;; - . IIrfJl I · 1

Proof. Let Then

rfJl E Sh,k,l

Therefore

1

rfJl (XI) = � 211" lA2r rfJ (x)e - il:&2 dX 2 . and so by Lemma 7.1, l i

II rfJ l t.. :$ lI rfJ l t�l I rfJI 1 2 :$ c (k:N ) - I rfJ U l '

which gives the desired result. Theorem 7.2.

•

11- ,

For any rfJ E Vh,k,N and 0 :$ r :$ .II rfJ lIl' :$ C ( kh + 1 ) r -I' I rfJl l r' 2 N

Proof. Let rfJl be as before. By Lemma 7.2,

11- =

0, 1.

(7.9)

Therefore 11- =

0, 1.

Chapter

7.

Combined Spectral Methods

(�) 2

285

It is easy to verify that

(� )

T - 2"

2T-2" E [2T I I ¢>t l l i'(A, ) :::; c I I ¢> I I : , p, = 0, 1 . . I I I� N _ Finally the desired result comes from induction and space interpolation. The L 2 (51)-orthogonal projection Ph,k,N : L; (51) Vh,k,N is such a mapping that for any v E L; (51) , (v - Ph,k,NV, ¢» = 0, V ¢> E Vh,k,N. Set f = mi n ( r , k + 1 ) . For simplicity, Ph,k denotes the L 2 (A 1 )-orthogonal projection on Sh,k, 1 , and PN denotes the L2 (A 2 )-orthogonal projection on VN, 2 ' Similarly, Pl.,k stands for the H 1 (A 1 )-orthogonal projection on 8h,k, 1 and P�:� stands for the HJ (A 1 ) orthogonal projection 'on 8£,10, 1 ' Theorem 7.3. If v E H; , S (51), r > ! and s � 0, th en 1 1 &� ¢> 1 I 2 :::; C

-+

IIv

-

Ph,k,NVIl :::; c(k) ( hr + N - ' ) I l v l l w,

• .

Proof. Let

Then Ph,k,NV(X) = Using Lemma 7.3, we obtain that I l v - Ph , k,NV I 1 2

E

I I I� N

E Ph, vI ( 1 ) e il", x

k

I'I� N

I l v l - Ph,k vt l l i' (A ' )

< c( k ) h 2 r

E

I I I� N

+E

•.

I I I> N

I l v t l l i' (Al )

I l v t l l �' (A, ) + CN-2B

E

I'I> N

I Ws l l v t l l i' (A, )

:::; c( k ) h 2 r l l v l l i' ( A. ,w (AI l ) + CN-2B l v l �' (A2 , L2 (A, ) ) '

In particular, for any v E H�:;(51) , r >

!

-

and s � 0,

The H 1 (n)-orthogonal projection Pl.,k,N : H; (n) for any v E H; (n) ,

( V (v - Pl,k,N V ) , V ¢» + (v , ¢»

-+

= 0,

Vh,k,N is such a mapping that V ¢> E Vh,k,N '

286

Spectral Methods and Their Applications

The H6 (n)-orthogonal projection for any E H6,p (n) ,

v

P�:�,N

:

( v (v - P�:�,N V) , V¢»

H6 ,p (n)

-+

V�k,N is such a mapping that v,.�k,N '

If v E M;, O ( n ) and r, s � 1, then Il v - Pl,k,N vt :s; c( k ) (hr- I + NI - . ) (hl -1' + NI'-I ) I v Il M" " If, in addition, for certain positive constants Cl and C2 , = 0,

V ¢>

E

Theorem 1.4.

p. =

0, 1 .

clN :s; h1 :s; C2 N,

then

(7. 10)

Proof. Let VI (Xl) be as before, and

v. (x) = LN (pl, k VI (Xl)) eil"' I) I �

By Lemma 7.4,

Il v - Pl, k ,N V I :

:s;

:s;

:s;

:s;

•.

I v - v. l I � v -P V L v -P V I I I �N ( 1 l l,k II I :'(A ) + 12 1 I l,k III :' (A1 ) ) + LN (I I vdlk1(A, ) + 12 1 I vdl i' (A, ) ) II I > c( k) h 2r- 2 ILN (i l vdl k' (A') + 12 1 I vd l k.- 1 (A, ) ) I I� + cN2 - 2 • LN 12 0- 2 ( 1 I vdl k1(A') + 12 1 I vdli2 (A1 ) ) III> 1 k) c( (hr + NI -O r Il v l �.. . . 1

S o the first conclusion i s true for

If v E M;" (O)

n

p. =

1 . The rest of the proof i s clear.

and r, s � 1 , then I v - P�:�,N vt :s; c( k ) (hr- 1 + NI -o ) (h l -1' + NI'-I ) I v I M" " If, in addition, (7.10) holds, then

Theorem 1 . 5 .

H6,p (n)

p.

(7. 1 1 ) _

= 0, 1 .

Chapter

7.

Combined Spectral Methods

287

Proof. Let VI(Xl) be as before and

v. (x) 1 L:1:5N (p�:�vl (xd) eil:r2 . Il v - p�:�,NV l l � c in! I v -wi t � cl v - v.l l · =

Then

W EVh,k,N

By an argument as in the der�va.tion of (7. 1 1 ) , we prove the first result for Finally by means of the duality and (7.10), we complete the proof.

Il p�:�,NV l wr,p . Let (7.10) hold, k and r, > ! . For any v H; (A 2 ' Hr (A 1 )), Il p�:�,NV I LOO � c( k) l I v I l H' (A2 ,W(Ad) · If, in addition, v then We can also bound the norm

�

Theorem 7.6.

E

1

s

E

fJ, =

1. _

HJ,,,(fl)

n

X;" (fl) ,

VI (Xl ) be as before and pl,k,NV (X) 1 L:1:5N vi (xl) eil:r2 . Obviously vi S� , k , l . By an argument as in the proof of Theorem we know that for any S� , k , l ' I VI - vil l �' (Ad 12 vI -vil l 12 (Ad � c( l I vl - l k1 (Ad 12 vI - 1 I 12 (A') ) · By taking P�:� v" we obtain from Lemma that Proof Let

E

=

E

=

6.6,

+ 11

+ 11

7.5

By means of the duality, It is clear that for fJ,

=

( 7.12) 0, 1 ,

Spectral Methods and Their Applications

288

We estimate the terms on the right-hand side of (7. 13). Firstly, we have

1 81 ViI I LOO(A,J :::; 1 81 Vi - nh,k 81 VdI LOO (A, ) + 1 81 v/ - nh,k 81 VdI Loo (A, ) + 1 81 VdI LOO (A, ) ' (7. 14) From Lemma 7. 1 , ( 7. 2 ) and (7.12), 1 81 vi - nh,k Bt vdI LOO (A, ) :::; c(k) h - t ( l i vi - vdI Hl (A, ) + 11 81 v/ - nh, k 81 vdI L2 (Atl ) < C(k) ( l I v dI H!(Al) + hr- t l vdlw+ l (Atl ) . By (7.3),

11 81 v/ - nh,k 81 VdI Loo (A1) :::; c(k) Il vdl w+l (A1) '

By imbedding theory,

11 81 VdILOO (A1 ) :::; I vdlw+ l (Al) '

The above statements yield that

:::; c(k) /EN Il vdl w+ l (Atl I I� 1 " " :::; c(k) ( I /EI �N ( 1 + 12 r ' ) ( I /EI �N ( 1 + [2 ) " I Vdl kr+ l (Atl ) (7.15) :::; c(k) l I v I l H'(A2 ,W+ l (Al)) ' 1

Similarly, it can be shown that

v , I /EI �N I iIl LOO (A, J :::; c(k)llvI l H'(A2 ,W (At l ) 1 E v I / I �N III I i I Loo (Atl :::; c(k) l I v I l H·+ (A2,W(At l ) ·

The combination of (7.13) and (7.15) -(7.17) completes the proof.

(7. 16) (7. 1 7) •

Finally we consider the combined Fourier pseudospectral-finite element approxi be the Fourier interpolation on the interval mation. Let associated with the set of interpolation points. Define = = Set 2� [ = = 2N 1 E

IN

A2 , h,k,N Ph,k IN INPh,k ' 1· dX N v(x)W(X) l (v, v v) ' II l i k i + "'2 EA2,N A1

(V, W)N Then for any v E e (A 2 ' L2(Al )), (v - Ih,k,Nv,
A2 ,N,

Chapter

7.

Combined Spectral Methods

Theorem. 7.7. [I v

Proof Let

E

H; , 8 (fl)

289

and r, s > ! , then

Eh,k ,N = Ph,k,N V - h,k ,NV, Then

By using Theorem 7.3 and the aim.

an

argument as in the proof of Theorem 2.4, we achieve •

Canuto, Maday and Quarteroni (1982) first proved T�eorem 7.3. Guo and Ma ( 1 991) obtained some results similar to Theorem 7.6. Here we prove them in a different way and improve the corresponding results.

7.3.

C ombined Legendre- Finite Element Approximations

In this section and the next section, we consider non-periodic problems with rectangu lar intersection of the domains. In this case, we can use the combined Legendre-finite element approximations. For the sake of simplicity, we only consider two-dimensional problems. Clearly it is not necessary to use such kind of combined approximations for two-dimensional problems. But it is easy to generalize the corresponding results to the three-dimensional problems. Let A = X A 2 . The integers x 2 < 1 } and fl = A l x l < 1 } A2 = 0 and N > O. The notations have the Pl. " and same meanings as in the previous section. Let lP N, 2 be the set of all polynomials of degree at most N defined on A 2 , and lP�, 2 = lPN, 2 nHJ (A 2 ) . For simplicity, let PN, PJr and denote the L 2 (A 2 )-orthogonal projection on lPN, 2 , the H I (A 2 )-orthogonal projection on lPN, 2 and the HJ {A 2 )-orthogonal projection on lP�, 2 ' respectively. Set = th,k,l ® lPN, 2 , = n HJ (fl). In the ® lPN, 2 and l1, ,, ,N = space several inequalities are valid.

k 2::

l {X l I l

pi/ Vh,k ,N Vh,k ,N ,

Theorem 7.8. For

,

{X I I 1 l h , !i2h,,, ,l , Sh,k ,l , S�,,, ,l ' Ph,k ,

Vh,k ,N Sh,k ,l

any ¢i E Yh,k,N and 2 5 p 5 00,

1 ¢i I LP 5 C ( k 2�2 ) ;-! 1 ¢i 1 ·

Vh, k,N

p�:f

Spectral Methods and Their Applications

290

Proof. For each fixed X 2 E A 2 , ¢>(Xb ') E Sh,k,l and for each fixed X l E A I , ¢>( ' , X 2 ) E lPN, 2 ' By Lemma 7. 1 and Theorem 2.7,

ck 11 ¢> II Loo :::; sup I I ¢> I I LOO (AI ) :::; . fL sup 11 ¢> lI v ( A I ) xEA2 x2 EA2

yh

� ( r sup ¢>2 (X) dX I ) t :::; ck: II ¢> I I . yh y h lA, x2 EA. c

<

Thus and the conclusion follows.

•

Theorem 7.9. For any ¢> E li,k, N and 0

Proof. Let

:::; r :::; 11 ,

N

1=0

¢>(x) = :L ¢>I(X I )L,(X 2 ) ' Then ¢>,(X I ) E Sh,k,l ' As in the proof of Theorem 7.2, we have (7.18)

On the other hand, 02 ¢>( X) =

N

:L ¢>1 (X I ) 02 L,(X 2 ) ' 1=0

By (2.49) and the same argument as in the proof of Theorem 2.8,

II02 ¢>(X I , · ) II L' (A. ) Therefore

N

:::; :L fi{l+l) I ¢>,(X I ) I :::; cN2 11 ¢>(XI , · ) ll v (A. ) · 1=0

r

II ¢> (X I ' · ) l I i' (A. ) dX I = cN4 1 1 ¢> 1 1 2 I I 02 ¢> 11 2 :::; cN4 lAl

which together with (7.18) imply the result for 11 = 1 and r = O. The proof is complete by induction and space interpolation. • The L 2 (n)-orthogonal projection Ph,k, N : L 2 (n) Vh,k,N is such a mapping that for any v E L 2 (n) , (v Ph,k,NV , ¢» = 0, 'if ¢> E Vh,k, N .

-

-+

Chapter

7.

Combined Spectrill Methods

Theorem 7.10.

If v E HT"' (!1) , r > !

291

and s :2: 0, then

Proof. Using Theorem 2.9 and Lemma 7.3, we find that Ilv

- Ph,k,NVi l

In particular, for

::; II·v - PN Ph , k Vl l ::; IIPN (v - Ph,k ) II + IIv - PNvi l ::; �( k ) ( hr llvllu(A" H'(A' ) + N- s llvllH' (A,p (A1» ) < c( k) ( hr + N -' ) IIvll w,"

v E H�"

•

(!1) ,

The HI (!1)-orthogonal projection for any v E H I (!1) ,

Pl,k,N

:

HI (!1)

-+ Vh,k,N is such a mapping that

(\7 (v -PI, k,N v) , \7 9'» + (v, 9'» 0,

The HJ (!1)-orthogonal projection any v E HJ (!1) ,

P�:�,N

=

:

HJ (!1)

V 9'> E

Vh,k,N .

-+ Vh�k,N is such a mapping that for

If v E MT, S (!1)

and r, s :2: 1, then fo r J.L = 0, 1, Il v - Pl,k,N v t ::; c( k) ( hr-I + h-I N- s + NI- S ) ( hl - 1L + NIL-I ) IIvIIM" " If in addition (7.10) holds, then

Theorem 7 . 1 1 .

Proof. Let v.

=

Ph,k P�V Vh,k,N . It is easy to see that E

Using Lemma 7.3, we obtain

IIv

-

Ph,kVl l �

<

12 IIv -Ph, k Vl l �' (AI ) dX 2 + 12 I 02 V - Ph,k 02 Vl l p (A' ) dX 2 (7.20) c( k ) h2r 2 ( IIv lli2 (A2,W (A,) + IIvllJf'(A2,W-'(A,) ) -

.

292

Spectral Metbods and Tbeir Applications

By Lemma 7.2 and Theorem 2.10,

Il ph,k (v - Pkv) I : i2 1l Ph,k (v - Pkv) I :, (A, ) dX2 2 ) dX 2 + lA I l ph 'k 82 (v - Pkv ) 1 1 P(A, 2 =

f

:::; c(k)h -2 N-2 ' ll v I H'(A2 ,P (A, » ) + c(k)N2 - 2s l v l 1- '(A2 P (A, )) '

Therefore the first result holds for E

Jl

= 1 . The remaining proof is clear.

Mr , s(!1) n HJ (!1)

•

and r, 2': 1, then for J-l = 0, 1, Il v -P�:Z,N v l lL :::; c(k) (hi'-I + h-I N-s + NI -S) (hl -1L + NIL-I ) I v I Mr" . If, in addition, (7.10) holds, then

Theorem 7 . 1 2 .

If v

s

Ph,kP]/V We now estimate I l p� 'Z N V I Wr, oo .

and following the same line as in the proof of = • Theorem 7 . 1 1 , we can prove this theorem easily.

Proof. By letting v*

Let (7.10) hold, k 2': 1 and r > ! . For any v I

Theorem 7.13.

Ht (A 2 ' W (A I )),

I

E

M�'& (!1) n HJ (!1) n

Il p�:Z,N v too :::; c(k)llvI I M� '� (n)nHt (A2 ,W(A')) ' If, in addition, k 2': 2 and v MM (!1) (!1) xr,t (!1), then Il p�:Z,NV l wl 'OO c(k)llvI I M�'�nxt'tnxr, i ' E

n

x � '�

n

:::;

Proof. We first have that

Il p�:Z,NV I LOO :::; Il p�:Z,NV - nh,k PNV I Loo + I nh,k PNVI l Loo .

(7.21)

Further, using Theorems 2.9, 7.8 and 7.12, and (7.2) yields that

Il p�:Z,N V - nh,k PNv too

:::;

:::;

C�N Il p�:Z,N V - nh,k PNv l

c(k) N ( I Ph:l k,N V - v l + Il v - nh,k PNVI I ) O v'h :::; c(k) l I v Il MH '

(7.22)

Chapter

7.

Combined Spectral Methods

On the other hand, we use an LOO ( A 2 ) -estimate in Section 2.4 to obtain that

I PNV Il L=(A2 ) ::; I PNV - v Il L=(A2 ) + I v Il L=(A2 ) ::; cNf - l'll v I HI'(A2 ) + cllvIl Ht(A2 ) ::; cl l vI I Ht(A2 ) '

293

(7.23)

By imbedding theory, (7.2) and (7.23) ,

I nh,k PNV Il L= ::; c IIn h,k PN VIl L=(A2 ,W(Al)) ::; c ll v Il Ht(A2 ,W(A' )) "

Then the first conclusion comes from (7.21), (7.22) and (7.24) . We now turn to prove the second result. Let = we deduce that By the definitions of and

(7.24) E

v. P�:Z,NV - P�:ZpJ./v V�k,N ' P�:Z P�:Z,N ' I v.l � (V (P�:Z,N V - P�:Zpj/v) , Vv.) = (V (v - P�:ZpJ./v) , Vv.) (V (v - pJ/v) , Vv.) + (VPj/ (v - P�:Z v) , Vv.) (01 (v - pj/ v) , 01 V.) + (02 (v - P�:Z v) , 02 V.) ::; Iv.I I ( 1 01 (v - pJ./ v) I + 1 02 (v - P�:Zv) I ) .

Hence by Theorem 2 . 1 1 and Lemma 7.5,

I v.I I ::; c(k) (h + N1 r I v l x " " i!

(7.25)

The combination of Theorem 7.8 and (7.25) yields

I v.llw1 , = ::; c(kVh)N I v.II I ::; c(k) l I vll x H '

( 7 .26)

On the other hand, we have that

1 I 0lP�:�p�o v I L= ::; Al + A2 + A3

where

I 0lP�:� P�ov - OI P�:Z pNv I L= ' A 2 = 1 I 0lP�:Z PNV - nh, k PNolv I L = , A3 = I nh, k PNOl VI l L= · Al =

By Theorems 2.9, 2 . 1 1 and 7.8, and Lemma 7.5,

Al

cj; I 0I Pi,'Z (p�Ov - PNV) I c(k )N Il pNl,ov - PNV I £2 , ' (A2 H (A, )) Vh ::; c(k) l I v Il Ht (A2 ,Hl (A'))" <

<

(7.27)

Spectral Methods and Their Applications

294

By Theorems 2.9 and 7.8, Lemma 7.5 and (7.2),

A2 � cUJ( (1I lh Pi:fpNV - OlPNvll + I OIPNV - nh, k Ot PNv l 1 ) � c(k) l I v l l p (A2 ,H! (A, » " By imbedding theory, (7.2) and (7.23) ,

A3 � c( k) l l v II H t (A2 ,H �+6(Al )) '

0

> O.

Similarly, we have that (7.28)

where

Bl Il pi:f82 Pko v - nh,k 82 pko v I Loo , B2 Il n h'k (82 PkOv - PN 02 V) I Loo ' B3 I n h,k PN02 VII Loo . = =

=

It can be verified that

and

Bl � cUJ( I (pi:f - nh,k ) (82 PkOv) 1I � c( k) l I v I I Hl (A2 ,H � (Al )) ' B2 � cUJ( ( l I 02 Pko v - 82 v I + I PN82 v - 82 vl I ) � c(k) l I v I I Hi (A2 P (At )) B3 � c(k) lI v I I Ht (A2 ,H! +6(At } ) '

8 > O.

By substituting the above estimates into (7.27) and (7.28) , we assert that

I pi:fpkO V l wl .OO � c( k ) l I v I I MH nx H nxr.t ·

(7.29)

Finally we complete the proof using (7.26) and (7.29) .

•

We now deal with the combined Legendre pseudospectral-finite element approx imation. Let be the Legendre-Gauss-Lobatto interpolation on the interval associated with the interpolation points x�i) and the weights w(i) , j N. Define = = Set

IN h,k ,N Ph,k IN INPh,k '

0� �

IIvllN

Then for any v E e

( A2 ' L2 ( Al)) ' (v h, k , N V , I/J ) N 0, V I/J Vh, k ,N ' -

=

E

=

(v, V )k ' 1

A2 ,

Chapter

7.

Combined Spectral Methods

Theorem 7.14. If v

E

Hr , 8(n)

295

and r, s > !, then

Proof. The conclusion follows from Lemma 7.3 and (2.63) .

7.4.

•

Combined Chebyshev -Finite Element Approximations

This section is devoted to the combined Chebyshev-finite element approximations. Let A I , A 2 and n be the same as in Section 7.3. Also, the notations Sh,k,l , Sh,k,l , SZ,k,l , IPN, 2 , IP�, 2' Vh,k,N , Vh,k,N , V� k,N and Ph,k , Pl,k , P�:� have the same meanings as in that xD - ' . Define section. Let w(x) = W(X 2 ) = (v, W )

=

(1 - ,

w in v(x)w(x)w(x 2 ) dx , Il v l w

=

,

(v, v) � . We still use the notations L�(n) , H: (n) , Hlijn) , W�,p (n) , H:, 8(n), H�:�(n) , M�" (n) and x:, 8(n) as before. Moreover, PN, PJv and pj/ stand for the L� (A 2 )-orthogonal projection on IP N, 2 , the H� (A 2 )-orthogonal projection on IPN, 2 and the H�, jA 2 ) orthogonal projection on IP�, 2 respectively. We first give some inverse inequalities. 7. 1 5 . 2 :::; p :::; 00 , E Vh,k,N

For any ¢J

Theorem

and

Proof. By Lemma 7.1 and Theorem 2.13, we have that ! c� sup 1 ¢J1 1 £2 (A, :::; c� ( f sup ¢J 2 (X) dXl ) :::; OO(A, L I ¢J I L"" < ",sup ¢JII I ) ) h ",. EA. .EA. h lA, ",.EA. V

< Therefore

V

Ck /f; (i, 11¢J (Xl o ·) llh(A.l dXl ) Ck /f; II¢Jl l w . !

=

( �1 ¢Jll ir.; :::; 11 ¢Jll t� I ¢JI I � :::; C k2t ) · II¢JII � 1

and the desired result follows.

Theorem 7.16. For any ¢J E Vh,k,N and 0 :::; r

:::;

fl ,

•

Spectral Methods and Their Applications

296

Proof. Let T/(x2) be the Chebyshev polynomial of degree 1 on 11. 2 , and

N

4>(x) = :E 4>1 (X l) TI ( X2) . 1=0

On the other hand,

N

024> (X) = :E 4>t(X l ) 02TI (X 2 ) ' 1=1

By (2.72) and an argument as in the proof of Theorem 2. 14,

Therefore The previous statements imply the result for I-' = 1 and r = O. The proof is complete by induction and space interpolation. • The L�(n)-orthogonal projection for any E L�(n) ,

v

Theorem 7 . 1 7. If

v

E

H�, ·(n) , r >

Ph,Ic,N

� and

:

s

L�(n)

� 0,

-+

Vh,Ic ,N is such a mapping that

then

Proof. Using Theorem 2.15 and Lemma 7.3, we find that

I v - Ph,.\:,N VI L

< <

I v - PN ph, ,, v l w :::; I PN (v - Ph, ,, ) 1 I w + I l v - PNv l w c(k ) ( hi" + N -' ) II v I l H;' " •

Chapter

7.

Combined Spectral Methods

297

v

In particular, for E H�::(o') ,

The H� (o')-orthogonal projection for any E H� (o') ,

v

Pl,k,N

:

H�(o')

�

Vh,k,N is such a mapping that

v

( (v - Pl,k,N V) , V r/Jt + (v , r/J)w = 0 , V r/J E Vh,k,N . The HJjo')-orthogonal projection P�:Z,N HL(o') � Vh� k,N is such a mapping that for any v E HJjO,) , (V (v - P�:Z,N V) , V( r/Jw )) 0 , V r/J E Vh�k,:.r · Theorem 7 . 1 8 . If v E M� S (O,) and r, s 2: 1, then :

=

'

If, in addition, (7.10) holds, then Proof. Let v. = Ph,k P� V E Vh,k,N . Then By Lemma 7 . 3 ,

Il v - Ph ,k VI I � ,w

S;

t2 (I I v - Ph, k V I �' (A,) + I o2 V - Ph, k 02 VI I �' (A, ) ) W(X 2 ) dX 2 ( 7. 30) c(k) h 2i'- 2 ( 1 I v l h (A2,w (A1» + I v I l Jf!(A2,W -l (Al» ) .

By Lemma 7.2, Theorem 2.16 and projection theorem,

Il ph,k (v - p� v) I : w

= S;

t, l ph,k (v - Pj. v) I :' (A,) W(X 2 ) dX 2 2 ) W(X 2 ) dX 2 + lA,r I l ph,k 02 (v - Pj.v) 1 P(A,

c(k) h -2 N- 2 8 1 I v ll ��(A2,L2 (Al )) + c(k) N2 - 2Sllvl l ��(A2,P (Al » '

Therefore the first result is true for

f.L

= 1 . The rest of the proof is clear.

( 7 . 31 ) •

298

Spectral Methods

If v E M�' · ( fl ) n HJ,w( fl )

an d

Their Applications f.L =

and r, s ;?: I , then for 0, I , I l v - Pl:�,NV II ,. ,w � c ( k ) ( h i'- l + h-l N - · + Nl-B) ( h l-,. + N,.-l) IvIM! If, in addition, (7.10) holds, then Theorem 7.19.

.• .

Proof. We have from Lemma 2.8 that

I l v - pl:�'N Vrl ,w � � wEViI�h,f IIv - w l l tw' •••N

Then by Lemmas 7.2 and 7.3, and Theorem 2.18, we can follow the same line as in the derivations of (7.3 0) and (7.31) to get the • conclusion. Take

w

=

v.

=

ph,/c pJ./ v

Vh� /c,N '

E

I l pl:�,N Vllw=.p . hold, k ;?: 1 and

We now estimate the norm Theorem 7.20.

H� ( A 2 , Hr ( A l )),

Let (7.10) E

r, s > ! .

E

For any v HJ ( fl ) n

I l pl:�,N V IILOO � c( k ) l IvI lHJ, nH':(A.,W(Ad ) ·

If, in addition, v M;,2 ( fl ) n X�'· ( fl ), then Il pl '� N V I I Wl,oo � c( k ) l Iv llM," 'nx" " Proof. We have Il pl:�,N V IIL oo � I l pl:�,N V - nh ,/cPNV IIL oo + IInh,/cPNVIlLoo ' ,

Moreover, using Theorems

Ilpl:�,N V - nh, k PNv l l Loo

� �

I

w

w

2.15, 7.15 and 7.19, and (7.2), we get that

� Ilp�:�,NV - nh,/cPNv lI w C( k ) � (l lp�:�,N - v l L + IIv - n h ,/cPNV l lw) � c( k ) lIvlh.w . C( k )

On the other hand, we use an LOO ( A 2 ) -estimate in Section 2.5 to obtain that for any f.L > 0,

I!PN V I l LOO(A.)

� �

c I!PNV - vIILOO(A.) + IIv Il LOO(A.) c (l + In N)N-" IIv l lw&. OO (A.) + IIv I l L OO(A.) '

Chapter

7.

Combined Spectral Methods

Taking Jl <

S

299

- � , we get

(7.32) Therefore, we have from imbedding theory that

The above statements lead to the first conclusion. We next prove the second result. Let = Lemma 2.8,

E

v. P�:�,N V - p�:�pi/v Vh�k,N ' Then by

� l I v.I I �,w

� � <

=

("1 (p�:�,N v - p�:�pi/v) , "1 (v.w)) ("1 (v - p�:�p�/v) , "1 (v.w)) (V (v - pi/v) , V (v. w) ) + ("1Pi/ (v - P�:� v ) , V (v.w) ) (81 (v - p�Ov) , 81 v.L + (82 (v - P�:�v) , 82 (Vow)) I v.l l ,w ( 1/ 81 (v - p�Ov) t + 1 82 (v - p�:�v) I U ·

Thus by Theorem 2.18 and Lemma 7.5,

from which and Theorem 7.15, we assert that

On the other hand,

Bj o PN, P� II ' I H� (A, ) ,

1 81P�:� p�ov I LOO � Al + A2 + A3, 1 82 P�:� p�Ov I Loo � Bl + B2 + B3

where Aj and are of the same forms as in (7.27) and (7.28) . The only difference is that are now the orthogonal projections associated with the norms and respectively. Therefore by Theorems 2.15, 2.18 and 7. 15, and Lemma 7.5,

Al

� c(k ) � 1 81P�:2 (p�O v - PNV) t � c(k)l I vI I H� (A2 ,H' (A, )) ,

By Theorems 2.15 and 7.15, Lemma 7.5 and (7.2),

A2 � C(k) � I 81 pl:2PNV - nh,k 81PNvt � c(k ) ll v I L� (A2 ,H2 (A&

1 · I La (A, )

300

By imbedding theory, (7.2) and (7.32) ,

Spectral Methods and Their Applications 6'

> O.

In the same manner, we verify that

B1 � B2 � B3 �

c(k ) ll vI I H!(A2 ,Hl (Al» ' c(k)ll v I H�(A2 ,L2 (A, ) ' c(k)ll v I Hi + 6 (A2 ,H � + ' (A, ) '

6'

> o.

The above six estimations together with (7.33) provide the bound of •

I l p�:2,N V l wl 'OO '

Finally we consider the combined Chebyshev pseudospectral-finite element ap proximation. Let IN be the Chebyshev-Gauss-Lobatto interpolation on the interval 11. 2 , associated with the interpolation points x� ) and the weights w (j ) , O � j � N. Define = Ph,kIN = INPh,k . Set N 1 (

h,k,N (V, W)N = � JfAI V (X t , X 2(j») W (X t , X 2j» ) (j) dx t , I vll N = (v, v) 1 · Then for any v E e (11. 2 , L2 (A 1 )), (v h,k,NV , ¢» N = 0, V ¢> E Vh,k,N ' Theorem 7 . 2 1 . If v E H�,S(f!) and r, s > �, then W

-

Theorems 7.17, 7.19 and 7.20 can be found in Guo, He and Ma ( 1995) .

Proof. The result follows from Lemma 7.3 and Theorem 2.19. 7.5.

•

Applications

In this section, we again take the two-dimensional unsteady Navier-Stokes equation as an example to describe the combined spectral-finite element methods. We focus on the combined Fourier pseudospectral-finite element schemes. But the main idea and the technique used can be generalized to non-periodic problems defined on a

Chapter

7.

Combined Spectral Methods

301

three-dimensional complex domain, and to other kinds .of combined spectral-finite element methods and combined pseudospectral-finite element methods presented in the previous sections. Let if, P and I' > 0 be the speed vector, the pressure and the kinetic viscosity. We consider the semi-periodic pr.oblem and require P ( x , t) to be in the space and A ( P ( x , t)) = for 0 :::; t :::; T. The L� ,in) , i.e. , P has the period for bilinear forms a ( v , w) and b ( v, w) are given by We adopt the artificial com pression ( .2 3 ) with the parameter (:J � O. When (:J = it is reduced to the original problem and so requires the incompressibility automatically. As explained in Section 6.3, a reasonable approximation of non-linear convective term plays an important role in preserving the global property of the genuine solution and in increasing the com putational stability and the total accuracy of n�merical solution. But it is not easy to do so for pseudospectral methods, since the interpolation and the differentiation do not commute. Besides, for the improvement of stability, a filtering will be used. Now let � 0 and N > 0 be integers. We use all the notations of Section 7.2. For simplicity, let be the set of scalar be the set of vector functions and functions, defined by

(6. 22) 271" .X 2 (6. 21 ).

6

0

0,

k

Vh,N

Wh,N

Wh,N = {w I W E Sh,lc,l VN,2 ,A(w) = O } . ®

The interpolation n h , lc+ 1 is denoted by nlc+ l for simplicity. Furthermore, for v (x)

N

=

L v/ (xl ) e i/",. , /=0

the filtering with single parameter a � 1 is given by

( ) v = � ( 1 - 1 � 1 " ) v/ ( xl ) ei/"'2 . 1 1 D(v , w , z) = 2 (( w . V') v , z) - 2 ( ( W . V')z, v) .

RN V = R

N a

Let

If V' . w = 0, then D( v , w , z) = (( w . V' ) v , z) . Thus we define the following trilinear form d(v, w , z) = 2 . V')z), v) . V') v ) , z) 2

1 (IN((w ·

1 (IN((w

We shall approximate the nonlinear term by d(v , v, z) . Clearly -

d (v , w , z) + d( z, w , v) = o.

(7.34)

302

Spectral Methods and Their Applications

Guo and Ma (1993) proposed a scheme for (6.22) . Let 7 be the mesh size in time. Uh,N and Ph,N are the approximations to U and P, respectively. 8, and 0 are parameters, 0 ::; 8, 0 ::; 1 . The combined Fourier pseudospectral-finite element scheme for (6.22) is to find Uh,N(X, t) E Vh,N, Ph,N (X, t) E Wh,N for all t E Rr(T ) such that (DrUh,N(t) , v ) (RNUh,N(t) 87 RNDrUh,N(t), RNUh,N(t), RNV) (Uh,N(t) DrUh,N(t), v ) - b (V, Ph,N(t) 0 7 DrPh,N(t » v E Vh,N, t E J4 (T), = (IN nk+1 J(t) , v ) , 0 [1 (DrPh,N(t), w) b (Uh,N(t) 7 DrUh,N(t), w) = 0, w E Wh,N, t E J4(T),

a-

a-,

+d + a-7 +

+/La

+

+

+

Uh,N(O) = IN nk+1 Uo , Ph,N(O) = IN nk+1 Po

V

(7.35)

where Po is determined by �Po = 'V . (f - (Uo ' 'V) Uo) .

a-

If 8 = = 0 = 0, then (7.35) is an explicit scheme. Otherwise a linear iteration is needed to evaluate the values of unknown functions at the interpolation points. In particular, if 8 = = 0 = �, then we get from (7.34) and (7.35) that 2 1 2 2

a-

II Uh,N(t) 1I =

+ f3 IIPh,N(t) 1 I + 2/L 7 Bellr(t-r) 1: IUh,N(S) + Uh,N(S + 7) 1 1

l I u h,N( 0 ) 11 2 + f3 I 1 Ph,N( 0 ) 1 I 2 + 7 1: ( IN nk+ 1 J(S) , Uh,N (S) + Uh,N (S + 7 » . BeRr(t - r)

Clearly it simulates the conservation (5._6 7) properly. We analyze the generalized stability of scheme (7.35). Let constant such that for any v E Vh,N ,

CI = cI ( k ) be a positive (7.36)

Assume that

a- > -21 '

or

A = 7 ( h - 2 + N2 ) < /LCI ( 12- 2a-)

Let c be a suitably small positive constant and qo > two cases, ( i ) 20a- ;::: 0 ( ii ) 20a- ::; 0

O.

(7.37) '

We consider the following

+ a- + 2c and 20 ;::: 1 + 2c + qo, + a- + 2c and -1 20 ;::: (1 + 2c + qo + /LACI( a- + 2c » ( 1 - /LACI G - a-))

(7.38) (7.39)

Chapter

7.

303

Combined Spectral Methods

Uh,N(O),Ph,N (O), !N nk+1 j and the right-hand side of the second uo , p, j and g , respectively, which induce the errors ( 7.35 ) , Ph, N u p for simplicity. Then from ( 7.35), it follows that Uh,N (DTU(t), v) + d (RNU(t) + 87" RNDTu(t), RNUh,N (t) + RNU(t), RNV) + d (RNUh,N (t) + 87" RNDTUh,N (�)' RNU(t), RNV) + J-Ia (u(t) +0"7" DTu(t), v) - b (v,p(t) + 97" DTP(t)) = (j(t) , v) , v E Vh,N , t E R, (T), ,8 (DTP(t), w ) + b (u(t) + 97" DTu(t), w) (g(t), w ) , V w E Wh,N , t E �( T). ( 7.40 ) Take v = u + 97" DTu in the first formula of ( 7 . 40 ). Since 1 297" (DTu(t), j (t))1 � e J-l7" 2 I D.Tu(t) 1 21 + c9e 2 1 Jet) 1 2- 1 ' J-l we get from (1.22) and ( 7 .34) that DT I l u(t) 11 2 + 7"( 29 - 1) I DTu(t) 1 I 2 + J-I(2 - e) l u (t) l � + J-I7"(0" + 9)DT l u (t) l � +J-I7" 2 (290" - - 9 - e) I DTU(t) l � - 2b (u(t) + 97" DTu(t),p + 97" DTP(t)) ( 7 .41) + t Fj (t) � C (le: 92 ) I l j(t) l � l

Suppoee that have the errors formula of denoted by and and of

{

=

0"

where

2d (RNUh,N (t) + 87"RNDTUh,N (t), RNU(t), RNU(t)) , 297" d (RNUh,N (t) + 87" RNDTUh,N (t), RNU(t), RNDTu(t)) + 27"( 9 - 8)d (RNU(t), RNUh:N (t), RNDTu(t)) , = 2 7"(9 - 8)d(RNU(t), RNU(t), RNDTu(t)) . Next take w p + 97" DTP in the second formula of ( 7. 4 0). We obtain that ,8DT I p (t)1 1 2 + ,87"(29 - 1 - e) I DTP(t)1 I 2 +2b(u(t) + 97"DTu(t),p(t) + 97"DTP(t)) � ,8 I p(t) 11 2 + (� + :;) I g (t) 1 2 . ( 7.42 ) Putting ( 7 .41) and ( 7. 4 2) together, we derive that F1 (t) F2 (t) F3 (t)

=

( 7 . 43)

Spectral Methods

304 We are going to estimate

an d

Their Applications

I Fj (t) l · Let v, w, z E Vh,N and

By virtue of (2.23), Theorems 2.1, and Theorem 2.4,

Bq(v, w , z) � l (wOqV, Z)NI + l (wOq Z, V)NI � c(k) lIwll (IIIN(zoqv) II + Il v llLoo lzh) � c(k)lI w ll Olzoq VII + II v IlLoo lzh ) · Moreover, for any 0

< 'Y <

!,

(i, II Oqv ll �� (A2 ) lIzll �� (A2 /Xl r � c(k) (i, IIOqvllk"(A.) lIzll�t-" (A ) dxl r � c(k) II Oqv llH"T( A. ,L2(A ,) Izll' 2 1

Ilzoq v ll �

1

Also by imbedding theory,

nr," ( n )

�

C(O) for

! r

+ !s < 2. Therefore

(7.44) Similarly (7.45) By Theorems 2.1 and 2.4, Lemma 4.1,(2.23) and integration by parts, we can prove that

Bq(v, w, z) = l(wOq V, Z )NI + l(oqIN(vw), z) 1 � c(k)(lvh ll w IlLoo + Ivwll) lIzll (7.46) � c(k) (II v llLoo l wll + I v h ll w llLoo ) IIzlI · Using Theorem 2.26, (7.44) and (7.45), it is not difficult to see that

t 1 cfJ l u(t) l � + c��) IIUh,N II ;(O,T;Ml,"+l) l I u(t) 1 I 2 , I F2 (t)1 � cfJT2 I D'Tu(t) l� + c��) (92 + (9 - 5) 2 ) IIUh,N II ;(O,T;Ml,"+l) lI u(t) 1I 2 . I Fl ( ) �

Furthermore, we use Theorem 2.26, Theorem 7.1 and (7.46) to get that

Chapter

7.

Combined Spectral Methods

305

By substituting the above estimates into ( 7.43 ) , we obta.in th at

DT (1Iu(t) 1 I 2 + f3 l 1 p (t) ln + T( 20 - 1 - 2c) ( I IDTu(t) 1 2 + f3 IIDTP( t) ln + 1-' lu(t) l � I-' T( U + O )DT lu(t) l � + I-' T 2 ( 20u - U - 0 - 2c) IDTU(t )l� � Al (lIu( t ) 11 2 + f3 l 1 u( t )1 I 2 ) + A2 ( t ) lu( t )l� + A3( t ) +

where

Al = c��) ( 1 + 02 + ( 0 hl) lIu h.N II �(o.T;M" 'Y+') + 1 , A2 ( t) = -1-'(1 - 2c) + c( k:�N (O'-'6) 2 I1u(t) 1I 2 , A3(t) = c��) IIRt) IL + (� + :�) Il y ( t) 11 2 . -

In the case ( 7.38 ) , the previous inequality reads

DT (1Iu(t) 11 2 + f3 l 1 p(t) 1I 2) + qoT (IIDTu(t ) 11 2 + f3 I1DTP(t) 11 2) + I-' lu(t) l � +I-'T( U + O) DT lu(t) l � � Al ( 1 Iu(t) 1 I 2 + f3 llp (t) 1I 2) + A2 (t) lu(t) l� + A3(t).

In the case ( 7.39 ) , we get from ( 7.36 ) and ( 7.37 ) that

( 7.47 )

T( 20 1 - 2c) (IIDTu(t) 1I 2 + f3 I DTP(t) 1 I 2) + I-'T 2 ( 20u - U - 0 - 2c) IDTU(t) l � � qoT (IIDTu(t) 1I 2 + f3 I1DTP(t) 1 I 2) and thus ( 7.47 ) is still valid. Let E( v, t) and p (t) be the same as in ( 6.35 ) and ( 6.36 ) . Then the summation of ( 7.47 ) for t E R,. ( T ) yields E (U , p , t) � p (t ) + T 2: (AIE (u , p, s) + A2 ( s ) lu( s ) ID · .E�(t-T) -

w,

Finally, we get from Lemma 3.2 the following result.

Assume that (i) (7.37) holds, and (7.38) or (7.39) is valid; I-'hc( k) (zi) for certam t l E RT( T ), p (t l) e t � , , T N (O 6)2

Theorem 7 . 2 2 .

..

.

b

_

Spectral Methods and Their Applications

306

(u , p, t ) $ cp ( t)eb1 t , b1 being some positive constant depending on k, (3, I-' and the norm Il uh,N ll c (o,T;Ml '1'+ l » E

'Y

> O.

We know from Theorem 7.22 that if (j > � , then the restriction o n A disappears and so we can use larger T . If (J = Ii > 2""- 1 for (j > then the restriction on also disappears and so the index of generalized stability, This improves the stability essentially. Indeed, it is the best result. On the other hand, since we adopt the operator the effect of the main nonlinear term is cancelled, i.e.,

�,

p e t)

s = -00.

d(v, w, z),

d (u(t), u(t) , u( t)) = o. This fact again shows the importance of a reasonable treatment with the leading nonlinear terms in the approximation of nonlinear problems. We are going to analyze the convergence of scheme (7.35) . In order to obtain the uniform optimal error estimation for (3 � we first consider the corresponding Stokes problem, which is itself very important in fluid dynamics. Let be the dual space of (HJ, p( O ) and = L�,p(O) . ( ' , - stands for the duality between and Let "I and = The Stokes problem is to find and such that

0,

V. pEW

V=

E V'

r

W E W e ' W. ,

{ b(au(,uw)v)=- b(v,p) w),

)

V'

uEV

V'

( Tf, v ) , 'v' v E V, (7.48) (e, 'v' w E W Let Vh,N be the trial function space of the speed vector u. We approximate u and p by u h,N E Vh,N and ph, N E Wh,N satisfying { l-'ab (u(h,uNh,N, w, )V)=-(e,b (w),v ,ph,N) = (Tf, v) , 'v''v' wv EE Vh,Wh,N'N (7.49) ' I-'

=

(7.50) is a combined Fourier pseudospectral-finite element approximation to (7.49) . Following the same line as in Girault and Raviart (1979), it can be shown that 7 . 5 has a unique solution provided that the following BB condition ( the inf-sup condition ) is fulfilled, sup 7.5 ) �

( 0)

vE V':.N

b(v, w) cllwll , 'v' w E Wh,N ' 11 V 11 1

---

( 0

Chapter

7.

Combined Spectral Methods

3 07

This condition is fulfilled for Vh,N = ( S�,k + ,l 181 VN, 2 , see Canuto, Fujii and Quar 2 teroni (1983) . If the components of trial functions belong to the different spaces Ah,N and Bh ,N , then it is denoted by Vh, N = Ah,N X Bh ,N. It was pointed out that ( 7.5 1 ) holds for Vh, N = S�, k + 2,l 181 VN, 2 X S�, k + l,l 181 VN, 2 . G o and Ma (1993) improved this result essentially, stated as in the following lemma.

)2

(

Lem m a 7.6.

If Vh,N

(7.51) is fulfilled.

) (

= S� , k + 1,l 181 VN, 2

(

) �

) X ( S�,l,l

181

VN, 2 )

and (7.10) holds, then

We define the linear operator P : V x W Vh,N X Wh ,N by P. ( u , p ) = ( P.u , Pop) = h (u ,N , ph ,N) where (uh,N , ph,N) is *the solution of (7.50 ) . Let f = min(r, k + 2). Using

-+

Lemma 7.6 and an abstract approximation result in Girault and Raviart (1979) , we obtain the following result. =

M; , 8 (f2) n HJ,p ( f2 ) , p E H;- l, 8- l (f2) n L5 ,p ( f2 )

Lemma 7.7.

If U J

E

Let Vh,N

Vh ,N '

(7.50) has a unique solution. Moreover if u and r, 2:: 1, then s

l, s- l (f2) n L2O (f2) Hrp , s(f2) n HQl,P (f2) ' p E Hr,P P

and r, s > 1 then -

By (7.52) and Theorem 2.26, it can be shown that if u H�" - l (f2) n L5 ,p ( f2 ) and 1 ::; ::; lX, then

s

We now deal with the convergence. Let (6.22) that

U*

E

=

P.U, p .

E

=

J

M; , S(f2) n HJ (f2) , p E

PoP. We deduce from

(D.,. U*(t), v) + d (RNU*(t) + 8r RND.,. U*(t), RN U*(t), RNV) v) - b (v, P*(t) + (hD.,. P *(t)) +/l a (U*(t) + O'rD.,.U*(t), 7 = (IN nk+ 1 f(t), v) + L Gj (v, t), V v E Vh ,N , t E R.,. (T) , j=l f3 (D.,. P *(t), w) + b (U·(t) + Or D.,.U*(t), w) = ( Gs(t), w) , V w E Wh ,N , t E R.,. (T), (7.54)

Spectral Methods and Their Applications

308 where

G1(v, t) = (D'TU(t) - 8tU(t), v) , G2 ( v, t) = (D'TU* (t) - D'TU(t), v) , G3 (V, t) = d (RN U*(t), RN U*(t), RNV ) - D (U(t), U(t), v) , G4(V, t ) = 07d ( RN D'TU* (t), RNU* (t), RNV ) , G5(V, t) = O'wra (D'TU*(t), v) , G6 (V, t) = -8rb (v, D'T P* (t)) , G7(V, t) = (f(t) - IN nk+ l f(t), v) , Gs( v, t) = (JD'T P* (t) . Let

U = Uh ,N - U* and F = Ph,N - P*. Then by (7.35) and (7.55) , we obtain that

(D'T U (t), v ) + d (RNU(t) + Or RN D'T U (t), RNU* (t) + RNU (t), Rv ) + d (RN U* (t) + Or RND'T U* (t), RNU (t), RNV) + p,a (U (t) + 0'7 DT U (t), v) -b (v, F(t) + 8rDTF(t) ) = - � Gj (v, t), V v E ltJ. ,N , t E R,. (T) , j= l (J (DTF(t), w) + b ( U (t) + 8r DT U (t), w ) = - (Gs(t), w) , V w E Wh,N , t E R,. ( T) , U ( O ) = IN nk+ 1 Uo - U*(O), F(O) = IN nk+ l Po - P *( O ) . 7

(7.55) We now estimate the terms on the right-hand side of (7.56) . Firstly by (4.24) and (7.53 ) ,

7 � II DTU(s) - OtU(s) 11:1 ::::; c(k)r 2 IJ UI I�(o,t;H- I)' seRr(t-T) r � IIDTU*(s) - DTU(s) 1I 2 aeAr (t-T) ::::; c(k) ( h 2f + N - 2 a ) ( IJ U II�(o,t;H" ') + II P II11(o,t;H.-I -I) ) . To estimate the nonlinear terms, we first let v, w, Z E Vh,N , ;::: I + � and I > O. Set .•

Aq = I «RNIN - I) ( RNW8qRNV ) , z ) l , Bq = I(IN ( RNW8qRNZ ) - RNW8qz , RNV ) I ,

S

q = 1 , 2, q = 1, 2.

By Theorem 2.26 and an argument as in the proof of Lemma 10 of Guo and Ma ( 1991 ) , we know that if 0: ;::: max ( s - p" 1 ) , P, = 0, ":"' 1 and s > !, then for any

Cbapter

7.

309

Combined Spectral Metbods

II RNIN (RN V RNW) - RNvRNwII HI' (A, ) :::; c(k) NI' -· (I v I H '(A, ) l w Il H"Y+ ! (A, + I w I H '(A, ) l I vI I H"Y + ! (AJ . )

(7.56)

By virtue of imbedding theory and (7.57) , we deduce t hat

Aq :::; I (RNIN - 1) (RNWOq RNV) 1I -1 1 I zl h < c( k) N - S (i, (I Oq vl � ' -l (A2) I w ll �"Y+ !. (A ') + l w I 1'-' (A2) I Oq v l �"Y+ ! (A2» ) dX l ) t 1I ;l h :::; c(k) N -S (IIOqvIl H ' -' (A2 P(A' ) Il w Il H"Y+! (A" H"Y + ! (A ') + I Oq v Il H"Y+ ! (A, P (A, )) I w I l H' -l (A" H"Y+! (Al)) ) I l zlh · Similarly,

Bq :::; I (RNIN (RN v RNw) - RN V RNW, Oq z ) 1 " :::; c(k) N - · (i , ( I vl � '(A') II w ll �"Y+ ! (A + I w I 1 '(A, ) I H �"Y+! (A ) dX l ) Ilzl h 2» 2) :::; c( k)N-S (1 I vII H '(A,P(A l)) ll w I H"Y + ! (A , H"Y + ! (A )) 2 ' +l I w Il H' (A2 P(Al)) I I v ll H"Y+ � (A2 ,H"Y+ ! (A' ) ) Il z l k 1

The above arguments imply .that

from which and (7.54), it follows that

(RNU* (t) , RNU*(t), RNV) - D (RNU* (t), RNU*(t), v)1 (7.57) :::; c(k)N - S ( 1 I U(t) I Ml " + I P(t) I H' -' (A2 P (A, )) f I l vlll. Similar to (7.44) and (7.45), we have from Theorem 2.26 and ( 7 . 53 ) that for r � 1 , I D (RNU*(t), RNU*(t), v) - D(U(t), U(t), v)1 :::; c(k) ( II U(t) I M',"Y + l + II U*(t)II Mlo"Y+ l ) II RNU*(t) - U(t) ll ll vl h (7.58) :::; c(k) ( hi' + N - s ) ( I U(t) I Mr" + I P(t)I I Hr - " , -, ) 2 1 I vl k Id

310

Spectral Methods and Their Applications

The estimates (7.58) and (7.59) lead to the bound for have from Theorem 2.26 and (7.45) that

I G3 ( v , t ) I I ' Furthermore we _

Moreover, by (7.53) and (7.54) ,

11 U* (s) l I �l"'+l II DrU* (s) 1I 2 � c(k) r2 ( 1 U 1 I &(o,t ;Ml ,..,+ l ) + I P I &(o,t ;H-r (A2 P (Ad » ) ( 1 I U Il h1 (o,t;Hl ) + I P l h1 (o,t ;L2 » ) ' We can estimate I Gs(v, t) 1 and I G6(v,t)1 in the same manner. Also by Theorem 2.4 and (7.2) , I G1(v , t ) 1 I � c( k ) ( hi' + N-' ) Il f(t) l w, . l v l . r3 L

.eR. (t-r)

In addition,

1 0 ( 0 ) 1 � c( k ) ( hi' + N -B ) ( ll UoII Hf + IIPoII Hf-1 , . -d , i P (O) i � c( k) ( 1 Uo lll + I PoI I H' + 6 (A. ,H ' + 6 (Al » ) ' 6 > O. .•

Finally we apply Theorem 7.22 to (7.56) , and the convergence follows.

Assume that (i) (7.10) and condition (i) of Theorem 7.22 are fulfilled; (ii) for certain constants r � 1 and s > �, U E C (0 , T', Mpr,. Jio,l p) HI ( 0 , T ,· Hr,B ) H2 (0 ' T · H - I ) P E e (0 , T', L02 ,1') HI (0 , T' Hr - l ,.- l ) f E e (O , T H; ) (iii) for certain t l E R,. ( T ) and certain positive constant b2 , ) eb2 t1 (r2 h2i' + N-2• + (3) - r Np.hc(k 6 ( 9)2 . Then for all t E Rr(tl) , I l uh,N (t) - U* (t) 1 I 2 � ceb2 (r2 h 2i' + N- 2• + (3) where b2 is a certain positive constant depending on k, f3, and the norms of U, P and f in the mentioned spaces.

Theorem 7.23.

n

i

n

i

"

n

n

,

,

P

<

+

t

+

_

p.

"

Chapter

7.

Combined Spectral Methods

311

We find that since we compare (ui., N , Ph,N) t o (U· , P·) , the solution of the cor responding Stokes equation (7.50) , the factor ! in the error estimates disappears. It improves the error estimates, and covers the special case {j = O. If we use this technique for the numerical method discussed in Section 6.3, we can also improve the corresponding results. On the other hand, if we take 6 :;:= 0 > � , then t l = T. It means the global convergence of scheme (7.35). Finally, we present some numerical results. We use (7.35) with (j = 0 = 1 , 6 = 0, k = 0 and the uniform mesh size h = it (MFPSFE) . We also use the corresponding finite element scheme (FE) with the solution U h . We compare the �rror U Uh,N and the error U Uh . In particular, we compare the errors for the nurrierical simulations with very small viscosity J.I. = 10 - 6 • For the description of the errors, let

-

-

and

We take the test function U = ( 821/J , -fA1/J ) ,

(

P = exp cos Xl

where

+ cos 2 2 + �) X

+

1/J = x� ( 1 - xd (exp(sin 3x 2 ) sin 6x 2 ) efo t • In all calculations, M = 14, N = 8 and T = 0.002. The numerical results with

various values of the artificial compressibility parameter {j and filtering parameter a are given in Tables 7.1 and 7.2. Evidently, the combined Fourier pseudospectral-finite element scheme provides better numerical results. It is indicated that a good choice of the filterings improves the stability. We also find that the errors decrease when {j decreases. All of the above facts agree with the theoretical analysis.

312

Spectral Methods and Their Applications

Table 7. 1 . The errors E (i ) (1.0) , ,8 = 10 - 4 • E ( 1 ) (1 .0) E ( 2) (1 .0) or 00 2.838E-1 3.426E-1 MFPSFE 3 9.229E-2 1 .411E-1 2 7.630E-2 1 . 165E-1 5.797E- 1 4.080E- 1 FE Table 7.2. The errors E (i ) (1.0) , ,8 = 10 - 5 • or E ( 1 ) (1 .0) E ( 2) ( 1 .0) 00 1 .243E- 1 1 .879E- 1 MFPSFE 4 5.914E-2 4.585E-2 3 5.931E-2 4.495E-2 1 . 81 5E - 1 2.948E - 1 FE The combined Fourier-finite element approximations were first used by Mercier and Raugel (1982). They have been used for various nonlinear problems, such as the steady Navier-Stokes equation in Canuto, Maday and Quarteroni ( 1984) , the unsteady Navier-Stokes equation in Guo and Cao (1991, 1992a) , Cao and Guo (1993a) , Guo and Ma (1991 , 1993), and Ma and Guo (1992), the compressible fluid flow in Guo and Cao (1992b). At the same time, the combined Chebyshev-finite element methods developed rapidly, with their applications to nonlinear problems, e.g. , see Guo, He and Ma (1995), He and Guo (1995) , and Guo, Ma and Hou (1996). For the convenience of computations, the Fourier spectral-finite difference approx imations are also used widely in practice. Some of early work- was due to Murdock (1977, 1986) , Ingham (1984, 1985), Beringen (1984), Milinazo and Saffman (1985) . In particular, Guo ( 1987, 1988b) set up a framework for the numerical analysis of this approach, used for the equations describing incompressible fluid flows, compressible fluid flows and atmospheric flows, see Guo (1987,1988b) , Guo and Xiong (1989) , and Guo and Huang (1993) . Also, the Fourier pseudospectral-finite difference approxi mations and their applications have been developed, e.g. , see Guo and Zheng (1992) , and Guo and Xiong (1994) .

CHAP TER 8 SP ECTRAL METHO D S O N THE SP HERICAL SURFACE

In the previous chapters, we discusseq various spectral methods in Descartes coor dinates with their applications to numerical simulations in fluid dynamics, quantum mechanics, heat conduction and other topics. However, in meteorology, ocean sci ence, potential magnetostatic field and some other fields, we have to consider prob lems defined on an elliptic plane, on a spherical surface or in a spherical gap. Several numerical algorithms have been developed for such problems, e.g. , see Boyd (1978a, 1978b), Haltiner and Williams ( 1980), Jarraud and Baede (1985) , Bramble and Pas ciak (1985) , and Dennis and Quartapelle (1985) . Some of them are related to the spectral methods on a spherical surface or in a spherical gap. But there were no rigorous approximation results for a long period as pointed out in Canuto, Hussaini, Quarteroni and Zang (1988) . Recently Guo and Zheng (1994) , and Guo (1995) de veloped a framework for the numerical analysis of spectral methods on the spherical surface. We shall present some results in this chapter. We first discuss the spectral methods and then the pseudospectral methods. The final part of this chapter is for the applications of these methods.

8.1.

Spectral Approximation on the Spherical S urface

A

This section is devoted to the spectral approximation on the spherical surface. Let and () be the lo:p.gitude and the latitude respectively. Denote by S the unit spherical surface,

S

= {(A, ()) 1 0 ::; A 211", -i ::; () i } · <

313

<

Spectral Methods and Their Applications

314

A

The differentiations with respect to and () are denoted by 0>. and 00. For a scalar function v, some commonly used differential operators are defined as

V' v =

Co� (} o>. v , oov) ,

>. >. w) , J(v , w) = -cos () (o voow - oovo 1 1 ue ( cos (} oov) + � 6. v = -2 . cos (} o>.v cos (} 1

'"

Let 1) ( S ) be the set of all infinitely differentiable functions with the regularity at () = ±�, and the period 211" for the variable 1)1 ( S ) stands for the duality of 1) ( S ) . We define the distributions in 1)1 ( S ) and their derivatives in the usual way, see Lions and Magenes (1972). If

A.

is vo>. w dS = - is zw dS, V w E 1)( S ) ,

,

A,

then we say that z is the derivative of v with respect to denoted by 0>. v etc. Furthermore, we can define the gradient, the Jacobi operator and the Laplacian in the sense of distributions. For example, if

is v 6.w dS = is zw dS, V w E 1)( S ) , then we say that

z = 6. v, etc. Now, let (v , w) = is vw dS, I lvl l = (v , v)� ,

and

L2( S ) = {v E 1)'( S ) I lI v ll

< oo } .

Furthermore, define

equipped with the semi-norm and the norm as

For any positive integer r, we can define the space Hr (s) with the norm I . Il r by induction. In particular, the norm I . 11 is equivalent to (11v112 II 6. v I 12)! . For any 2 real r 2: the space Hr (s) is defined by the interpolation between the spaces H[r]

0,

+

Chapter 8. Spectral Methods on the Spherical Surface

315

the space H: (S) is the duality of H - r (s) . Particularly, and H[r] + l . For r < 2 O H (S) = L (S) and Il v l l o = Il v ll . It can be verified that for any v , w H 2 (S) ,

0,

E

(8 . 1)

- (�v , w ) = (\7v , \7w ) .

. The space C(!1) i s the set of all continuous functions o n S, with the norm I I · 1 1 0 . Lemma 8 . 1 . For

C(S) .

any

0 � J.L � r , HT (S)

'-t

HI-'(S) . For

any

r >

0, W+1 (S)

'-t

Proof. The first assertion follows immediately from their definitf�ns. We now prove

the second one. Let B be the unit ball in the three-dimensional Euclidean space, and w be a function defined on B. We denote by Tw the restriction to S of w . We can take HT + 1 (S) to be the trace space of HT + � (B) with the norm II v IIW+l (S) =

By imbedding theory, W + � (B)

On the other hand, for any v and

'-t

inf

weHr + � (B) Tw=v

Il w li H "+ � (B l "

C(B) , and so for any w E Hr + � (B),

E HT+1 (S), there exists w E HT+� (B) such that Tw = v

Consequently, Il v l i o(s) = sup Iv(x) 1 = sup Iw(x) 1 xES xES

su� Iw(x) 1 � c Il w(x) I i H "+ � (B) � 2cll v ll w+I (S) . � xEB

This implies the assertion. Let

L� (S) = Lemma 8 . 2 . For

any

•

{ v E L2 (S) I A (v) = O } ,

vE

Hl (S) n L6(S),

A(v) =

is v dS.

Spectral Methods and Their Applications

316

Co being a positive constant independent of v .

Proof. By the Poincare inequality, we claim that and the desired result follows. Lemma 8.3. If v

E

L 2 (S), w

•

E

Hr+ 1 (8)

and

r

> 0,

then

I l v w ll :s c ll v ll ll w ll r+ 1 '

Proof. By Lemma 8 . 1 , •

L I (x)

We next turn to the L 2 (8)-orthogonal system. Let be the Legendre poly nomial of degree I. The normalized associated Legendre function is given by

{

+ l ) (m - I)! (1 X 2 ) t fi Lm ( X ) , . I,m ( X ) = (2m2(m + I) ! L I,m (X) = L _I,m (x), Moreover, the spherical harmonic function Yi, m (..\ , 0) is

L

_

x

for 1 2 0, m 2 for I < 0,

m

III,

2 I ll .

(8.2)

(8.3)

The set of these functions is the L2 (S)-orthogonal system on 8, i .e. , 21r 1r = 0) cos 0

la l� Yi,m (>', O) "f/',m'(>" 2

dOd>' 81 ,118m,ml•

The spherical harmonic expansion of a function v E L 2 (8) is 00 v= L L

v (>', O) 1= - oo m� 1 1 1 I,mYi,m

where the coefficient

VI,m is given by VI,m = l" l: v ( >. , O) "f/,m (>', 0) cos 0 dOd>'. 2

Chapter 8. Spectral Methods on the Spherical Surface

It can be checked that

- sYl,m ( \ 0) = m ( m

317

+ 1 )1I,m (A, 0) .

(8.4)

So 1I ,m ( A , O ) is the eigenfunction of the operator - � on S, corresponding to the . eigenvalue m ( m 1 ) . Thus in the space Hr ( s ) , the normJ v ll r is equivalent to

+

mr ( m

( f :E

1= - 00 m?: 111

VN = span {1I ,m 1 1 1 1

+ 1 )" I VI,m I 2 )

1

2

(8.5)

Now let N be any positive integer, and define the finite dimensional space VN as

:::; N, I I I :::; m :::; M ( l)} . Clearly, VN depends on the choice of M ( l ) . Usually we take M ( l ) N or N + I l l . For fixing the discussion, we take M ( l ) = N . Let VN b e the subset of VN containing �

all real-valued functions. In this space, we have the following inverse inequality. Theorem 8 . 1 . For any 1>

E

VN and

j.t ,

0 :::; r :::;

:E :E JI,m1l,m (A, 0) . 1> = 1=-N m=111

Proof. Let

N

N

Then by (8.5) ,

< <

c

N

N

L L

m l' ( m

+ 1 ) 1' I JI,m l 2

1= - N m=111 N N cN21' - 2r L L m r ( m 1=- N m=111

+ 1) " I JI,m l 2 :::; cN21'- 2r ll 1> I I ; · •

The L 2 (S)-orthogonal projection PN : L 2 (S) any v E L 2 (S), or

equivalently,

PNV(A, 0) =

N

N

L L

1= - N m=111

-+

VN is such a mapping that for

vI,m1l,m (A, O ) .

318

Spectral Methods and Thei� Applications

Theorem 8 . 2 . For any v E Hr (s) and 0 :::;

p.

:::; r,

(8. 5), L mJ.l ( m + 1)J.l l vl ,m I 2 + L L mJ.l ( m + 1)J.l l vl ,m I 2 l i v - PNvl l ! I=LN-N m=N+ NL L 1 mJ.l(m + 1)J.l l vl,m I 2 + I LI>N m=L1 1 mJ.l(m + 1 )J.l l vl,m I 2 1=-Nm=N+l 1 1>Nm=N+l 1=-oom=N+l mr (m + 1)' I VI,m I 2 The proof of the above two theorems is due to Guo and Zheng (1994) .

Proof. By

00

:::; c

c

c

00

:::; c

00

00

:::; cN 2 J.1 - 2r :L

8.2 .

:L

00

00

:::; cN2 J.1 - 2 r l l v l l � ·

•

Pseudospectral Approximation on the Spherical Surface

(j ) w A ( - 1,1). SN {(..x ( ") 0 (; » ) l ..x ( " ) - } + 0(;) arcsin x (;) - The discrete inner point ( . , · ) N and the discrete norm I . l i N are defined by (V ,W)N + 1 E2N �N (.�(k) ,O(i») (..x(k) ,O(i» ) w(i) , I v i N The interpolation IN C(S) VN is such a mapping that for any C(S) , INv 1=N-N m=N1 1 VI,mY/,m (..x , O), (8. 6 ) where 2N N (..x(k) , OW) ,m (..x(") , O(i») w(i) . VI,m + 1 k=03=0

We now present another approximation on the spherical surface, based on the in terpolation. Let x U ) and be the Legendre-Gauss interpolation points and the weights on the interval = Define SN as a set of grid points, =�

,

=

=

2N

271"

, 0 < k < 2N, 0 < -

vE

--t

2

_71" _

2N

""

L..J �

v

J.

<

N .

= (v , v) � .

w

= :L :L

:

Clearly

=

v

2N

=

l'

fil

Chapter 8. Spectral Methods on the Spherical Surface

The degree of freedom of SN is ( 2 N N

1

3 9

+ l ) (N + 1), while the dimension of VN

I ) 2 1 + 1) = (N + 1) 2 . Thus, IN v ( A ( k) , (}Ul) 1=0

#- v ( A ( k) , (}U) ) generally. Consequently,

is

IN is not an interpolation in the usual sense. It is actually the projection from C(S) onto VN , corresponding to the discrete inner product ( . , · )N. This fact can be seen in the proof of the following lemma. Lemma 8.4. For any v

E

C(S) and
(i) IN
m ::; N,

::;

m' ::; N, 2N N ( Yil ,m, Yil' ,m ) N = 2N1+ 1 " " ei (I - I/ ) .>.(k) L I ,m (sin (}U) ) L I I ,m I (sin (}U) ) wU) .

In fact, for all 1 1'1

---

,

Clearly

L..J J=O L..J k=O

1 "2N ei(I_I/ ) ,(k) 2N + 1

___

L..J k=O

_

-

8I, l"

m + m'

On the other hand, L I ,m( x)L I ,m/( X) is an algebraic polynomial of degree ::; 2N. By (2.34) , the Legendre-Gauss quadrature formula with N points is exact for it. Hence we have from the orthogonality of {LI ,m( X)} that

+1

� LI,m (xU)) LI,ml (xU)) wU)

Therefore for all I II

::;

m

::;

N and 1 1'1 ::;

=

tl LI,m(x)L/,m/( x) dx

=

8m,m/

m ::; N,

(8.7) Consequently, IN Y/, m = Y/,m ' Next, we examine the second conclusion. Let
N

=

N

I: I: JI ,mY/,m( A , (}). 1= - N m=l lI

(8.8)

Spectral Methods and Their Applications

320

By (8.6) and (8.7) ,

N

N

N

N

N

N

2: l'=-N 2: m=lll 2: m'=1l' 2: I V" m �" ,m' ( Yi,m , Y,',m') N 2: 2: Vl ,m ( Yi,m , 4» = l=-N l=-N m=I'1 _

(INv, 4» N . Furthermore,

N

N

N

N

2: 2: (v, Yi, m) N �" m = 2: I: V" m�" m

l=-N m=lll (INv, 4» = (INv, 4» N ·

l=-N m=lll

•

Lemma 8 . 5 .

For any v E C(S), we have that

(i) II INv lI = II INv ll N $ IIv IlN ,

(ii) IIv - INv l l N = inf IIv - 4> IIN . Proof. It follows from Lemma 8.4 that t/>e VN

Then the first conclusion follows immediately. The second one follows from the projection theorem and the last result of Lemma 8.4. • Lemma 8.6.

I

l

For any 4> E VN , we have that

(i) cos o 8� 4>

L = I co� B 8� 4> I ,

(ii) 11884>IIN = 11 884>11 .

Proof. We see from (8.8) that N 1 - 8� 4> = I: I: il �" m eiIAL " m ( sin B) ( cos B) - l , cos B m=O lll$m N " L.. ,, 4>' " m e'., � 8",L " m ( sin B ) cos B. L.. m=O lll$m

8.

321

Chapter Spectral Methods on the Spherical Surface

I:;I,m ' denote the sum in the above formulae. Then

Let

I� 12 cos B OA

� ) ( cos B OA

� �

2N - 2, and so the Legendre-Gauss quadrature formula is exact' for it . Thus we obtain

that

1 - O

=

I:;I,m 'I'I:;,m''ll'�l,m �I" m' (eilA LI,m (sin B) (cos B) -I , eil'A LI',m, (sin B) (cos B) -I ) /I co� B OA

On the other hand, o",L I ,m (x)o",L I ,m' (x) ( 1 is an algebraic polynomial of degree m m ' :$ 2N. Therefore the second conclusion can be proved in the same manner . •

+

Theorem 8 . 3 .

For any v

E

Hr (s) , o :$

J-t

:$

r and r > I ,

II v - INv l I 1-' :$ CNI-'+I+5- r ll v l I "

being an arbitrary positive constant. Proof. By Lemma 8 . 1 , INv is well defined. 8

We first let

conclusion (ii) of Lemma 8.5 that Since INv , PNv

E

J-t

= O.

Il v - INv l i N :$ IIv - PNv li N :$ e II v - PNv l l c '

It follows from (8.9)

+ II INv - PNv l l = II v - PNv ll + II INv - PNv ll N II v - PNv l l + I l v - INv ll N + Il v - PNv ll N :$ e Il v - PNv llc ·

VN , we have from Lemma 8.4 and (8,9) that

II v - INv l l :$ II v - PNv l l <

Further, we get from Lemma 8 . 1 and Theorem 8.2 that for 8 > 0, II v - INv l l :$ e II v - PNv 1 1 1

:$ eN 1 +5 -r l l v l l r .

If J-t > 0, then we deduce from Theorem 8.1 and (8.10) that II PNv - INv l l 1-' :$ eN I-' I! PNV - INv l l

+5

(8.10)

:$ eNI-' ( l i v - PNv l l + II v - INv l D :$ eNI-'+1+5- r ll v l l r .

This estimate together with Theorem 8.2 imply the desired result. All results in this section come from Cao and Guo ( 1 997) .

•

Spectral Methods and Their Applications

322

B.3.

Applications

In this section, we take the vorticity equation as an example to describe the spectral methods on the surface. Let H, 'Ill' and JL > 0 be the vorticity, the stream function and the kinetic viscosity, respectively. The vorticity equation on the spherical surface S is of the form Ot H J(H, '111' ) - JLIlW = I, (A, 9) E S, 0 < t � T, (8. 1 1 ) -Ilw = H, (A, 9) E S, 0 � t � T,

{

+

(A, 9) E S

H(O) = Ho,

where I and Ho are given functions. I t i s natural t o assume that all functions have the period 211' for the variable A, and are regular at the pole points 9 = ± � . For fixing '111' , we also require that W(A, 9, t) E LW1) for 0 � t � T. The weak form of (8. 1 1 ) is to find H E L2(0, Tj HI(S)) n LCO(O, Tj L2(S)) and 'Ill' E L2(0, Tj HI(S) n L�(S)) such that

{

(OtH(t) + J(H(t), w(t)), v) + JL( V H(t), Vv) = ( f (t), v), v v E HI (S), 0 < t � T, ( V w(t), Vv) = (H(t), v), V v E HI (S), 0 � t � T, (A, 9) E S. H(O) = Ho ,

(8.12) The existence and uniqueness of the solution of (8.12) was discussed in Zen (1979) . In particular, if Ho E LOO (S) and I E LOO(O, Tj LOO(S)), then (8. 12) possesses a unique solution and H E L2(0, Tj HI(S» n HI (O, T, H-I(S)). The solution is conservative. To prove it, we first show that for any v, z E Hr + I (S), w E HI (S ) and r > 0,

(J(v, w), z) + (J(z, w), v) = O. Indeed, the integration by parts yields

(8. 13)

{" L: Z (O>.V09W - O>.W09V) d9dA " - fo 2 L: (O>.Z09W - 09Z0>.W) d9dA - l" Z (A, i) v (A, i) O>.W (A, i) dA + l" Z (A, -i) v (A, -i) O>.W (A, -i) dA. From Courant and Hilbert (1953) , the regularity of w implies that w approaches the limits independently of A, as 9 -+ ± � . This means that O>.W = 0 at 9 = ± � , and so (J(v, w), z ) =

2

2

v

Chapter 8. Spectral Methods on the Spherical Surface

323

H

(8.13) follows. Next let v = in (8.12) and integrate the resulting equation from a to t . We obtain from (8.13) that II H(t) II 2 + 2 p.

l I H (s ) l� ds

= IIHo ll 2 + 2

1o\j(s ) , H(s)) ds.

(8. 14)

Now let V� = VN n L�(S) . and lIT are approxiiQated by TiN and ""N . The spectral scheme for (8. 12) is to find TiN( ). , 0, t) E VN , ""N( ).. , 0, t) E V� for all t E R,. ( T ) such that

{

H

(DT TiN (t) + J (TiN(t) + CrDTTiN(t) , ""N(t) ) , 4» 4> E VN , t E RT ( T ) , - I' (t. (TiN (t) + UrDTTiN(t)) , 4» = (J(t), 4» , 4> E V� , t E RT ( T ) , - ( t.""N(t) , 4» = ( TiN (t) , 4» , ( ).. , O) E S TiN ( a ) = PNHo ,

V V

(8.15)

where c and u are the parameters, a � c, u � 1 . If both of them are zero, then (8.15) is an explicit scheme. Otherwise a linear iteration is needed at each time t E R.,. ( T ) . We now consider the existence and uniqueness of the solution of (8.15) with c i- a or u =F a. Let F (t) = TiN(t - r) - r ( l - c ) J (TiN(t - r ) , ""N (t - r )) +p.r(l - u )t.TiN(t - r) + rJ(t - r ) .

Then for any t E R.,. ( T ) ,

Clearly i t i s a linear algebraic system for the coefficients o f the spherical harmonic expansion of TiN (t) . Hence we only have to verify that TiN(t) is a trival solution when F (t) == a. In fact, by taking 4> = TiN (t) and using (8. 13), It implies TiN (t) == a , whence TiN (t) is determined uniquely at each time t E R,. ( T ) . In particular, if c = u = � , then we have from (8.13) and (8.15) that p.r ll TiN(t) II 2 + 2

L..J

"

= l l TiN( a ) 11 2 + r

ITiN (S) + TiN(S + r ) ll2

BERr(t-T)

�

BE Rr(t-T)

(J( ) , TiN (S) + TiN (S +. r )) .

s

(8.16)

324

Spectral Methods and Their Applications

This is a reasonable ana!ogy of (8. 14) . f We now analyze the generalized stability of scheme (8. 15). Suppose that and the right-hand side of (8.15) have the errors j and respectively. They and denoted by induce the errors of and Then

17N(O), �o , g, �N ¢N. 17N 1/;N, (DT�(t) + J (� (t) + 8TDT�(t) , 1/;N(t) + ¢(t)) + J (17N(t) + 8TDT17N(t), ¢ (t)) , rp) - It (� ( �(t) + (7TDT�(t)) , rp) = (l(t), rp) , V rp E VN, t E RT(T), � - ( ¢ (t), rp) = ( �(t) + g(t), rp) , V rp E V� , t E RT(T), A (¢ (t)) = 0 , t E RT(T), � ( O ) = �o , (A, 8) E S. ( 8. 1 7 ) Let e be an undetermined positive constant. By taking rp = 2 � + eT DT fJ in the first formula of (8. 1 7 ) , we get from ( 1 . 22 ) and (8.13) that DT 11�(t)1 1 2 + T(e - 1 ) I DT�(t) 1 2 + 21t 1�(t) l ; + itT (0" + �) I DT�(t) l; + ItT 2 (eO" - 0" - n I DT�(t) l ; + � Fj (t) = (l(t) , 2�(t) + eTDT�(t)) (8.18) where

Fl(t) 2 ( J (17N(t) + 8TDT17N(t), ¢(t)) , �(t)) , F2 (t) eT (J (17N(t) + 8TDT 17N(t), ¢ (t)) , DT � (t)) , F3 (t) T(e - 28) (J ( � (t), 1/;N(t)) , DT�(t)) , F4(t) T(e - 28) (J (� (t), ¢ (t)) , DT �(t)) . Further, we put rp = ¢ in the second formula of (8. 17), and then obtain that =

=

=

=

By Lemma 8.2, we find that

from which and Lemma

8 . 2,

(8 . 1 9)

Chapter

8.

325

Spectral Methods on the Spherical Surface

e

We are going to estimate I Fj (t) l . Let > O. By find that for , >

0,

I FI (t) 1

$ $ $

( 8.13 ) , ( 8.19) and Lemma 8.1, we

2 1(J (7} (t) , t,b(t)) , 71N(t) + 6TDT71iv (i)) I ep, 1 7} (t)l � + e: II 71N I I �(O,T;LOO ) 1 t,b (t) l : ep, 1 7} (t)l � + e: I171N I I �(O,T;H"Y+') ( U 7}(t) 112 + II g (t) 11 2) .

Similarly,

8.3 yields T (e - 26) 2 I I .,pN I I C(o,T;H"Y+2) 2 171 (t) II2 . 1 F3(t) 1 $ e T I I DT71 (t) II 2 + C e

Furthermore, the use of Lemma

_

_

Since

8.1 that _ T 2 (e - 26)2 II 1/N I 1 C(O,T;L2) 2 1 1/- (t ) 1 I2 · IF3(t) 1 $ e 'T I I DT71 (t) I I 2 + c N "1 e Also, by Lemma 8.3, Theorem 8.1 and ( 8. 2 0 ) , T(e 2 ) 2 1F4 (t) 1 $ e T IIDT 7}N I I 2 + C � 6 1 t,b (t) I :+ I 7} (t)l � 2 $ eT I IDT7}(t) 11 2 + C'T (e � 26) 2 ( 1 I 7}(t) II; + I Ig (t) II;) 17} (t) l � $ eT I I DT 7}(t) 11 2 + cTN2"1 (ee - 26) 2 (1 I 7} (t )1 I 2 + I l g (t)1 1 2 ) 1 7} (t) I � . Clearly, I (i(t) , 7}(t) + e� DT7}(t)) 1 $ eT II DT7} (t) 112 + C 1 7} (t)1 1 2 + C ( 1 + T;2 ) I j(t) 1 2 . By substituting the above estimates into ( 8.18 ) , we claim that DT 1 I �(t) 11 2 + T( e - 1 - 3e ) I I DT�(t) 112 + p, 1 � (t)l � + p,T (0" + � ) DT 1 � ( t) l � +p,T2 (eO" - 0" - � - e) I D �(t )l � $ Al 1 I � (t) 11 2 + A2 (t) 1 �(t )l � + A3(t) ( 8. 22)

we deduce from Theorem

T

Spectral Methods and Their Applications

326

l + e I1 77N ll 2c O,T;H'Y+ 1 ) , Al c ( 1 + � ( ) CTN2-y C - ( e 2 6 ) 2 ( I1 77N l c2 (O , T ; L2 ) + 1 I � (t) 1 I 2 + I l g ( t )1I 2 ) , A2 (t ) . = - p, + p, + Te ) 1 J-(t ) 1 2 + C ( 1 + e2 ) I1 77N ll c2 O,T;H"I+ 1 II g- (t )1I 2 . A3( t) = C ( 1 + e ( ) p, Let qo > 0 and c be suitably small. We choose e in three different cases. (i) u > !. We take 2 u + 2c ) e � 6 = max 1 + qo + 3c, 2 u ( 1 .

where

=

c

-

c

_

T(e - 1 3c ) IIDT�(t )1I 2 + p,T 2 (eu - u - � - ) IDT�(t) l � � qoT IIDT�(t )1I 2 . (8.23) (ii) u = !. According to Theorem 8 . 1 , there exists a positive constant c] such that I v ll � � c]N2 11 v1l 2 for all v VN. For this reason, we take e � 6 = 1 + qo + 3c + P,C] T N2 G + ) Then

c

-

E

c

and so (8.23) also holds.

( 111. . ) U .

< "2

1 and T N2

<

2

P,C] (1 - 2u ) . Then we take 1 + qo + 3c + j.L C] T N2 ( u + ) e-6= 1 + p, c] T N 2 ( u - !) >

c

.

whence (8.23) is still valid. Now let

E(v , t) = II v(t )11 2 + T z: T ( l v( s ) l � + qoT IIDTv( s )1I 2 ) , ( ) p e t ) = 11� (O) 11 2 + p,T (u + �) I�( O )I� + T z: A3(S). p,

seRr t-

seRr (t-s)

.R,. ( t ) yields E (� , t ) � p (t ) + T z: (AIE ( i7 , s ) + A2 (S)E2 (� , S)) .

Then the summation (8.22) over

seRr(t-T)

Applying Lemma 3.2 to (8.24), we claim the result as follows.

(8.24)

Chapter 8. Spectral Methods on the Spherical Surface

Assume that (i) N2,,! is suitably small, being an arbitrary small positive constant; r.") > -1 or N2 CII-' ( 1 2- 2U ) ,. U 2 ·· (iii) I l g (t) 1 2 ::; T�b 2"! and p (tl) eb2 t' ::; TNbs "! for some tl E Rr (T). Then for all t E R,. (tl)

Theorem 8.4. T

{ n

T

327

I

<

( 8.25 )

where b1 - bs are some positive constants depending on I-' and I1 77N l o(O,T ;H�+ ' ) '

{

We now consider a special case, i.e. , 28 2:: 6 , 28 6, 28 2:: 6 , We can take e = 28 and s o A 2 == - I-'

2::

(t)

+

for u > l , ( 8.26 ) for u = l , for (j < C l-' . It leads t o the following result.

�.

If (8. 26) and condition (ii) of Theorem 8.4 are fulfilled, then for all p (t) and t E R,. (T) , the estimate (8. 25) is valid.

Theorem 8 . 5 .

We know again from Theorem 8.5 that a suitable implicit approximation of non linear term can improve the stability essentially. Indeed in the case of Theorem 8.5, scheme ( 8. 15) has the generalized stability index = -'- 00 . In the final part of this section, we deal with the convergence of ( 8.15 ) . First of all, we derive from ( 8.4) that for v

s

PN fl ( >' , fJ) v

E H2 ( S),

m ( m l)vl , m Y/, m (>', fJ) 111:SN m �111 2: 2: VI ,m flY/,m ()., fJ) = flPN V(>', fJ). ( 8.27) I I I:SN m �111 Then we have from ( 8.12) and ( 8.27) that

- 2: 2:

+

HN = PNH and WN = PNW. (DrHN(t) + J (HN(t) + 8T DrHN(t), WN(t)) , r/J) + I-' (V (HN(t) + UT DrHN(t)) , V r/J) = (� Gj (t) + f(t), r/J) + (VG4 (t), Vr/J ) , 'V r/J E VN, t E Rr(T), (VWN(t) ; V r/J) = (HN(t), r/J ) , 'V r/J E V�, t E R,. (T ) , A (W N(t)) = 0, t E R,.(T), = HN(O) PNHo , ()., fJ) E S

Let

( 8.28 )

Spectral Methods and Their Applications

328 where

Gl(t) = DrHN(t) - OtHN(t), G2 (t) = J (HN(t), WN(t » - J (H(t), w (t » , G3 (t) = DTJ (DrHN(t), WN(t » , G4 (t) = J.LO'TDrHN(t). Next, set fI = 1IN - HN and q, = 'ifJN - WN. We get from (8.15) and (8.28) that (DrfI(t) + J (fI(t) + DTDrH(t), WN(t) + q, (t) ) +J (HN(t) + DTDrHN(t) , q, (t) ) , t/» - J.L ( (fI(t) + O'T DrfI(t) ) t/» = - (� Gj(t), t/» - ( V'G4 (t), V't/» , V t/> VN, t Rr( T ) - (Aq, (t), t/» = (fI(t) , t/» , V t/> V�, t Rr(T), A ( q, (t) ) = 0 , t Rr(T), fIC O) = 0, (A, 9) S. By an argument as in the proof of Theorem 8.4, we can deduce an estimate similar to (8.24) . But 1IN and ii are replaced by HN and H respectively, while p et) is replaced A

E

E

E

E

E

,

,

E

by

As before, we have

I: I Gl(s) 1 I 2 � CT 2 I 1 H I �2 (O .t ;£2 ) ' T ERT(t 8 -r)

By Lemma 8 . 1 , Theorem 8.2 and a similar estimate as (8.21) , we verify that for any 'Y > 0,

I G2 (t) 1 I 2 � c I WN I �+2 I 1 HN(t) - H(t) lI� + c Il H(t) lI� I WN(t) - w (t) I �+2 � cN-2r ( I W(t) I ;+2 I 1 H(t) I �+1 + I w (t) II�+'Y+2 I1 H(t) l I n � cN-2r ( I H(t) II ; II H(t) I �+1 + II H(t) II�+'YI I H(t) I �) .

By virtue of Lemma 8.3 and Theorem 8.2, it is easy to see that T

I: I G (s)1 I 2 8ERT(t-r) 3

�

< <

CT2 I 1 wN II�(O.T;H"'�2 ) II HN I �' (O.T;H') CT 2 I 1 W II&(O.T;H"'+2 )II H I �' (O.T;Hl ) cT 2 1 1 H 1 I &(O.T;H.., ) II H I �' (O.T;H' ) '

Chapter 8. Spectral Methods on the Spherical Surface

Finally

329

IG4 (t) l � ::; C7 2 1I H (t) 1 1'(t,t+T;H') ' ( 72 N-2r ) as long as the related norms of H exist.

Therefore p CJ + statements lead to the following conclusion. =

The above

Assume that (i) (8.26) or condition (i) of Theorem 8.4 is valid; (ii) condition (ii) of Theorem 8.4 is fulfilled; (iii) For r > O , H E C (O , Tj W+l ) Hl (O , Tj Hl ) H2 (O , Tj L2 ) . Then for all t E RT (T),

Theorem 8.6.

n

n

J-t

b4 being a positive constant depending on and the norms of H in the mentioned spaces. The above numerical analysis comes from Guo (1995) . The spectral methods on the spherical surface have also been used for the barotropic vorticity equation in Guo and Zheng (1994), and for a system of nonlinear partial dif ferential equations governing the fluid flows with low Mach number in Guo and Cao (1995) . The applications of the pseudospectral methods on the surface can be found in Cao and Guo (1997) .

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