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Acta Numerica (2011), pp. 1–119 doi:10.1017/S096249291100002X

c Cambridge University Press, 2011 Printed in the United Kingdom

Topics in structure-preserving discretization∗ Snorre H. Christiansen Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway E-mail: [email protected]

Hans Z. Munthe-Kaas Department of Mathematics, University of Bergen, N-5008 Bergen, Norway E-mail: [email protected]

Brynjulf Owren Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway E-mail: [email protected] In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subﬁelds in the numerical approximation of ordinary and partial diﬀerential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial diﬀerential equations, followed by a discussion of how time integrators can be designed to preserve ﬁrst integrals in the diﬀerential equation using discrete gradients and discrete variational derivatives. Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes. The theory of mixed ﬁnite elements is developed in terms of special inverse systems of complexes of diﬀerential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite ∗

Colour online available at journals.cambridge.org/anu.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren piecewise polynomial ﬁnite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

CONTENTS 1 2

Introduction Integration methods based on Lie group techniques 3 Schemes which preserve ﬁrst integrals 4 Spatial symmetries, high-order discretizations and fast group-theoretic algorithms 5 Finite element systems of diﬀerential forms Appendix References

2 5 21 32 65 108 112

1. Introduction The solution of partial diﬀerential equations (PDEs) is a core topic of research within pure, applied and computational mathematics. Both measured in the volume of published work1 and also in terms of its practical inﬂuence on application areas such as computational science and engineering, PDEs rank above all other mathematical subjects. Historically, the ﬁeld of numerical solutions of PDEs has its roots in seminal papers by Richard Courant and his students Kurt Friedrichs and Hans Lewy, in the early twentieth century. Among the nearly 4000 academic descendants of Courant we ﬁnd a large fraction of the key contributors to the ﬁeld, such as Friedrichs’ student Peter Lax, who received the Abel Prize in 2005 for his ground-breaking contributions to the theory and application of PDEs. Fundamental properties of PDE discretizations, which have been recognized as crucial ever since the days of the Old Masters, include accuracy, stability, good convergence properties and the existence of eﬃcient computational algorithms. In more recent years various aspects of structure preservation have emerged as important in addition to these fundamental properties. 1

A count of all mathematical publications from 2001 to 2010, sorted according to the AMS Mathematics Subject Classiﬁcation, reveals that number 35 PDEs ranks highest of all primary topics with 46 138 entries; in second place is 62 Statistics, with 39 176.

Topics in structure-preserving discretization

3

Within the topics of ordinary diﬀerential equations and time integration, a systematic study of structure-preserving discretizations was undertaken by Feng Kang in Beijing, starting in the 1980s. During the past decade a systematic study of the preservation of various geometric structures2 has evolved into a mature branch of numerical analysis, termed the geometric integration of diﬀerential equations (Hairer, Lubich and Wanner 2006, Leimkuhler and Reich 2004, Sanz-Serna and Calvo 1994). Experience has shown that the preservation of geometric properties can have a crucial inﬂuence on the quality of the simulations. In long-term simulations, structure preservation can have a dramatic eﬀect on stability and global error growth. Examples of such structures are symplecticity, volume, symmetry, reversibility and ﬁrst integrals. In short-term simulations it is frequently seen that discretization schemes designed with structure preservation in mind enjoy small errors per step, and hence become eﬃcient for shorter time simulations too. An important class of problems is that of partial diﬀerential equations whose solutions may be subject to blow-up in ﬁnite time. We have seen that schemes which are designed to inherit certain symmetries of the continuous problem tend to perform well in capturing such ﬁnite-time singularities in the solution. For spatial PDEs, a parallel investigation of compatible discretizations has been undertaken (Arnold, Bochev, Lehoucq, Nicolaides and Shashkov 2006a). The equations of mathematical physics (describing ﬂuids, electromagnetic waves or elastic bodies, for instance) have been presented in geometric language, easing the construction of discretizations which preserve important geometric features, such as topology, conservation laws, symmetries and positivity structures. When well-posedness of the continuous PDE depends on the conservation or monotonicity of certain quantities, such as energy, it seems equally important for the stability of numerical schemes that they enjoy similar properties. Rather than approximately satisfying the exact conservation law, as – it could be argued – any consistent scheme would do, it seems preferable, in order to obtain stable methods, to exactly satisfy a discrete conservation law. For PDEs written in terms of grad, curl and div acting on scalar and vector ﬁelds, one is led to construct operators acting on certain ﬁnite-dimensional spaces of scalar and vector ﬁelds, forming a complex which, in spatial domains with trivial topology, is an exact sequence. More generally, various discretizations of the de Rham sequence of diﬀerential forms have been introduced, and the successful ones are related to constructs of combinatorial topology such as simplicial cochain complexes (Arnold, Falk and Winther 2010).

2

A geometric structure is understood as a structural property which can be deﬁned independently of particular coordinate representations of the diﬀerential equations.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

This survey paper takes a view of PDEs analogous to looking at the moon through a telescope with very high magniﬁcation. In the vast lunar landscape we will focus on a small number of craters with particularly beautiful properties, and leave the rest of the lunar surface out of our main focus. One goal of the paper has been to tie together recent time integration techniques, in particular Lie group techniques and integral-preserving integration, and combine these with recent developments in structure-preserving spatial discretizations. The paper consists of four main parts: Sections 2–5. Section 2 covers integration methods based on Lie group techniques. We provide an introduction to the theory of Lie group integrators and describe how computational algorithms can be devised, given a group action on a manifold and coordinates on the acting group. Various choices of coordinates are discussed. Another type of Lie group integrator consists of those based on compositions of ﬂows, and we discuss some applications of these methods to the time integration of PDEs. We focus in particular on applications outside the class of exponential integrators for semilinear problems. In Section 3 we discuss integral-preserving schemes. These methods apply to any PDE which has a known ﬁrst integral, for instance an energy functional. We demonstrate how the method of discrete variational derivatives can be used to devise integral-preserving time integration schemes for PDEs. We consider, in particular, schemes that are linearly implicit, and therefore need the solution of only one linear system per time step. We also discuss methods which apply discrete variational derivatives, or discrete gradients, to conserve an arbitrary number of ﬁrst integrals. In Section 4 we cover spatial symmetries, high-order discretizations and fast group-theoretic algorithms. The topic of this section is the exploitation of spatial symmetries. A recurring theme is that of linear diﬀerential operators commuting with groups of isometries acting on the domain. We investigate high-order spectral element discretization techniques based on simplicial subdivisions (into triangles, tetrahedra and simplexes in general) using high-order multivariate Chebyshev bases on the triangles. The eﬃcient computation of matrix (and operator) exponentials is crucial for time integrators based on Lie group actions. We survey recent work on high-order discretization techniques, fast Fourier algorithms based on group-theoretic concepts and fundamental results from representation theory. In Section 5 we discuss ﬁnite element systems for diﬀerential forms. A framework for mixed ﬁnite elements is developed, general enough to allow for non-polynomial basis functions and a decomposition of space into noncanonical polytopes, but restrictive enough to yield spaces with local bases, interpolators commuting with the exterior derivative, and exact sequences where desirable. A smoothing technique gives Lq -stable commuting quasiinterpolation operators, from which various error estimates can be derived.

Topics in structure-preserving discretization

5

2. Integration methods based on Lie group techniques The use of Lie group techniques for obtaining solutions to diﬀerential equations dates back to the Norwegian mathematician Sophus Lie in the second half of the nineteenth century. In Lie’s time these were used exclusively as analytical tools; however, more recently it has become increasingly popular to include Lie groups as an ingredient in numerical methods. The use of Lie groups in the numerical approximation of diﬀerential equations can be divided into two categories: one in which the aim is to preserve symmetries or invariance of the continuous model, and another in which Lie groups are used as building blocks for a time-stepping procedure. In this section we shall give a brief introduction to the mathematical machinery we use, and we will focus on the second category, that of using Lie groups as a fundamental component when designing numerical time integrators. We shall give a short introduction to the basics of Lie group integrators; for more details consult Iserles, Munthe-Kaas, Nørsett and Zanna (2000) and Hairer et al. (2006). The variety of studies related to Lie group integrators is now too large to cover in an exposition of this type. For this reason we shall focus on a selected part of the theory, and mostly consider integrators designed for nonlinear problems. Important classes of schemes that we shall not discuss here are methods based on the Magnus expansions and Fer expansions, usually applied to linear diﬀerential equations. This work developed in the 1990s, in large part due to Iserles and Nørsett: see, e.g., Iserles and Nørsett (1999). There are several excellent sources for a summary of these methods and their analysis, for instance the Acta Numerica article by Iserles et al. (2000), the monograph by Hairer et al. (2006) and the more recent survey by Blanes, Casas, Oteo and Ros (2009), which also contains many applications of these integrators. 2.1. Background and notation Let M be some diﬀerentiable manifold and let X (M ) be the set of smooth vector ﬁelds on M . We consider every X ∈ X (M ) as a diﬀerential operator on the set of smooth functions F(M ) on M . Thus, in local coordinates (x1 , . .∂. , xm ) in which X has components X1 , . . . , Xm , we write X = i Xi (x) ∂xi , so X[f ] = df (X), f ∈ F(M ) is the directional derivative of f along the vector ﬁeld X. The ﬂow of a vector ﬁeld X ∈ X (M ) is a one-parameter family of maps exp(tX) : Dt → M , where Dt ⊂ M . For any x ∈ Dt we have exp(tX)x = γ(t), where γ(t) ˙ = X|γ(t) ,

γ(0) = x,

t ∈ (a(x), b(x)), a(x) < 0 < b(x).

The domain for exp(tX) is the set Dt = {x ∈ M : t ∈ (a(x), b(x))}; for further details see, e.g., Warner (1983, 1.48).

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

The chain rule for the derivative of compositions of maps between manifolds is (ψ ◦ φ) = ψ ◦ φ , φ : M → N, ψ : N → P. From this, we easily obtain the useful formula X[f ◦ φ](x) = φ (X|x )[f ](y),

φ : M → N, X ∈ X (M ), f ∈ F(N ), (2.1) for x ∈ M, y = φ(x).

The Lie–Jacobi bracket on X (M ) is deﬁned simply as the commutator of vector ﬁelds Z = [X, Y ] = XY − Y X. With respect to coordinates (x1 , . . . , xm ) it has the form Zi = [X, Y ]i =

m j=1

Xj

∂Yi ∂Xi − Yj . ∂xj ∂xj

This bracket makes X (M ) a Lie algebra; the important properties of the bracket is that it is bilinear, skew-symmetric and satisﬁes the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0,

∀X, Y, Z ∈ X (M ).

It is an easy consequence of (2.1) that φ ([X, Y ]) = [φ (X), φ (Y )],

X, Y ∈ X (M ), φ : M → N.

(2.2)

We say that Y is φ-related to X ∈ X (M ) if Y |φ(x) = φ (X|x ) for each x ∈ M . Note that Y is not generally a vector ﬁeld on N : since φ does not have to be either injective or surjective, we must treat Y as a pullback section of φ∗ T N over M . In the particular case when φ is a diﬀeomorphism, there is a unique Y ∈ X (N ) which is φ-related to X; this vector ﬁeld is then called the pushforward of X with respect to φ: Y = φ∗ X = φ (X) ◦ φ−1 . A Lie group G is a diﬀerentiable manifold which is furnished with a group structure such that the multiplication is a smooth map from G × G to G, and the map g → g −1 , g ∈ G is smooth as well. We brieﬂy review some important concepts related to Lie groups. Every Lie group G has a Lie algebra associated to it, which can be deﬁned as the linear subspace g ⊂ X (G), of right-invariant vector ﬁelds on G equipped with the Lie–Jacobi bracket. By right-invariance we mean invariance under right translation. Consider the diﬀeomorphism Rg : G → G deﬁned as Rg (h) = h · g. A vector ﬁeld X ∈ X (G) is right-invariant if it is Rg -related to itself, i.e., Rg∗ X = X. By (2.2) it follows that the Lie–Jacobi bracket between two right-invariant vector ﬁelds is again right-invariant. For every v ∈ Te G, where e is the identity element, there is a unique Xv ∈ g such that Xv |g = Rg (v), and for X ∈ X (G) there is an element vX ∈ Te G given as vX = X|e . This shows that g is isomorphic to Te G, and it may often be convenient to represent

Topics in structure-preserving discretization

7

g as Te G rather than as right-invariant vector ﬁelds on G. We next let the group act locally on a manifold. A left group action on a diﬀerentiable manifold M by a Lie group G is a map G × M → M , which we denote as y = g · x ∈ M or simply y = gx for g ∈ G and x ∈ M . The group action must satisfy the conditions e · x = x, ∀x ∈ M,

g · (h · x) = (g · h) · x.

The group action is said to be free if g · x = x ⇒ g = e, so that the only group element which leaves x ∈ M ﬁxed is the identity element. If it is true that for every pair of points x ∈ M, y ∈ M there exists a group element g such that y = g · x, the action is called transitive. We usually just need a local version of transitivity, requiring that for every x ∈ M , G · x contains some open neighbourhood of x. The orbit of the group action containing x is the set Ox = {g · x | g ∈ G}. It is sometimes useful to restrict the action to an orbit when transitivity is a desired property. Similarly, if one needs the group action to be free, it may be useful to extend the action to multispace M ×r M (r copies of M ) or to a suitable jet space whenever M takes the form of a ﬁbred space. Let G be a Lie group with Lie algebra g. Assume that G acts locally on the manifold M , and set Λx (g) = g · x for g ∈ U ⊆ G, x ∈ M where U is some open neighbourhood of the identity element. For every ﬁxed v ∈ g there is a vector ﬁeld Xv ∈ X (M) deﬁned by d Xv |x = λ∗ (v)|x = Λx (v) = γ(t) · x, dt t=0

where γ(t) is any smooth curve on G such that γ(0) = e, γ(0) ˙ = v. So λ∗ is a Lie algebra homomorphism of g into its image in X (M ). For Lie group integrators, it is important that the action is locally transitive; in this case it is true that for every x ∈ X λ∗ (g)|x = Tx M. This property means, in particular, that for any smooth vector ﬁeld F ∈ X (M ) there exists a map f : M → g such that F |x = λ∗ (f (x))|x ,

(2.3)

a formulation called the generic presentation of ODEs on manifolds by Munthe-Kaas (1999). If the action is free then the map f (x) is unique for a given vector ﬁeld F . Suppose further that the Lie algebra g is of dimension d and let e1 , . . . , ed be a basis for g. Let Ei = λ∗ (ei ), i = 1, . . . , d. We ¯ = span{E1 , . . . , Em }, that is, call the set {E1 , . . . , Ed } a frame,3 and let g 3

In the literature, a ‘frame’ is often used as a local object requiring that E1 |x , . . . , Ed |x is a basis for Tx M . We do not impose this condition here per se, as we ﬁnd it useful to have a global representation of vector ﬁelds on M .

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

the linear span of the frame ﬁelds over R or C. Now we can represent any smooth vector ﬁeld X ∈ X (M ) by means of d functions fi : M → R: F |x =

d

fi (x) Ei |x .

(2.4)

i=1

The set of functions fi is uniquely given only if the action is free. There are two alternative ways to proceed, following either the terminology introduced by Crouch and Grossman (1993) or that of Munthe-Kaas (1995, 1998, 1999). In the former case, one introduces a freeze operator Fr : M × X (M ) → g¯ relative to the frame. It is deﬁned by Xp |x := Fr(p, X)|x = fi (p)Ei |x . (2.5) i

The frozen vector ﬁeld Xp has the property that it coincides with the unfrozen ﬁeld X at the point x = p, i.e., Xp |x = X|x . A main assumption underlying many Lie group integrators is that ﬂows of frozen vector ﬁelds can be calculated, or in some cases approximated with acceptable computational cost. 2.2. Integrators based on coordinates on the Lie group In what follows, we shall take g to be Te G. Consider a diﬀeomorphism deﬁned from some open subset of U ⊆ g containing 0, i.e., Ψ : U → G and we require that Ψ(0) = e, Ψ0 (v) = v, ∀v ∈ g. For convenience, we shall work with a right-trivialized version of Ψu , setting ◦ dΨu , Ψu = RΨ(u)

dΨu : g → g.

Let the group act locally on the manifold M ; suppose that the action is deﬁned on a subset Gp ⊆ G, where Ψ(U ) ⊂ Gp for every p ∈ M . For any p ∈ M and u ∈ U , set λp (u) = Ψ(u) · p, and let f : M → g represent the vector ﬁeld F ∈ X (M ) through the form (2.3). We deﬁne the vector ﬁeld F˜p on X (U ) by F˜p = dΨ−1 (2.6) u f (λp (u)). u

A simple calculation (Munthe-Kaas 1999, Owren and Marthinsen 2001) shows that F is λp -related to F˜ such that, for any u ∈ U , we have F |λp (u) = λp u (F˜p |u ). This local relatedness between vector ﬁelds F of the form (2.3) whose ﬂows are to be approximated, and vector ﬁelds on the algebra of the acting group serves as the key underlying principle of many Lie group integrators. The idea is that near any point p ∈ M one may represent curves in the form y(t) = λp (σ(t)), and if the diﬀerential equation for y(t) is given by F , then a diﬀerential equation for σ(t) is that of F˜ . Most of the known integrators for ODEs have the property that when the solution belongs to

Topics in structure-preserving discretization

9

some linear space, the numerical method will provide approximations for the solution belonging to the same linear space. So one may now approximate solutions to the ODE given by F˜ , and obtain numerical approximations in the linear space g. The map λp will then map these approximations onto M by deﬁnition. One may typically choose the initial value in each step to be p, so that when solving for σ in g one sets σ(tn ) = 0. Suppose that a one-step method map which preserves linear structure is denoted Φh,F˜p , where h is the time step. Algorithm 2.1. for n = 0, 1, . . . do p ← yn σn+1 ← Φh,F˜p (0) yn+1 ← λp (σn+1 ) end for We may be more speciﬁc and for instance insist that the scheme Φh,F˜ to be used in the Lie algebra is an explicit Runge–Kutta method, with weights bi and coupling coeﬃcients aji , 1 ≤ j < i ≤ s. In this case we can phrase the scheme as follows for integrating a system in the form (2.3) from t0 to t0 + h, with initial value y(t0 ) = y0 ∈ M . Algorithm 2.2. (Runge–Kutta–Munthe-Kaas) for i = 1 → s do j ui ← h i−1 j=1 ai kj ki ← f (Ψ(ui ) · y0 ) ki ← dΨ−1 ui (ki ) end for v ← h sj=1 bj kj y1 = Ψ(v) · y0 An obvious beneﬁt of schemes of this form is that they will preserve the manifold structure, a feature which cannot be expected when M is modelled as some embedded submanifold of Euclidean space. If the manifold happens to belong to a level set of one or more ﬁrst integrals, then the Lie group integrators will automatically preserve these integrals. There are, however, other interesting situations, when the group action is used as a building block for obtaining a more accurate representation of the exact solution than is possible with other methods. The idea behind Lie group integrators can also be seen as a form of preconditioning. Coordinate maps. The computational cost of the Lie group integrators is an important issue, and the freedom one has in choosing the coordinate map Ψ may be used to optimize the computational cost. The generic choice for Ψ(u) is of course the exponential map Ψ(u) = exp u. This choice is called

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

canonical coordinates of the ﬁrst kind. If the Lie group and its Lie algebra are realized as matrices, we have exp u =

∞ uk k=0

k!

.

Issues related to computing the matrix exponential have been thoroughly debated in the literature, going back to the seminal paper of Moler and Van Loan (1978), and its follow-up, Moler and van Loan (2003). For dense n × n matrices one would normally expect a computational complexity of approximately C n3 , where C will depend on several factors, as the tolerance for the accuracy, the size of the matrix elements and the conditioning of the matrix. The methods presented in these papers, however, do not usually respect the Lie group structure, such that an approximation g˜ ≈ exp u, u ∈ g will not belong to the group, i.e., g˜ ∈ G; this is a crucial issue as far as exact conservation is concerned. In practice, one is left with two alternatives. (1) Apply a standard method which yields g = exp u to machine accuracy. For relevant examples with Lie group integrators, the factor C typically lies in the range 20–30 (Owren and Marthinsen 2001). (2) Apply some approximation which is not exact, but which respects the Lie group structure, i.e., g˜ ≈ exp u with g˜ ∈ G. This approach has been pursued by Celledoni and Iserles (2000, 2001) for approximation by low-rank decomposition, as well as Zanna and Munthe-Kaas (2001/02) and Iserles and Zanna (2005) by means of the generalized polar decomposition. These approaches still have a computational cost of C n3 for Lie algebras whose matrix representation yields dense matrices, but the constant C may be smaller than for the general algorithms. It is interesting to explore other possible choices of analytic matrix functions than the exponential. One then replaces the exponential map with some local diﬀeomorphism from a neighbourhood of 0 ∈ g to G, usually required to map 0 → e. But it turns out that the exponential map is the only possible choice of analytic map that works for all Lie groups: this is asserted, for instance, using a result by Kang and Shang (1995). Lemma 2.3. Let sl(d) denote the set of all d × d real matrices with trace equal to zero and let SL(d) be the set of all d × d real matrices with determinant equal to one. Then, for any real analytic function R(z) deﬁned in a neighbourhood of z = 0 in C satisfying the conditions: R(1) = 1 and R (0) = 1, we have that R(sl(d)) ⊆ SL(d) for some d ≥ 3 if and only if R(z) = exp(z). They prove this result by taking the Lie group SL(d) of unit determinant d × d matrices, d ≥ 3, whose Lie algebra sl(d) consists of d × d matrices with

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Topics in structure-preserving discretization

trace zero as an example: see also Hairer et al. (2006, p. 102). There are, however, exceptions when certain speciﬁc Lie groups are considered. For instance, if a matrix group GJ and its corresponding Lie algebra gJ can be characterized as GJ = {A ∈ GL(d) : A JA = J},

gJ = {a ∈ gl(d) : a J + Ja = 0}, (2.7)

for some ﬁxed d × d matrix J, it turns out that every analytic function R(z) satisfying R(z)R(−z) = 1

(2.8)

will have the property a ∈ gJ ⇒ R(a) ∈ GJ . Such groups include the symplectic group Sp(d), the orthogonal groups O(d), SO(d), and the Lorenz groups SO(, d − ). One of the most popular choices of maps that satisﬁes (2.8) is the Cayley transformation Ψcay (z) =

1+z . 1−z

This transformation was used by Lewis and Simo (1994), and later by Diele, Lopez and Peluso (1998) for the group SO(d), Lopez and Politi (2001) for general groups of the form (2.7), and Marthinsen and Owren (2001) for linear equations. If one does not require the map to be realizable as an analytic function of the Lie algebra matrix, there are other choices, for instance the canonical coordinates of the second kind. This map requires the use of a basis for g, say v1 , . . . , vs where s = dim g. Then the map is deﬁned as Ψccsk : v = α1 v1 + · · · + αs vs → exp(α1 v1 ) · exp(α2 v2 ) · · · exp(αs vs ). Here, the choice as well as the ordering of the basis is clearly an issue. In Owren and Marthinsen (2001) this coordinate map was, together with a generic basis of the Lie algebra, known as the Chevalley basis. The main advantage of this map is that when matrices are used, natural choices of bases are frequently realized as sparse matrices whose exponentials may be explicitly known. As an example, one may consider the special linear group SL(d), which can be realized as the set of d × d matrices with unit determinant. The Lie algebra is then the set of trace-free d × d matrices, and the Chevalley basis is obtained by choosing d − 1 trace-free diagonal matrices, say ei+1 e i+1 − ei ei , 1 ≤ i ≤ d − 1 together with all matrices ei ej , j = i. Also, hybrid variants are possible, where, for instance, the Lie algebra is decomposed into a direct sum of subspaces, say g = g1 ⊕ g2 ⊕ · · · ⊕ gs¯,

s¯ j=1

dim(gj ) = s,

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

and a coordinate map can be realized as Ψhyb : v = v1 + · · · + vs¯ → exp(v1 ) · · · exp(vs¯),

vi ∈ gi .

In fact, such coordinate maps derived from the generalized polar decomposition were considered in Krogstad, Munthe-Kaas and Zanna (2009). Computing the inverse diﬀerential. In the transformation of the vector ﬁeld to the Lie algebra, one needs to compute the inverse of the diﬀerential dΨ−1 u v. For the case of the exponential mapping, it was found by Baker (1905) and Hausdorﬀ (1906) that it can be expanded in an inﬁnite series of commutators. Deﬁning the operator adu (v) = [u, v], for any u and v in g, we have ∞ Bk k −1 ad (v). (2.9) dexpu v = k! u k=0

Here, the constants Bk are the Bernoulli numbers, B2k+1 = 0, k ≥ 1, and the ﬁrst non-zero ones are B0 = 1, B1 = −1/2, B2 = 1/12, B4 = −1/720. We will not dwell on the convergence properties of this series (discussed, for instance, in Varadarajan (1984)), the reason being an observation made by Munthe-Kaas (1999). The series (2.9) may be truncated at any point to yield an approximation in the Lie algebra g. Since one usually requires that the initial point of each time step in the Lie algebra, σ0 = 0, it follows that the term of index k will be of at least order hk as h → 0. It is therefore suﬃcient to truncate the series at an index that is consistent with the convergence order of the time-stepper Φh,F˜ . An extensive analysis of the number of terms that must be kept in such commutator expansions was presented in Munthe-Kaas and Owren (1999), but see also McLachlan (1995). Considering the coordinate maps Ψccsk and Ψhyb , their inverse diﬀerential maps are studied in Owren and Marthinsen (2001) and Krogstad et al. (2009) respectively. In these cases, the maps are computed exactly, but by carefully studying the structural properties of the respective Lie algebras it is possible to ﬁnd computationally inexpensive algorithms: typically, if the 3/2 ). dimension of the Lie algebra is d, the cost of computing dΨ−1 u (v) is O(d Choosing the group and the group action. A particularly interesting case for a global group action is the homogeneous space. If the group G is acting transitively on M , then the manifold is called a homogeneous space. In this case it is well known (see, e.g., Bryant (1995)) that M is naturally diﬀeomorphic to G/Gp , where Gp is the isotropy or stabilizer subgroup of G, Gp = {g ∈ G : g · p = p}. The simplest case is when the action is free, so that Gp = e for all p ∈ M , meaning M ∼ = G. In this case the function f in (2.3) is unique, and there is no isotropy. In the case that there is a non-trivial isotropy group, the choice

Topics in structure-preserving discretization

13

of f is not unique, in fact f may be replaced by f + w, where w(x) ∈ ker λ∗ for every x ∈ M . It turns out that the choice of w aﬀects the numerical integrator. We proceed to give some examples frequently seen in applications: see Munthe-Kaas and Zanna (1997). Suppose that the Lie group G = SO(d) is a group of orthogonal d × d matrices. Its Lie algebra is so(d), the set of d × d matrices which are real and skew-symmetric. The manifold M is a connected open subset of the d × r matrices, and we give two examples. (1) Let the Lie group action be left multiplication g · m where · is now just matrix–matrix multiplication. We may now write the ODE vector ﬁeld as: F |y = f (y) · y,

f : M → g.

(a) M is the set of vectors in Rd with unit length, identiﬁed with the (d − 1)-dimensional sphere S d−1 . (b) M is the set of d × r matrices with orthonormal columns, identiﬁed with the Stiefel manifold Vr (Rd ). (2) r = d, and the Lie group action is the conjugation g · m = gmg , the right-hand side is just products of matrices. We may choose M to be some isospectral set, such as the set of all matrices with a given prescribed Jordan form; one may also restrict to the set of symmetric matrices with a ﬁxed set of eigenvalues. Another interesting choice is the aﬃne group, ﬁrst discussed in the context of Lie group integrators in Munthe-Kaas (1999). One may start with the group G = GL(d) Rd , the semidirect product between the general linear group and Rd . G is the group of all aﬃne linear maps acting on Rd ; we have (A, a) · y = Ay + a. The group product is given as (A, a) · (B, b) = (AB, Ab + a). The corresponding Lie algebra is g = gl(d) Rd and the Lie bracket is given as [(A, a), (B, b)] = ([A, B], Ab − Ba). The exponential mapping can be expressed in terms of the matrix exponential exp(A, b) = (exp(A), dexpA (b)), where

∞

dexpA (b) =

1 exp(A) − I b= Aj b. A (j + 1)! j=0

Munthe-Kaas made the observation that this action has a large isotropy group: the dimension of the group is d(d + 1) whereas the manifold M = Rd

14

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

only has dimension d. Setting λp (A, a) = exp(A, a)·p = exp(A)p+dexpA (a), we ﬁnd λ∗ (A, a)(y) = Ay + a, such that the isotropy algebra at y consists of all elements in G of the form (A, −Ay). Suppose that the ODE vector ﬁeld is F . Then at one extreme one may choose f (y) = (0, F |y ), in which case one would recover the method Φh,F in the algorithm described above. However, as suggested in Munthe-Kaas (1999), one may instead add to this the element (J, −Jy), J ∈ GL(d) from ker λ∗ (.)(y), to obtain the function f (y) = (J, F |y − Jy), and a natural choice for J would be the Jacobian of the vector ﬁeld F evaluated at some point near y. One may also replace the group GL(d) by a subgroup: the group SO(d) yields the special Euclidean group SE(d), whereas in applications to PDEs a popular choice is to select some oneparameter subgroup exp(tL), L ∈ gl(d). The Lie subalgebra elements are now of the form (αL, a) with ﬁxed L; thus one may just write (α, a) ∈ R×Rd , and the Lie bracket of this subalgebra is simply [(α, a), (β, b)] = [0, L(αb − βa)].

(2.10)

The isotropy group Gp is now only one-dimensional; its Lie algebra is spanned by the element (1, −Lp). Time-symmetric Lie group integrators. For many problems it is of interest to apply integrators which are symmetric in time or self-adjoint. For onestep methods, say y1 = φh (y0 ), this means that if the integrator is applied backwards in time, i.e., with step size −h and y1 as input, the output y0 results, such that φh = φ−1 −h . For Lie group integrators, one may consider as an example the Runge–Kutta–Munthe-Kaas scheme based on the midpoint rule, using the coordinate map Ψ = exp. This scheme is an implicit counterpart to the method given in Algorithm 2.2: u=

h k, 2

k = f (exp(u) · y0 ),

y1 = exp(hk) · y0 .

Here we have omitted the correction dexp−1 u , since it is not going to alter the order of the method. Following Zanna, Engø andMunthe-Kaas (2001), we deﬁne the midpoint approximation y 1 = exp 12 hu · y0 ; the method may 2 be written in the form h h f y 1 .y 1 , y0 = exp − f y 1 .y 1 , y1 = exp 2 2 2 2 2 2

Topics in structure-preserving discretization

15

which is easily seen to be symmetric. Unfortunately, Lie group integrators based on coordinates on the Lie group are not generally symmetric even if the underlying scheme on the Lie algebra is a symmetric method. Zanna et al. (2001) considers how a recentering of the coordinate system for the Lie group can be used to obtain symmetric integrators. Rather than letting Ψ(0) = e be the base point of the coordinate chart, one may choose a θ ∈ g and deﬁne a new base point q := Ψ(θ)−1 · y0 . The curve y(t) ∈ G is now represented locally by a curve σ(t) ∈ g, as y(t) = Ψ(σ(t)) · q = Ψ(σ(t)) · Ψ(θ)−1 · y0 . This shift of coordinate chart results in the same diﬀerential equation for σ(t), but with initial value σ(0) = θ corresponding to y(0) = y0 . An implicit Runge–Kutta-type method generalizes Algorithm 2.2 as follows:   σi = θ + h sj=1 aji kj ,    −1 i = 1, . . . , s, (2.11) ki = f (Ψ(σi ) · Ψ(θ) · y0 ),     ki = dΨ−1 σi (ki ), v = θ + h si=1 bi ki , y1 = Ψ(v) · Ψ−1 (θ) · y0 . The idea is now to choose the centre θ depending on the stages ki and h in such a way that the resulting method is symmetric whenever the underlying Runge–Kutta scheme is symmetric. From Zanna et al. (2001) we ﬁnd that this can be achieved by setting θ = θ(h; k1 , . . . , ks ) and requiring it to satisfy θ(−h; ks , . . . , k1 ) = θ(h; k1 , . . . , ks ) + h

s

bi ki .

(2.12)

i=1

There is of course no unique way of satisfying (2.12). An obvious choice which has been called the geodesic midpoint is h i b ki , 2 s

θ(h, k1 , . . . , ks ) = −

i=1

but in fact, one may choose more generally θ(h; k1 , . . . , ks ) = −h

s i=1

w i ki ,

where ws+1−i + wi = bi ,

i = 1, . . . , s.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

A concrete example of a symmetric scheme in the format (2.11), using the geodesic midpoint for θ and the 2-stage Gauss method is h θ = − (k1 + k2 ), 4√ 3 σ1 = −h k2 , 6 k1 = f (Ψ(σ1 ) · Ψ(θ)−1 · y0 ),

√

3 k1 , 6 k2 = f (Ψ(σ2 ) · Ψ(θ)−1 · y0 ),

σ2 = h

k2 = dΨ−1 k1 = dΨ−1 σ1 (k1 ), σ2 (k2 ), h v = (k1 + k2 ), y1 = Ψ(v) · Ψ(θ)−1 · y0 . 4 McLachlan, Quispel and Tse (2009) recently studied symmetric and symplectic schemes which are linearization-preserving at ﬁxed points, meaning that if y¯ is such that f (¯ y ) = 0 in the initial value problem y = f (y), y(0) = y0 then the integrator has the Taylor expansion about y¯:

y1 − y¯ = exp(hf (¯ y ))(y0 − y¯) + O(y0 − y¯2 ). Symmetric schemes for a class of exponential integrators applied to semilinear problems were proposed by Celledoni, Cohen and Owren (2008). 2.3. Lie group integrators based on compositions of ﬂows A pioneering article by Crouch and Grossman (1993) suggests applying compositions of ﬂows of frozen vector ﬁelds (2.5) to construct integration methods for vector ﬁelds which have been expressed in the form (2.4). They propose such generalizations both of multistep methods and of explicit Runge– Kutta methods. The latter takes a step from t0 to t0 + h as follows. Algorithm 2.4. (Crouch–Grossman) Y1 ← y0 F1 ← Fr(Y1 , F ) for r = 2 → s do 1 Yr ← exp(har−1 r Fr−1 ) · · · exp(har F1 )(y0 ) Fr ← Fr(Yr , F ) end for y1 ← exp(hbs Fs ) · · · exp(hb1 F1 )(y0 ) Here, br , akr are the weights and the coupling coeﬃcients of the Runge– Kutta method. Explicit third-order with s = 3 were presented in Crouch and Grossman (1993), and Owren and Marthinsen (1999) presented a general theory of order conditions and examples of fourth-order methods with four stages. A potential drawback of the Crouch–Grossman methods is that they require the computation of a large number of ﬂows of frozen vector ﬁelds in every time step. This led Celledoni, Marthinsen and Owren

Topics in structure-preserving discretization

17

(2003) to consider a version of these methods where ﬂows of arbitrary linear combinations of frozen vector ﬁelds were considered. The methods take the following form. Algorithm 2.5. (Commutator-free methods) Y 1 ← y0 F1 ← Fr(Y1 , F ) for r = 2 → s do k k F ) · · · exp(h Yr ← exp(h k αr,J k k αr,1 Fk )(y0 ) Fr ← Fr(Yr , F ) end for y1 ← exp(h k βJk Fk ) · · · exp(h k β1k Fk )(y0 ) Note that when the vector ﬁelds commute, the expressions for Yr and y1 collapse to k k ar Fk (y0 ), y1 = exp b Fk (y0 ), Yr = exp k

k

where akr

=

J j=1

k αr,j ,

k

b =

J

βjk .

j=1

These coeﬃcients correspond to the akr , bk in the Runge–Kutta–MuntheKaas methods, and they must in particular satisfy the classical order conditions for Runge–Kutta methods. The key idea of the commutator-free methods is to keep the number of exponentials or ﬂows as low as possible in each stage. In Celledoni et al. (2003) the following fourth-order method was presented: Y 1 = y0 , Y2 = exp

1

F1 = Fr(Y1 , F ),

hF1 (y0 ), F2 = Fr(Y2 , F ), F3 = Fr(Y3 , F ), (2.13) Y3 = exp 2 hF2 (y0 ), 1 Y4 = exp − 2 hF1 + hF3 (Y2 ), F4 = Fr(Y4 , F ), h h (3F1 + 2F2 + 2F3 − F4 ) (y0 ). y1 = exp 12 (−F1 + 2F2 + 2F3 + 3F4 ) exp 12 21

Note that if the number of exponentials is counted according to the format of Algorithm 2.5, we would have two exponentials in the expression for Y4 , but since the rightmost exponential coincides with Y2 , we eﬀectively compute only one exponential in each stage Yr , r = 2, 3, 4 and two exponentials in y1 , and thus the total number of ﬂow calculations in this method is 4. In comparison, the methods of Crouch and Grossman would require 10 ﬂow calculations. A complete description of the order theory for this type of method was given in Owren (2006). Wensch, Knoth and Galant (2009)

18

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

extend the idea of re-using ﬂow calculations, as with Y2 in (2.13), to allow for the composition of exponentials to be applied to more general expressions involving y0 and the stage values Yi . For the case of linear diﬀerential equations, one can design commutator-free methods based on the Magnus series expansions: see Blanes and Moan (2006). 2.4. Applications of Lie group integrators to PDEs The Lie group integrators described in the preceding sections can be used as time-steppers in solving partial diﬀerential equations. It should come as no surprise that these integrators are intimately linked to a well-known class of methods known as ‘exponential integrators’, which go back to the early 1960s in pioneering work by Certaine (1960), Nørsett (1969) and Lawson (1967). An excellent account of these methods was presented in a recent Acta Numerica article by Hochbruck and Ostermann (2010), and we refer the reader to that article and the references therein for a detailed account of exponential integrators. There are several important papers which study analytical properties of exponential integrators, in particular for parabolic problems. An important issue is to describe the convergence order in the presence of unbounded operators: see, for instance, Hochbruck and Ostermann (2005) and Ostermann, Thalhammer and Wright (2006) and the references therein. In studying such time integrators one possibility is to apply the time-stepper without discretizing in space, thereby interpreting the method as a discrete trajectory in some abstract space of inﬁnite dimension. Another option is to ﬁrst do a spatial discretization to obtain a ﬁnite-dimensional system, and then apply the time-stepping scheme to this ﬁnite-dimensional problem. The most widespread application of exponential integrators is perhaps to the semilinear problem ut = Lu + N (t, u). (2.14) Here L is a linear operator, and N (t, u) is a nonlinear function or mapping. In many interesting cases, L may be a diﬀerential operator of higher order than those appearing in N (t, u); we refer to Minchev (2004) for an extensive treatment of this case. Such a problem lends itself to a formulation by the action of the aﬃne group as described above; then any Lie group integrator can be applied. Cox and Matthews (2002) found an exponential integrator for (2.14) which turned out to be identical to the general commutatorfree Lie group integrator in Celledoni et al. (2003) applied with the aﬃne action adapted to (2.14), when the Lie algebra g is a subalgebra of the aﬃne Lie algebra consisting of elements (αL, b) as described above. Also, Krogstad (2005) derived exponential integrators with excellent properties for semilinear problems. In what follows we shall consider the application of Lie group integrators to some non-standard cases, diﬀering from (2.14).

Topics in structure-preserving discretization

19

Celledoni–Kometa methods. Whereas the most common application of exponential integrators is to problems of the form (2.14), there are other examples in the literature. One is a class of schemes proposed by Celledoni and Kometa (Celledoni 2005, Celledoni and Kometa 2009). They consider convection–diﬀusion problems of the form ∂u + V(u) · ∇u = ν∇2 u + f, (2.15) ∂t where the convection term V(u) · ∇u is dominating, and 0 < ν 1. A semidiscretized version of (2.15) is obtained as yt − C(y)y = Ay + f,

y(0) = y0 .

(2.16)

Here the nonlinear convection term is assumed to take the form C(y)y for some matrix C(y) and the diﬀusion operator is similarly replaced by a constant matrix A. A typical strategy is now to treat the convection term and the diﬀusion term separately, using an explicit and possibly high-order approximation for the convection term, and an implicit integrator for the comparably stiﬀ diﬀusion term. This is the idea behind the IMEX methods, for instance: see Kennedy and Carpenter (2003) for a general theory. Note that the matrix or linear operator A will usually be symmetric and negative deﬁnite: therefore implicit schemes can be implemented eﬃciently in such a splitting. The approach of Celledoni and Kometa (2009) is to use a non-trivial ﬂow calculation for the convection part. A model example of a method in the class they consider for (2.16) is y1 = exp(h C(y0 ))y0 + h Ay1 + h f. This scheme can be generalized by adding stages in a Runge–Kutta fashion and by introducing compositions of exponentials in a similar way to that of commutator-free Lie group integrators. This leads to a semi-Lagrangian approach in which linearized versions of the convection part of the problem are solved exactly by the method of characteristics, for example. The general form of the method applied to (2.16) is   Yi = φi y0 + h sj=1 ai,j φi,j AYj ,   k k  i = 1, . . . , s, φi = exp h k αi,J C(Yk ) · · · exp h k αi,1 C(Yk ) ,     φi,j = φi φ−1 j , ¯ 0 + h s bi φ¯i AYj , y1 = φy i=1 k φ¯ = exp h β k C(Yk ) · · · exp h β C(Yk ) , k

¯ −1 . φ¯i = φφ i

J

k

1

Examples of schemes of order two and three are provided in Celledoni and Kometa (2009).

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

Commutator-free methods. Wensch et al. (2009) introduce the multirate inﬁnitesimal step method for diﬀerential equations formulated on Euclidean space of the form y = f (y) + g(y). (2.17) The method is deﬁned by the authors as follows. (1) Deﬁne the initial points of the stage ﬂows as Zni (0) = yn + αij (Ynj − yn ). j

(2) Compute the stages by solving the following equations exactly: 1 Zni (τ ) = γij (Ynj − yn ) + βij f (Ynj ) + di g(Zni (τ )), h j

j

Yni = Zni (h), and the update stage, yn+1 , can be interpreted as an additional stage Yn,s+1 . It is thus assumed that one can calculate exactly ﬂows of vector ﬁelds of the form y = c + f (y), (2.18) and the explicitness of the scheme is ensured by requiring that αij = βij = γij = 0 for j ≥ i. Furthermore, one assumes that the method is balanced, meaning that di = j βij for every i. The order analysis can be treated by the theory developed in Owren (2006), and third-order methods are derived under some simplifying assumptions, setting the γij = 0 and demanding that precisely one αij = 0 for each i. When g(y) ≡ 0, the scheme reduces to a standard Runge–Kutta method. In practical situations one may relax the requirement that the ﬂow of (2.18) is solved exactly, but one may instead approximate it with high order of accuracy, for instance by using much smaller time steps than the h which features in the method format above – hence the name multirate methods. The main application of these methods in Wensch et al. (2009) is the 2D Euler equations for atmosphere processes, in which the two main scales are the fast acoustic waves and the slower advection terms: ∂ρ = −∇ · (ρu), ∂t ∂ρu ∂p = −∇ · (u ⊗ ρu) − ∇Θ − f (ρ), ∂t ∂Θ ∂Θ = −∇ · (Θu), ∂t p = ΘR(p/p0 )κ .

Topics in structure-preserving discretization

21

Here u = (u, w) denotes the horizontal and vertical velocity, ρ is the density, Θ = θρ where θ is the potential temperature, p is the pressure and R, p0 , κ are physical parameters. The spatial discretizations used are third-order upwind diﬀerences for the advection terms and centred diﬀerences for the pressure terms.

3. Schemes which preserve ﬁrst integrals The development of energy-preserving ﬁnite diﬀerence schemes go all the way back to the seminal work of Courant, Friedrichs and Lewy (1928) where the celebrated energy method was presented for the ﬁrst time. Energy is an example of a ﬁrst integral, a functional which is preserved along exact solutions of the PDE. In the early days of integral-preserving methods, the focus was mostly on stability issues, the preservation of energy, for instance, is an invaluable tool in proving stability of numerical methods for evolutionary problems. More recently some of the interest has shifted to the conservation property itself, considering not only stability issues, but also the ability of the numerical solution to inherit structural properties of the continuous system. In this section we shall discuss some rather general procedures that can be used to construct numerical schemes which exactly preserve one or more ﬁrst integrals of the partial diﬀerential equations. The framework we present includes, but is not limited to, Hamiltonian PDEs. That is, we shall consider systems which can be written in the form δH , (3.1) δu where D is a skew-adjoint operator, which is applied to the variational derivative of a ﬁrst integral H. In general, the operator D may depend on the solution itself as well as its partial derivatives with respect to spatial variables. In the literature one can also ﬁnd several examples in which ‘dissipative versions’ of (3.1) are studied, where the operator D is replaced by an operator which causes the functional H to decrease monotonically along exact solutions, the aim then being to design a numerical method which has the same dissipation property. Here we shall consider only the conservative case, and our aim is to derive numerical schemes which, at each time level t = tn , produce a numerical approximation U n to the exact solution u(tn , ·) of (3.1), satisfying H(U n ) = H(U 0 ) for all n ≥ 1, or Hd (un ) = Hd (u0 ) where Hd is a suitable spatially discretized version of H and un is the corresponding space discretization of U n . We shall also assume that the spatial domain X and boundary conditions are of such a type that (possibly repeated) integration by parts does not cause boundary terms to appear. In the rest of this section we deﬁne M to be a ﬁxed connected open subset M ⊂ X × U , where X is the space of independent variables x1 , . . . , xp , and ut = D

22

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

U is the space of dependent variables u1 , . . . , uq . The spaces of prolongations M (n) ⊂ U (n) are then open subsets of the corresponding jet spaces. Following Olver (1993), we deﬁne the C-algebra of smooth diﬀerential functions A from M (n) to C. The space Aq is the set of q-plets of diﬀerential functions. Usually, the order n of the prolongation is of minor importance to the arguments presented here, and thus we just write F [u] or F ((uαJ )) for F ∈ Aq . We shall make use of operators D = D[u], whose adjoint D∗ [u] is deﬁned via P D∗ Q

DP Q = X

X

for all compactly supported diﬀerential functions P and Q. The operator D is skew-adjoint if D∗ = −D. We consider the subclass of evolution equations which can be written in the form ∂u δH = F [u] = D[u] [u], ∂t δu

(3.2)

where D[u] is skew-adjoint. The functional H[u] is usually of the form H[u] = G[u]. (3.3) Equations of this form include the set of Hamiltonian PDEs, but here we do not need to assume that the operator D[u] deﬁnes a Poisson bracket. The general formula for the variational derivative expressed in terms of the Euler operator is included here for later use: δH δu is an m-vector depending on uαJ for |J| ≤ ν where ν ≥ ν, deﬁned via (Olver 1993, p. 245) δH ∂ · ϕ dx = H[u + ϕ], (3.4) ∂ =0 Ω δu for any suﬃciently smooth m-vector of functions ϕ(x). One may calculate δH δu by applying the Euler operator to G[u]; the α-component is given by δH α = Eα G[u], (3.5) δu where Eα =

|J|≤ν

(−1)|J| DJ

∂ , ∂uαJ

(3.6)

so that the sum ranges over all J corresponding to derivatives uαJ featuring in G. We have used total derivative operators: ∂uα ∂ J . D J = D j1 . . . D jk , D i = ∂xi ∂uαJ α,J

23

Topics in structure-preserving discretization

Another example of a Hamiltonian PDE is a generalized version of the Korteweg–de Vries equation ut + uxxx + (up−1 )x = 0, which can be cast in the form (3.2), with 1 2 1 p u − u dx, H[u] = 2 x p

D=

(3.7) ∂ . ∂x

(3.8)

Clearly, the functional H[u] is conserved along solutions of (3.2), since d δH δH δH H[u] = [u] ut = [u]D[u] [u] = 0, dt δu δu δu because D is skew-adjoint. The idea behind conservative integration schemes is to construct numerical methods which exactly reproduce this property as the numerical method advances from time tn to tn+1 , i.e., assuming that the solution to (3.2) at time t = tn is approximated by U n , we ask the scheme to satisfy H[U n ] = H[U 0 ], ∀n ≥ 1. (3.9) Alternatively, (3.9) may be imposed for some approximation Hd to H. The methodology we present here has its source in the ODE literature. One of the most prevalent approaches is that of discrete gradients, examples of which were proposed by LaBudde and Greenspan (1974), and treated systematically by Gonzalez (1996) and McLachlan, Quispel and Robidoux (1999). A ﬁnite-dimensional counterpart to (3.2) is the system y˙ = S(y)∇H(y) = f (y),

y ∈ Rm ,

(3.10)

where S(y) is an antisymmetric matrix. The idea is to introduce a discrete approximation to the gradient, letting ∇H : Rm ×Rm → Rm be a continuous map satisfying H(u) − H(v) = ∇H(v, u) (u − v), ∇H(u, u) = ∇H(u). The existence of such discrete gradients is well established in the literature, and their construction is not unique: see, for instance, the monograph by Hairer et al. (2006). The averaged vector ﬁeld (AVF) gradient, for example, is deﬁned by 1 ∇AVF (v, u) = ∇H(ξu + (1 − ξ)v) dξ. (3.11) 0

Once a discrete gradient has been found, one immediately obtains an integral-preserving method, simply letting y n+1 − y n = S(y n , y n+1 ) ∇(y n , y n+1 ). ∆t

24

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

Here ∆t is the time step, and S(y n , y n+1 ) is some approximation to the matrix S in (3.10); one would normally require that S(y) = S(y, y). Discrete gradient methods are usually implicit. The generalization of this approach to PDEs is immediate, and has been developed independently of the ODE literature by Furihata, Matsuo and co-authors (Furihata 1999, Furihata 2001a, Furihata 2001b, Furihata and Matsuo 2003, Matsuo 2007, Matsuo 2008, Matsuo and Furihata 2001, Matsuo, Sugihara, Furihata and Mori 2000, Matsuo, Sugihara, Furihata and Mori 2002, Yaguchi, Matsuo and Sugihara 2010), and recently presented in its full generality in Dahlby and Owren (2010). One may deﬁne a discrete variational derivative of the Hamiltonian H[u] to be a continuous (diﬀerential) function of two arguments δH u and v δ(v,u) satisfying the relations δH H[u] − H[v] = (u − v) dx, (3.12) δ(v, u) Ω δH δH = . δ(u, u) δu

(3.13)

To obtain a conservative scheme we simply replace the skew-adjoint operator D[u] in (3.2) by some approximation D[v, u] satisfying D[u, u] = D[u], and deﬁne the method as U n+1 − U n δH = D[U n , U n+1 ] . (3.14) n ∆t δ(U , U n+1 ) The AVF scheme can of course also be interpreted as a discrete variational derivative method, where 1 δH δHAVF = [ξu + (1 − ξ)v] dξ. (3.15) δ(v, u) 0 δu The fact that (3.15) veriﬁes the condition (3.12) is seen from the elementary identity 1 d H[u] − H[v] = H[ξu + (1 − ξ)v] dξ. (3.16) 0 dξ The derivative under the integral is written d d H[v + (ξ + ε)(u − v)] H[ξu + (1 − ξ)v] = dξ dε ε=0 δH = [ξu + (1 − ξ)v](u − v) dx. Ω δu Now substitute this into (3.16) and interchange the integrals to obtain (3.12). We illustrate the discrete variational derivative method to the Korteweg– de Vries equaton (3.7)–(3.8) using (3.15). The variational derivative is

Topics in structure-preserving discretization δH δu

25

= −uxx − up−1 such that 1 δHAVF =− (ξu + (1 − ξ)v)xx + (ξu + (1 − ξ)v)p−1 dξ δ(v, u) 0 uxx + vxx 1 up − v p =− − . 2 p u−v

Thus, taking D = D = U n+1 − U n + ∆t

∂ ∂x , the conservative method for (3.7) reads n+1 + U n Uxxx 1 ∂ (U n+1 )p − (U n )p xxx = 0. + 2 p ∂x U n+1 − U n

3.1. Linearly implicit methods for polynomial Hamiltonians A major principle behind the conservative schemes is that they preserve the (discretized) ﬁrst integral exactly. This means that for an implicit scheme, the nonlinear equation to be solved in each time step must be solved exactly or to machine accuracy, and this may sometimes cause the overall method to be expensive compared to other approaches. A possible remedy would be to look for an iteration method having the property that the ﬁrst integral is preserved in every iteration; another possibility is to design schemes which are linearly implicit by construction, thereby requiring only one linear solve for every time step. In the second of these two approaches, discussed in Dahlby and Owren (2010), one may apply a polarization technique if the Hamiltonian is of polynomial type, meaning that the diﬀerential function G in (3.3) is a multivariate polynomial in the jet space coordinates u(n) . The idea is to substitute G[u] by a function of k ≥ 2 indeterminates G[w1 , . . . , wk ], satisfying G[u, . . . , u] = G[u] G[w1 , w2 , . . . , wk ] = G[w2 , . . . , wk , w1 ]

(consistency),

(3.17)

(cyclicity).

(3.18)

In particular, whenever G[u] = G((uαJ )) is a multivariate polynomial of degree p, one may construct G[w1 , . . . , wk ] satisfying (3.17)–(3.18), which is also multi-quadratic as long as k ≥ (p + 1)/2. We next replace the skewadjoint operator D[u] by some skew-adjoint approximation D[w1 , . . . , wk−1 ] of k − 1 arguments, satisfying D[u, . . . , u] = D[u] D[w1 , w2 , . . . , wk−1 ] = D[w2 , . . . , wk−1 , w1 ] Deﬁning

(consistency), (cyclicity).

(3.19) (3.20)

H[w1 , . . . , wk ] =

Ω

G[w1 , . . . , wk ],

we may now deﬁne a polarized version of the discrete variational derivative in a similar way to (3.12)–(3.13), but it is now a function of k +1 arguments.

26

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

We refer to Dahlby and Owren (2010) for the general deﬁnition and give only the particular expression for the AVF case: 1 δH δH = [ξwk+1 + (1 − ξ)w1 , w2 , . . . , wk ] dξ. (3.21) δ(w1 , . . . , wk+1 ) 0 δw1 δH , is deﬁned as Here the variational derivative on the right-hand side, δw 1 before, considering H as a function of its ﬁrst argument only, leaving the others ﬁxed. The following method was proposed in Dahlby and Owren (2010):

U n+k − U n δH = kD , k∆t δ(U n , . . . , U n+k )

n ≥ 0.

(3.22)

Remark 3.1. Note that the method (3.22) is a multistep method, and thus approximations U 1 , . . . , U k−1 to the ﬁrst time steps are required. Remark 3.2. The procedure is linear, in the sense that if the Hamiltonian can be split into a sum of two terms H1 + H2 , the method can be applied to each term separately. We summarize some important properties of the scheme (3.22), the proof of which can be found in Dahlby and Owren (2010). Theorem 3.3. • The scheme (3.22) is conservative in the sense that H[U n , . . . , U n+k−1 ] = H[U 0 , . . . , U k−1 ],

∀n ≥ 1,

for any polarized Hamiltonian function H satisfying (3.17)–(3.18). • The polarized AVF scheme (3.22) using (3.21) has formal order of consistency two for any polarized Hamiltonian satisfying (3.17)–(3.18), and skew-symmetric operator D satisfying (3.19)–(3.20). • If G[w1 , . . . , wk ] is multi-quadratic in all its k arguments, (wJα )j , j = 1, . . . , k, then the polarized AVF method (3.22) is linearly implicit. To illustrate the linearly implicit scheme just presented, consider (3.7) with p = 6, the mass critical generalized Korteweg–de Vries equation. We then have 1 1 G[u] = u2x − u6 . 2 6 As indicated in Remark 3.2, we can treat the two terms separately. For the second term, one needs to use k ≥ 3: choosing k = 3 leads to the unique multi-quadratic choice G2 [w, v, u] = G2 (w, v, u) = − 16 u2 v 2 w2 . In the ﬁrst term we have several options, but for reasons to be explained later, ¯ := D = ∂/∂x. we choose G1 [w, v, u] = 16 (u2x + vx2 + wx2 ). As before, we let D

27

Topics in structure-preserving discretization

This now leads to the scheme

n + U n+3 U n+3 − U n Uxxx ∂ U n + U n+3 n+1 2 n+2 2 xxx ) (U ) = 0. + + (U 3∆t 2 ∂x 2

Thus the scheme is linear in U n+3 , and preserves exactly the time-averaged Hamiltonian 1 n 2 n n+1 n+2 (Ux ) + (Uxn+1 )2 + (Uxn+2 )2 H[U , U ,U ]= 6 2 1 + U n U n+1 U n+2 dx. 6 We remark that some care should be taken when choosing the polarization. Since the resulting method is a multistep scheme, it has spurious solutions which might be unstable. Considering as a test case the term H[u] = u2x , letting D be a constant operator having the exponentials as eigenfunctions, such as in diﬀerentiation operators with constant coeﬃcients, one may choose the one-parameter family of polarizations 2 ux + vx2 1 θ H[v, u] = + (1 − θ)ux vx dx. 2 2 Dahlby and Owren (2010) remark that this leads to a stable scheme only if θ ≥ 12 . 3.2. Preserving multiple ﬁrst integrals The preservation of more than one ﬁrst integral may often be of interest. For instance, in PDEs which have soliton solutions, a well-known technique for proving stability of the exact solution is based on using the preservation of two speciﬁc invariants: see the articles by Benjamin and co-authors (Benjamin 1972, Benjamin, Bona and Mahony 1972). A procedure for preserving multiple ﬁrst integrals by a numerical integrator for ODEs was described in McLachlan et al. (1999). Their method was based on ﬁnding a presentation of the ODE vector ﬁeld f (y) by means of a completely skewsymmetric rank q + 1 tensor ﬁeld S, writing y˙ = f (y) = S(·, ∇H1 , . . . , ∇Hq ),

(3.23)

where H1 , . . . , Hq are q known ﬁrst integrals. This form eﬀectively encodes the preservation of all the q ﬁrst integrals, since, for 1 ≤ i ≤ q, d Hi (y) = ∇Hi (y) · f (y) = S(y)(∇Hi (y), ∇H1 (y), . . . , ∇Hq (y)) = 0. dt Proof of the existence of the formulation (3.23), along with constructive examples, is provided in McLachlan et al. (1999). The discrete gradient method is now easily implemented. Given a discrete gradient, ∇Hi (y n , y n+1 )

28

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

for each of the ﬁrst integrals, one may, for instance, deﬁne the method via y n+1 − y n ¯ n , y n+1 )( · , ∇H1 , . . . , ∇Hq ), = S(y ∆t for a suitable approximation S¯ to S. Another method for preserving multiple ﬁrst integrals was presented in Minesaki and Nakamura (2006) for a subclass of Hamiltonian ODEs called St¨ackel systems. In a recent paper by Dahlby, Owren and Yaguchi (2010), the problem of retaining several ﬁrst integrals simultaneously has been considered. Again we explain this approach for ODEs, or in the case when our PDE has already been discretized in space to yield a system y˙ = f (y), y ∈ Rm , with ﬁrst integrals H1 , . . . , Hq . Using the Euclidean structure on Rm one may consider the immersed submanifold Mc ⊂ Rm , a leaf of the foliation induced by the integrals M = Mc = {y ∈ Rm : H1 (y) = c1 , H2 (y) = c2 , . . . , Hq (y) = cq }. The tangent space to this manifold may be characterized as the orthogonal complement to span(∇H1 , . . . , ∇Hq ). In order to adapt this setting to the situation with discrete tangents, the discrete tangent space was deﬁned relative to two points, suﬃciently close to each other in M × M . For all y in a neighbourhood of a point p ∈ Mc , let T(p,y) M = {η ∈ Rm : ∇Hj (p, y), η = 0, 1 ≤ j ≤ q} for the Euclidean inner product ·, · on Rm . A vector η = η(p,y) ∈ T(p,y) M is called a discrete tangent vector. Note that this deﬁnition causes T(y,y) M = Ty M . The following observation is immediate. Any integrator satisfying y n+1 − y n ∈ T(yn+1 ,yn ) M

(3.24)

preserves all integrals, in the sense that Hi (y n+1 ) = Hi (y n ), 1 ≤ i ≤ q. The obvious approach to adopt in order to obtain a method satisfying (3.24) is to combine a non-conservative scheme with projection as follows. For simplicity, let φ∆t denote a one-step method of order p, φ∆t : Rm → Rm . For any y ∈ M , let y also denote its injection into Rm . Then consider un+1 = φ∆t (yn ),

yn+1 = yn + P(yn+1 , yn )(un+1 − yn ),

(3.25)

where P(yn , yn+1 ) is a smooth projection operator onto the discrete tangent space T(yn ,yn+1 ) M . An alternative scheme is yn+1 = yn + ∆tP(yn , yn+1 )ψ∆t (yn+1 , yn ), where ψ∆t is an integrator which can be written in the form yn+1 = yn + ∆t ψ∆t (yn+1 , yn ).

(3.26)

Topics in structure-preserving discretization

29

This method is itself assumed to be of order p, that is, y(t + ∆t) − y(t) − h ψ∆t (y(t + ∆t), y(t)) = O(∆tp+1 ).

(3.27)

The following result was proved in Dahlby et al. (2010). Theorem 3.4. If the non-conservative methods φ∆t , ψ∆t are of order p, then so are the schemes (3.25) and (3.26): y(t + h) − y(t) − P(y(t), y(t + ∆t))(un+1 − y(t)) = O(∆tp+1 ),

(3.28)

where un+1 = φ∆ t(y(t)), and y(t + h) − y(t) − hP(y(t + h), y(t))ψh (y(t + h), y(t)) = O(hp+1 ).

(3.29)

An alternative approach to projection is to derive methods based on a local coordinate representation of the manifold. Based on the assumption that the q ﬁrst integrals are all independent, we can deﬁne a coordinate chart centred at a point p ∈ M as follows. Suppose y ∈ M is suﬃciently close to p such that all discrete gradients ∇H1 (p, y), . . . , ∇Hq (p, y) form a linearly independent set of vectors in Rm . Let Tp (y) be a smooth map from the manifold M into the set of orthogonal m × m matrices T : M → O(m) such that its last q columns form a basis for the linear span of the discrete gradients. We now deﬁne the coordinate map implicitly to be η m−q χp : R → M : χp (η) = y : y − p = Tp (y) . (3.30) 0q That these maps induce an atlas on M is proved in Dahlby et al. (2010). One may now express the original diﬀerential equation in terms of η: η˙ = −T (χp ◦ η)DTp (f (χp ◦ η))η + T (χp ◦ η)f (χp ◦ η). (3.31) χp ◦η

The method proposed takes one step as follows. (1) Let η0 = 0. (2) Take a step with any pth-order method applied to the ODE (3.31), using p = y n in (3.30). The result is η1 . (3) Compute y n+1 = χ(η1 ). The method deﬁned in this way will clearly be of order at least p. Using this approach, one clearly needs to compute DTp because it is needed explicitly in (3.31), and this map may also be useful in computing the implicitly deﬁned coordinate map. It is shown in Dahlby et al. (2010) how DTp |y (v)η can be computed with a complexity of order O(mq 2 + q 3 ). 3.3. Discretizing in space The adaptation of the approaches just presented to PDEs which have already been discretized in space is of course straightforward: the integral in

30

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren Continuous calculus

Discrete calculus

First integral H[u] = G[u]

-

Discrete ﬁrst integral Hd [u] = (Gd [u])k ∆x k

?

? Discrete variational δHd derivative δ(v, u)

Variational derivative δH δu

?

?

PDE ut = D[u]

-

δH δu

Finite diﬀerence scheme δHd u − un ¯ =D ∆t (δun , δun+1 ) n+1

Figure 3.1. The Furihata formalism. This illustration is taken from Furihata (1999).

(3.3) may be replaced by a quadrature rule, resulting in a discrete version, for instance, Hd : Rm → R. The skew-adjoint operator D must be replaced by a skew-symmetric m × m matrix, Dd . We now consider ﬁnite diﬀerence approximations. The function space to which the solution u belongs is replaced by a ﬁnite-dimensional space with functions on a grid indexed by Ig ⊂ Zd . We use boldface symbols for these functions. Let there be Nr grid points in the space direction r so that N = N1 · · · Nd is the total number of grid points. We denote by uα the approximation to uα on such a grid, and by u the vector consisting of (u1 , . . . , um ). We will replace each derivative uαJ by a ﬁnite diﬀerence approximation δJ uα , and replace the integral by a quadrature rule. We then let bi (Gd ((δJ u)))i ∆x. (3.32) Hd (u) = i∈Ig

Here ∆x is the volume (length, area) of a grid cell and b = (bi )i∈Ig are the

Topics in structure-preserving discretization

31

weights in the quadrature rule. The discretized Gd has the same number of arguments as G, and each input argument as well as the output are vectors in RN . We have here approximated the function uαJ by a diﬀerence approximation δJ uα , where δJ : RN → RN is a linear map. As in the continuous case, we use square brackets, say F [u], as shorthand for a list of arguments involving diﬀerence operators F [u] = F (u, δJ1 u, . . . , δJq u). We compute 1 ∂Gd bi [ξu + (1 − ξ)v]dξ(δJ (uα − vα )) ∆x Hd [u] − Hd [v] = ∂δJ uα i 0 J,α i∈Ig δHd = ,u − v , (3.33) δ(v, u) where

δHd = δJ B δ(v, u) J,α

1 0

∂Gd α α [ξu + (1 − ξ)v ] dξ , ∂uJ

B is the diagonal linear map B = diag(bi ), i ∈ Ig , and the discrete inner product used in (3.33) is u, v = uαi viα . α,i∈Ig

Notice the resemblance between the operator acting on Gd in (3.33) and the continuous Euler operator in (3.6). We make the following assumptions. (1) The spatially continuous method (3.14) (using (3.15)) is discretized in space, using a skew-symmetric Dd and a selected set of diﬀerence quotients δJ for each derivative ∂J . (2) Considering (3.5) and (3.6), the choice of discretization operators δJ used in ∂G/∂uαJ [u] is arbitrary, but the corresponding DJ is replaced by the transpose δJ . In this case, using the same Dd , an identical set of diﬀerence operators in discretizing H (3.32), and choosing all the quadrature weights bi = 1, the resulting scheme is the same. Letting er denote the rth canonical unit vector in Rd , we deﬁne the most common ﬁrst-order diﬀerence operators ui+er − ui , (δr+ u)i = ∆xr ui − ui−er , (δr− u)i = ∆xr ui+er − ui−er . (δr1 u)i = 2∆xr

32

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

These diﬀerence operators are all commuting, but only the last one is skewsymmetric. However, for the ﬁrst two we have the useful identities (δr+ ) = −δr− ,

(δr− ) = −δr+ .

Higher-order diﬀerence operators δJ can generally be deﬁned by taking compositions of these operators: in particular, we shall consider examples in the next section using the second and third derivative approximations δr2 = δr+ ◦ δr− ,

δ 3 = δ 1 ◦ δ 2 .

We may now introduce numerical approximations Un representing the fully discretized system: the scheme is δHd Un+1 − Un = Dd . n ∆t δ(U , Un+1 ) The conservative schemes based on polarization are adapted in a straightforward manner, introducing a function Hd [w1 , . . . , wk ] which is consistent and cyclic, and a skew-symmetric map Dd depending on at most k − 1 arguments. The scheme is then Un+k − Un δHd = kDd . k∆t δ(Un , . . . , Un+k )

(3.34)

This scheme conserves the function Hd in the sense that Hd [Un+1 , . . . , Un+k ] = Hd [U0 , . . . , Uk−1 ],

n ≥ 0.

4. Spatial symmetries, high-order discretizations and fast group-theoretic algorithms In this chapter we survey some group-theoretic techniques applied to the discretization and solution of PDEs. Inspired by recent active research in Lie group and exponential time integrators for diﬀerential equations, we will in Section 4.1 present algorithms for computing matrix and operator exponentials based on Fourier transforms on ﬁnite groups. As a ﬁnal example, we consider spherically symmetric PDEs, where the discretization preserves the 120 symmetries of the icosahedral group. This motivates the study of spectral element discretizations based on triangular subdivisions. In Section 4.2 we introduce novel applications of multivariate non-separable Chebyshev polynomials in the construction of spectral element bases on triangular and simplicial subdomains. These generalized Chebyshev polynomials are intimately connected to kaleidoscopes of mirrors acting on a vector space, i.e., groups of isometries generated by mirrors placed on each face of a triangle (2D), a tetrahedron (3D) or in general simplexes. Mathematically this is described by the theory of root systems and Weyl groups,

Topics in structure-preserving discretization

33

which we will review. Group-theoretic techniques lead to FFT-based fast computational algorithms. This is well known in the case of commutative groups, but it turns out to be important to understanding such ideas in the setting of non-commutative groups as well. Lie group integrators enjoy a number of nice geometrical properties, of which we will here focus on their symmetry and equivariance properties. Fundamental to the theory of diﬀerential equations is the equivariance of the solution curves with respect to any diﬀeomorphism φ : M → M acting on the domain. Let φ∗ F denote the pushforward of the vector ﬁeld F , i.e., (φ∗ F )(z) = T φ · F (φ−1 (z)) for all z ∈ M, where T φ denotes the tangent map (in coordinates, the Jacobian matrix). Then the two diﬀerential equations y (t) = F (y(t)), y(0) = y0 and z (t) = (φ∗ F )(z(t)), z(0) = φ(y0 ) have analytical solution curves related by z(t) = φ(y(t)). In particular, if φ∗ F = F , we say that φ is a symmetry of the vector ﬁeld, and in that case φ maps solution curves to other solution curves of the same equation. For numerical integrators it is in general impossible to satisfy equivariance with respect to arbitrary diﬀeomorphisms, since this would imply an analytically correct solution. (There always exists a local diﬀeomorphism that straightens the ﬂow to a constant ﬂow in the x1 direction, and this is integrated exactly by any reasonable numerical method.) The equivariance group of a numerical scheme is the largest group of diffeomorphisms under which the numerical solutions transform equivariantly. It is known that the equivariance group of classical Runge–Kutta methods is the group of all aﬃne linear transformations of Rn . Lie group integrators based on exact computation of exponentials have equivariance groups that include the Lie group G on which the method is based; hence, if some elements g ∈ G are symmetries of the diﬀerential equation, then G-equivariant Lie group integrators will exactly preserve these symmetries. However, if the exact exponential is replaced with approximations, care must be taken not to destroy G-equivariance and symmetry preservation of the numerical scheme. In the case of PDEs, symmetry preservation also depends on symmetry-preserving spatial discretizations. Equivariance is the foundation of Fourier analysis. Classical Fourier analysis can be deﬁned as the study of linear operators L which are equivariant with respect to an (abelian) group of translations acting on a domain, i.e., L◦τ = τ ◦L for all translations τ (t) : t → t + τ . The fact that exponential functions are the eigenfunctions of translation operators, exp(2πiλ(t + τ )) = exp(2πiλτ ) exp(2πiλt), and hence also eigenfunctions for L, is the explanation for the omnipresence of Fourier bases in computational science. The more specialized cos and sin bases appear naturally when boundary conditions are taken into account, as symmetrization and skewsymmetrization of the exponentials with respect to the reﬂection t → −t. Chebyshev polynomials again occur naturally through a change of variables

34

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

cos(t) → y. The availability of fast computational algorithms (FFT-based) for all these bases again relies in a crucial manner upon the underlying group structures provided by the equivariance. Products of equivariant operators lead naturally to the convolution product, which is diagonalized by Fourier transforms. Many important operators are equivariant with respect to larger noncommutative groups of transformations, such as the important Laplacian ∇2 which is equivariant with respect to every isometry of the domain.4 The generalized Fourier theory of linear operators equivariant with respect to non-commutative groups of transformations is not as widely known as the abelian theory within computational science. Convolution products also occur naturally in this setting. However, due to non-commutativity, exponential bases are no longer suﬃcient to diagonalize convolutions. Instead one must also use the higher-dimensional representations of the group, which can be understood as homomorphisms of the group into a space of (unitary) matrices. Exponentials are one-dimensional representations of abelian groups. In the natural bases obtained from representation theory, equivariant operators are block-diagonalized. This is a starting point for constructing fast algorithms for matrix exponentials, eigenvalue computations and linear equations in cases where the operator is equivariant. We start our presentation with a discussion of matrices equivariant with respect to a ﬁnite non-abelian group acting upon the index set. This is a good example for understanding some basic concepts of non-commutative harmonic analysis, without having to consider more technical analysis questions which occur in the case of inﬁnite groups. 4.1. Introduction to discrete symmetries and equivariance The topic of this subsection is the application of Fourier analysis on groups to the computation of matrix exponentials. Assuming that the domain is discretized with a symmetry-respecting discretization, we will show that by a change of basis derived from the irreducible representations of the group, the operator is block-diagonalized. This simpliﬁes the computation of matrix exponentials. The basic mathematics here is the representation theory of ﬁnite groups (James and Liebeck 2001, Lomont 1959, Serre 1977). Applications of this theory in scientiﬁc computing have been discussed by a number of authors: see, e.g., Allgower, B¨ohmer, Georg and Miranda (1992), Allgower, Georg, Miranda and Tausch (1998), Bossavit (1986), Douglas and Mandel (1992) and Georg and Miranda (1992). Our exposition, based on the group algebra, is explained in detail in ˚ Ahlander and Munthe-Kaas (2005). 4

The Laplacian can indeed be deﬁned as the unique (up to constant) second-order linear diﬀerential operator which commutes with all isometries.

Topics in structure-preserving discretization

35

G-equivariant matrices. A group is a set G with a binary operation g, h → gh, inverse g → g −1 and identity element e, such that g(ht) = (gh)t, eg = ge = g and gg −1 = g −1 g = e for all g, h, t ∈ G. We let |G| denote the number of elements in the group. Let I denote the set of indexes used to enumerate the nodes in the discretization of a computational domain. We say that a group G acts on a set I (from the right) if there exists a product (i, g) → ig : I × G → I such that ie = i for all i ∈ I, i(gh) = (ig)h for all g, h ∈ G and i ∈ I.

(4.1) (4.2)

The map i → ig is a permutation of the set I, with the inverse permutation being i → ig −1 . An action partitions I into disjoint orbits Oi = {j ∈ I | j = ig for some g ∈ G},

i ∈ I.

We let S ⊂ I denote a selection of orbit representatives, i.e., one element from each orbit. The action is called transitive if I consists of just a single orbit, |S| = 1. For any i ∈ I we let the isotropy subgroup at i, Gi be deﬁned as Gi = {g ∈ G | ig = i}. The action is free if Gi = {e} for every i ∈ I, i.e., there are no ﬁxed points under the action of G. Definition 4.1.

A matrix A ∈ CI×I , is G-equivariant if

Ai,j = Aig,jg

for all i, j ∈ I and all g ∈ G.

(4.3)

The deﬁnition is motivated by the result that if L is a linear diﬀerential operator commuting with a group of domain symmetries G, and if we can ﬁnd a set of discretization nodes I such that every g ∈ G acts on I as a permutation i → ig, then L can be discretized as a G-equivariant matrix A: see Allgower et al. (1998) and Bossavit (1986). The group algebra. We will establish that G-equivariant matrices are associated with (scalar or block) convolutional operators in the group algebra. Definition 4.2. The group algebra CG is the complex vector space CG where each g ∈ G corresponds to a basis vector g ∈ CG. A vector a ∈ CG can be written as a= a(g)g where a(g) ∈ C. g∈G

The convolution product ∗ : CG × CG → CG is induced from the product in G as follows. For basis vectors g, h, we set g ∗ h ≡ gh, and in general,

36

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

a(g)g and b = h∈G b(h)h, then a(g)g ∗ b(h)h = a(g)b(h)(gh) = (a ∗ b)(g)g, a∗b=

if a =

g∈G

g∈G

where

h∈G

(a ∗ b)(g) =

g∈G

g,h∈G

a(gh−1 )b(h) =

h∈G

a(h)b(h−1 g).

(4.4)

h∈G

Consider a G-equivariant A ∈ Cn×n in the case where G acts freely and transitively on I. In this case there is only one orbit of size |G| and hence I may be identiﬁed with G. Corresponding to A there is a unique A ∈ CG, given as A = g∈G A(g)g, where A is the ﬁrst column of A, i.e., A(gh−1 ) = Agh−1 ,e = Ag,h .

Cn

(4.5)

Similarly, any vector x ∈ corresponds uniquely to x = g∈G x(g)g ∈ CG, where x(g) = xg for all g ∈ G. Consider the matrix vector product Ag,h xh = A(gh−1 )x(h) = (A ∗ x)(g). (Ax)g = h∈G

h∈G

If A and B are two equivariant matrices, then AB is the equivariant matrix, where the ﬁrst column is given as (AB)g,e = Ag,h B h,e = A(gh−1 )B(h) = (A ∗ B)(g). h∈G

h∈G

We have shown that if G acts freely and transitively, then the algebra of G-equivariant matrices acting on Cn is isomorphic to the group algebra CG acting on itself by convolutions from the left. In the case where A is G-equivariant with respect to a free, but not transitive, action of G on I, we need a block version of the above theory. Let Cm× G ≡ Cm× ⊗CG denote the space of vectors consisting of |G| matrix blocks, each block of size m × . Thus A ∈ Cm× G can be written as A= A(g) ⊗ g where A(g) ∈ Cm× . (4.6) g∈G

The convolution product (4.4) generalizes to a block convolution ∗ : Cm× G × C×k G → Cm×k G given as A(g) ⊗ g ∗ B(h) ⊗ h = (A ∗ B)(g) ⊗ g, A∗B = g∈G

where

(A ∗ B)(g) =

A(gh−1 )B(h) =

h∈G

and

A(h)B(h−1 g)

g∈G

h∈G

denotes a matrix product.

h∈G

A(h)B(h−1 g),

(4.7)

Topics in structure-preserving discretization

37

If the action of G on I is free, but not transitive, then I splits into m orbits, each of size |G|. We let S denote a selection of one representative from each orbit. We will establish an isomorphism between the algebra of G-equivariant matrices acting on Cn and the block-convolution algebra Cm×m G acting on Cm G. We deﬁne the mappings µ : Cn → Cm G, ν : Cn×n → Cm×m G by µ(y)i (g) = yi (g) = y ig ∀i ∈ S, g ∈ G, ν(A)i,j (g) = Ai,j (g) = Aig,j ∀i, j ∈ S g ∈ G.

(4.8) (4.9)

In ˚ Ahlander and Munthe-Kaas (2005) we show the following. Proposition 4.3. Let G act freely on I. Then µ is invertible and ν is invertible on the subspace of G-equivariant matrices. Furthermore, if A, B ∈ Cn×n are G-equivariant, and y ∈ Cn , then µ(Ay) = ν(A) ∗ µ(y), ν(AB) = ν(A) ∗ ν(B).

(4.10) (4.11)

To complete the connection between G-equivariance and block convolutions, we need to address the general case where the action is not free, and hence some of the orbits in I have reduced size. One way to treat this case is to duplicate the nodes with non-trivial isotropy subgroups; thus a point j ∈ I is considered to be |Gj | identical points, and the action is extended to a free action on this extended space. Equivariant matrices on the original space is extended by duplicating the matrix entries, and scaled according to the size of the isotropy. We deﬁne µ(x)i (g) = xi (g) = xig ∀i ∈ S, g ∈ G, 1 Aig,j ∀i, j ∈ S g ∈ G. ν(A)i,j (g) = Ai,j (g) = |Gj |

(4.12) (4.13)

With these deﬁnitions it can be shown that (4.10)–(4.11) still hold. It should be noted that µ and ν are no longer invertible, and the extended block convolutional operator ν(A) becomes singular. This poses no problems for the computation of exponentials since this is a forward computation. Thus we just exponentiate the block convolutional operator and restrict the result back to the original space. However, for inverse computations such as solving linear systems, the characterization of the image of µ and ν as subspaces of Cm G and Cm×m G is an important issue for ﬁnding the correct solution (˚ Ahlander and Munthe-Kaas 2005, Allgower, Georg and Miranda 1993). The generalized Fourier transform (GFT). So far we have argued that a symmetric diﬀerential operator becomes a G-equivariant matrix under discretization, which again can be represented as a block convolutional operator. In this subsection we will show how convolutional operators are

38

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

block-diagonalized by a Fourier transform on G. This is the central part of Frobenius’s theory of group representations from 1897–1899. We recommend the monographs by F¨ assler and Stiefel (1992), James and Liebeck (2001), Lomont (1959) and Serre (1977) as introductions to representation theory with applications. Definition 4.4. Cd×d such that

A d-dimensional group representation is a map R : G → R(gh) = R(g)R(h)

for all g, h ∈ G.

(4.14)

Generalizing the deﬁnition of Fourier coeﬃcients we deﬁne for any A ∈ ˆ ∈ Cm×k ⊗ Cm×k G and any d-dimensional representation R a matrix A(R) d×d C as ˆ A(R) = A(g) ⊗ R(g). (4.15) g∈G

Proposition 4.5. (Convolution theorem) For any A ∈ Cm×k G, B ∈ Ck× G and any representation R we have ˆ ˆ (A ∗ B)(R) = A(R) B(R).

(4.16)

Proof.

The statement follows from ˆ ˆ A(R)B(R) = A(g) ⊗ R(g) B(h) ⊗ R(h) g∈G

=

h∈G

A(g)B(h) ⊗ R(g)R(h) =

g,h∈G

=

A(g)B(h) ⊗ R(gh)

g,h∈G

A(gh−1 )B(h) ⊗ R(g) = (A ∗ B)(R).

g,h∈G

Let dR denote the dimension of the representation. For use in practical computations, it is important that A ∗ B can be recovered by knowing (A ∗ B)(R) for a suitable selection of representations, and furthermore that their dimensions dR are as small as possible. Note that if R is a represen˜ tation and X ∈ CdR ×dR is non-singular, then also R(g) = XR(g)X −1 is ˜ a representation. We say that R and R are equivalent representations. If ˜ ˜ there exists a similarity transform R(g) = XR(g)X −1 such that R(g) has a block diagonal structure, independent of g ∈ G, then R is called reducible, otherwise it is irreducible. Theorem 4.6. (Frobenius) For any ﬁnite group G there exists a complete list R of non-equivalent irreducible representations such that d2R = |G|. R∈R

39

Topics in structure-preserving discretization

Deﬁning the GFT for a ∈ CG as a(g)R(g) a ˆ(R) =

for every R ∈ R,

(4.17)

g∈G

we may recover a by the inverse GFT (IGFT): 1 a(g) = dR trace (R(g −1 )ˆ a(R)). |G|

(4.18)

R∈R

For the block transform of A ∈ Cm×k G given in (4.15), the GFT and the IGFT are given componentwise as Ai,j (g)R(g) ∈ CdR ×dR , (4.19) Aˆi,j (R) = g∈G

Ai,j (g) =

1 dR trace (R(g −1 )Aˆi,j (R)). |G|

(4.20)

R∈R

Complete lists of irreducible representations for several common groups are to be found in Lomont (1959). Applications to the matrix exponential. We have seen that via the GFT, any G-equivariant matrix is block-diagonalized. Corresponding to an irreducible ˆ representation R, we obtain a matrix block A(R) of size mdR × mdR , where m is the number of orbits in I and dR is the size of the representation. Let Wdirect denote the computational work, in terms of ﬂoating-point operations, for computing the matrix exponential on the original data A, and let Wfspace be the cost of doing the same algorithm on the corresponding block diagonal ˆ Thus GFT-based data A. 3 3 3 d2R , Wfspace = cm3 d3R Wdirect = c(m|G|) = cm R∈R

and the ratio becomes O(n ) : 3

Wdirect /Wfspace =

R∈R

R∈R

d2R

3 / d3R . R∈R

Table 4.1 lists this factor for the symmetries of the triangle, the tetrahedron, the 3D cube and the maximally symmetric discretization of a 3D sphere (icosahedral symmetry with reﬂections). The cost of computing the GFT is not taken into account inthis estimate. There exist fast GFT algorithms of complexity O |G| log (|G|) for a number 2 | , the of groups, but even if we use a slow transform of complexity O |G total cost of the GFT becomes just O m2 |G|2 , which is much less than Wfspace .

40

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

Table 4.1. Gain in computational complexity for matrix exponential via GFT. Domain Triangle Tetrahedron Cube Icosahedron

G

|G|

{dR }R∈R

D3 S4 S4 × C 2 A5 × C 2

6 24 48 120

{1, 1, 2} {1, 1, 2, 3, 3} {1, 1, 1, 1, 2, 2, 3, 3, 3, 3} {1, 1, 3, 3, 3, 3, 4, 4, 5, 5}

Wdirect /Wfspace 21.6 216 864 3541

Example 4.7. (Equilateral triangle) The non-commutative group of smallest order is D3 , the symmetries of an equilateral triangle. There are six linear transformations that map the triangle onto itself: three pure rotations and three rotations combined with reﬂections. In Figure 4.1(a) we indicate the two generators α (rotation 120◦ clockwise) and β (right–left reﬂection). These satisfy the algebraic relations α3 = β 2 = e, βαβ = α−1 , where e denotes the identity transform. The whole group is D3 = {e, α, α2 , β, αβ, α2 β}. Given an elliptic operator L on the triangle such that L(u◦α) = L(u)◦α and L(u◦β) = L(u)◦β for any u satisfying the appropriate boundary conditions on the triangle, let the domain be discretized with a symmetryrespecting discretization: see Figure 4.1(b). In this example we consider a ﬁnite diﬀerence discretization represented by the nodes I = {1, 2, . . . , 10}, such that both α and β map nodes to nodes. In ﬁnite element discretizations one would use basis functions mapped to other basis functions by the symmetries. We deﬁne the action of D3 on I as (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)α = (5, 6, 1, 2, 3, 4, 9, 7, 8, 10), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)β = (2, 1, 6, 5, 4, 3, 7, 9, 8, 10), and extend to all of D3 using (4.2). As orbit representatives, we may pick S = {1, 7, 10}. The action of the symmetry group is free on the orbit O1 = {1, 2, 3, 4, 5, 6}, while the points in the orbit O7 = {7, 8, 9} have isotropy subgroups of size 2, and ﬁnally O10 = {10} has isotropy of size 6. The operator L is discretized as a matrix A ∈ C10×10 satisfying the equivariances Aig,jg = Ai,j for g ∈ {α, β} and i, j ∈ S. Thus we have, e.g., A1,6 = A3,2 = A5,4 = A4,5 = A2,3 = A6,1 . D3 has three irreducible representations given in Table 4.2 (extended to the whole group using (4.14)). To compute exp(A), we ﬁnd A = ν(A) ∈ C3×3 G from (4.13) and ﬁnd Aˆ = GFT(A) from (4.19). The transformed maˆ 1 ) ∈ Cm×m and A(ρ ˆ 2 ) ∈ Cm×m ⊗C2×2 ˆ 0 ), A(ρ trix Aˆ has three blocks, A(ρ 2m×2m C , where m = 3 is the number of orbits. We exponentiate each of these blocks, and ﬁnd the components of exp(A) using the inverse GFT (4.20).

41

Topics in structure-preserving discretization Table 4.2. A complete list of irreducible representations for D3 .

ρ0 ρ1 ρ2

α

β

1 1 √ −1/2 − 3/2 √ 3/2 −1/2

1 −1 1 0 0 −1 7

4

5

α 3

6

10

β 9

(a)

2

1

8

(b)

Figure 4.1. Equilateral triangle with a symmetry-preserving set of 10 nodes.

We should remark that in Lie group integrators, it is usually more important to compute y = exp(A) · x for some vector x. In this case, we ˆ i ))·ˆ x(ρi ), and recover y by inverse GFT. Note that compute yˆ(ρi ) = exp(A(ρ x ˆ(ρ2 ), yˆ(ρ2 ) ∈ Cm ⊗C2×2 C2m×2 . Example 4.8. (Icosahedral symmetry) As a second example illustrating the general theory, we solve the simple heat equation u t = ∇2 u on the surface of a unit sphere. See Trønnes (2005) for details of the implementation. The sphere is divided into 20 equilateral triangles, and each triangle is subdivided into a ﬁnite diﬀerence mesh respecting all the 120 symmetries of the full icosahedral symmetry group (including reﬂections). To understand this group, it is useful to realize that ﬁve tetrahedra can be simultaneously embedded in the icosahedron, so that the 20 triangles correspond to the (in total) 20 corners of these ﬁve tetrahedra. From this one sees that the

42

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

icosahedral rotation group is isomorphic to A5 , the group of all 60 even permutations of the ﬁve tetrahedra. The 3D reﬂection matrix −I obviously commutes with any 3D rotation, and hence we realize that the full icosahedral group is isomorphic to the direct product C2 × A5 , where C2 = {1, −1}. The irreducible representations of A5 have dimensions {1, 3, 3, 4, 5}, and the representations of the full icosahedral group are found by taking tensor products of these with the two one-dimensional representations of C2 . The fact that the full icosahedral group is a direct product is also utilized in faster computation of the GFT. This, however, is not of major importance, since the cost of the GFT in any case is much less than the cost of the matrix exponential. 4.2. Spatial symmetries and high-order discretizations In this subsection we will discuss novel techniques for spatial discretizations based on reﬂection groups and multivariate Chebyshev polynomials. Given high-order time integration methods, such as Lie group methods, it is desirable to also use high-order discretizations in space. We will see that it is important to employ discretizations respecting spatial symmetries, both for the quality of the discretization error and also for eﬃciency of linear algebra computations such as matrix exponentials, eigenvalue computations and solution of linear systems. Spectral methods are important discretization techniques, where special function systems are used to expand the solution on simple (e.g., boxshaped) domains. For PDEs with analytic solutions spectral discretization techniques enjoy exponentially fast convergence (Canuto, Hussaini, Quarteroni and Zang 2006). On more general domains spectral element methods are constructed by patching together simple subdomains. Typically the basis functions on the subdomains arise as eigenfunctions for a self-adjoint linear operator, such as the exponential Fourier basis on periodic domains and orthogonal polynomials (Legendre or Chebyshev) on bounded domains with homogeneous boundary conditions. Most of the popular orthogonal polynomial systems used in current spectral (and spectral-element) methods are special cases of Jacobi polynomials, which are the solutions of singular Sturm–Liouville problems on a bounded domain, i.e., they are eigenfunctions of a second-order linear diﬀerential operator which is self-adjoint with respect to a weighted inner product, where the weight function is singular on the boundary. A problem with standard polynomial-based spectral element methods is a lack of ﬂexibility with respect to geometries. On box-shaped domains it is easy to construct polynomial bases by tensor products of univariate polynomials. Box-shaped domains are, however, diﬃcult to patch together in more general geometries, and diﬃcult to match with domain symmetries,

Topics in structure-preserving discretization

43

e.g., for problems on spherical surfaces. For these reasons, it is desirable to explore high-order function systems based on triangles (2D), tetrahedra (3D) and in general simplexes for higher dimensions. 4.2.1. Root systems and Laplacian eigenfunctions on triangles and simplexes For certain triangular or simplicial domains with homogeneous Dirichlet or Neumann boundary conditions, the eigenfunctions of the Laplacian are known explicitly. We will see later that these systems are closely related to reﬂection groups (crystallographic groups and Coxeter groups) and root systems. In order to motivate a discussion of reﬂection groups, we start with an important particular example, the eigenfunctions of the Laplacian on an equilateral triangle . What is the sound of an equilateral drum? A detailed discussion of this example is also found in Huybrechs, Iserles and Nørsett (2010). Without loss of generality, the triangle has corners in the origin [0; 0], √ √ we assume that √ √ λ1 = [1/ 2; 1/ 6] and λ2 = [0; 2/ 3], shown as the shaded domain in Figure 4.4(a), labelled A2 (see p. 50). Let {sj }3j=1 denote reﬂections of R2 ˜ denote the full group of isometries of about the edges of the triangle. Let W 2 3 R generated by {sj }j=1 . This is an example of a crystallographic group,5 a group of isometries of Rd where the subgroup of translations form a lattice in Rd . From this fact we will derive the Laplacian eigenfunctions on . Recall that a lattice L in Rd is the Z-linear span of d linearly independent vectors in Rd , L = spanZ {α1 , . . . , αd }; thus L is an abelian group isomorphic to Zd . The reciprocal lattice L∗ = {λ ∈ Rd : (λ, α) ∈ Z for all λ ∈ Λ and α ∈ L}, where (·, ·) is the standard inner product on Rd . We have L∗ = spanZ {λ1 , . . . , λd }, where {λ1 , . . . , λd } is the dual basis of {α1 , . . . , αd }, i.e., (λj , αk ) = δj,k . The reciprocal lattice serves as the index set for the Fourier basis for L-periodic functions on Rd . Consider the L-periodic domain (a d-torus) Td = Rd /L and the space of square-integrable periodic functions L2 (Td ). The Fourier basis for L2 (Td ) is the L2 -orthogonal family of exponential functions indexed by the dual lattice, {exp(2πi(λ, y))}λ∈L∗ . ˜ is the Back to our equilateral triangle, where the translation lattice of W lattice L = spanZ {α1 , α2 } < R2 √ √ √ √ generated by the vectors α1 = ( 2, 0) and α2 = (−1 2, 3/ 2). The unit 5

More speciﬁcally it is an aﬃne Weyl group, to be deﬁned below.

44

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

cell of L can be taken as either the rhombus spanned by α1 and α2 or the hexagon indicated in Figure 4.4. The hexagon is the Voronoi cell of the origin in the lattice L, i.e., its interior consists of the points in R2 that are closer to the origin than to any other lattice points. As a ﬁrst step in our construction, we consider the L-periodic eigenfunctions of the Laplacian ∇2 . Since (λj , αk ) = δj,k , the dual lattice is L∗ = spanZ {λ1 , λ2 } and the periodic eigenfunctions are ∇2t e2πi(λ,t) = −(2π)2 λ2 e2πi(λ,t)

for all λ ∈ L∗ , t ∈ R2 /L.

We continue to ﬁnd the Laplacian eigenfunctions on by folding the expo˜ be the linear subgroup which leaves the origin ﬁxed nentials. Let W < W (the symmetries of ): W = {e, s1 , s2 , s1 s2 , s2 s1 , s1 s2 s1 }, where e is the identity and si , i ∈ {1, 2} act on v ∈ R2 as si v = v − 2(αi , v)/(αi , αi ). We deﬁne even and odd (cosine- and sine-type) foldings of the exponentials 1 2πi(λ,wt) 1 2πi(w λ,t) e = e , (4.21) cλ (t) = |W | |W | w∈W

sλ (t) =

1 |W |

w∈W

w∈W

det(w)e2πi(λ,wt) =

1 det(w)e2πi(w λ,t) , |W |

(4.22)

w∈W

where, in our example, |W | = 6. The Laplacian commutes with any isometry, in particular ∇2 (f ◦ w) = (∇2 f ) ◦ w for w ∈ W . Hence, reﬂected exponentials and the functions cλ (t) and sλ (t) are also eigenfunctions with the same eigenvalue. Lemma 4.9. The eigenfunctions of ∇2 on the equilateral triangle with Dirichlet (f = 0) and Neumann boundary conditions (∇f · n = 0) are given by ∇2t cλ (t) = −(2π)2 λ2 cλ (t), ∇2t sλ (t) = −(2π)2 λ2 sλ (t), respectively, for λ ∈ spanN {λ1 , λ2 }, the set of all non-negative integer combinations of λ1 = [1/2; 1/6] and λ2 = [0; 2/3]. The set spanN {λ1 , λ2 } ⊂ L∗ contains exactly one point from each W -orbit: see the discussion of Weyl chambers below. An important question is how good (or bad!) these eigenfunctions are as bases for approximating analytic functions on . It is well known from the univariate case that a similar construction, yielding the eigenfunctions cos(kθ) and sin(kθ) on [0, π], does not give fast convergence. For an analytic function f (θ) where f (0) = f (π) = 0, the even 2π-periodic extension is piecewise smooth, with only C 0

Topics in structure-preserving discretization

45

continuity at θ ∈ {0, π}. Hence the Fourier cosine series converges only as O(k −2 ). There are several diﬀerent ways to achieve spectral convergence O(exp(−ck)). One possible solution is to approximate f (θ) in a frame (not linearly independent) consisting of both {cos(kθ)}k∈Z and {sin(kθ)}k∈Z+ . Another possibility is to employ a change of variables x = cos(θ), yielding (univariate) Chebyshev polynomials of the ﬁrst and second kind: Tk (x) = cos(kθ), Uk (x) = sin((k + 1)θ)/ sin(θ). These polynomials (in particular the ﬁrst kind) are ubiquitous in approximation theory and famous for their excellent approximation properties. We will develop the corresponding multivariate theory and see that we have similar possibilities for constructing spectrally converging frames and bases. Root systems and aﬃne Weyl groups. Let V be a ﬁnite-dimensional real Euclidean vector space with standard inner product (·, ·). The construction above can be generalized to all those simplexes ⊂ V with the property ˜ generated by reﬂecting about its faces is a that the group of isometries W crystallography group. All such simplexes are determined by a root system, a set of vectors in V which are perpendicular to the reﬂection planes of W passing through the origin. We review some basic deﬁnitions and results about root systems. For more details we refer to Bump (2004). Definition 4.10. A root system in V is a ﬁnite set Φ of non-zero vectors, called roots, that satisfy the following conditions. (1) The roots span V. (2) The only scalar multiples of a root α ∈ Φ that belong to Φ are ±α. (3) For every root α ∈ Φ, the set Φ is invariant under reﬂection in the hyperplane perpendicular to α, i.e., for any two roots α and β, the set Φ contains the reﬂection of β, sα (β) := β − 2

(α, β) α ∈ Φ. (α, α)

(4) (Crystallographic restriction.) For any α, β ∈ Φ, we have 2

(α, β) ∈ Z. (α, α)

Condition (4) implies that the obtuse angle between two diﬀerent reﬂection planes must be either 90◦ , 120◦ , 135◦ or 150◦ . This is a fundamental fact in crystallography, implying that rotational symmetries of a crystal must be either 2-fold, 3-fold, 4-fold or 6-fold. The rank of the root system is the dimension d of the space V. Any root system contains a subset (not uniquely deﬁned) Σ ⊂ Φ of so-called simple

46

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

An g

g

g

g

g

g> g

Cn g

g

g

g< g g

Dn g

g

g @g

E8 g

g

G2 g< g

Bn

g

g

g E6 g

g

g

g

g

g E7 g

g

g

g

g

g

g

g

g

g

g> g

g

g

F4

g

g

g

Figure 4.2. Dynkin diagrams for irreducible root systems.

positive roots. This is a set of d linearly independent roots Σ = {α1 , . . . , αd } such that any root β ∈ Φ can be written either as a linear combination of αj with non-negative integer coeﬃcients, or as a linear combination with nonpositive integer coeﬃcients. We call Σ a basis of the root system Φ. A root system is conveniently represented by its Dynkin diagram. This is a graph with d nodes corresponding to the simple positive roots. Between two nodes j and k, no line is drawn if the angle between αj and αk is 90◦ , a single line is drawn if it is 120◦ , a double line for 135◦ and a triple line for 150◦ . It is only necessary to understand the geometry of irreducible root systems when the Dynkin diagram is connected. Disconnected Dynkin diagrams (reducible root systems) are trivially understood in terms of products of irreducible root systems. For irreducible root systems, the roots are either all of the same length, or have just two diﬀerent lengths. In the latter case a marker < or > on an edge indicates the separation of long and short roots (short < long). Since the work of W. Killing and E. Cartan in the late nineteenth century it has been known that Dynkin diagrams of irreducible root systems must belong to one of four possible inﬁnite cases, An (n > 0), Bn (n > 1), Cn (n > 2), Dn (n > 3), or ﬁve special cases, E6 , E7 , E8 , F4 , G2 , shown in Figure 4.2. We say that two root systems are equivalent if they diﬀer only by a scaling or an isometry. Up to equivalence, there corresponds a unique root system to each Dynkin diagram. We will see that the root system and also the Weyl group can be explicitly computed (as real vectors and matrices) using from the Cartan matrix deﬁned below. A root system Φ is associated with a dual root system Φ∨ deﬁned such that a root α ∈ Φ corresponds to a co-root α∨ := 2α/(α, α) ∈ Φ∨ . If all roots have equal lengths then Φ∨ = Φ (up to equivalence), i.e., the root system is self-dual. For the cases with two root lengths we have Bn∨ = Cn , F4∨ = F4 and G∨ 2 = G2 . The Cartan matrix is deﬁned as Cj,k = (αj , αk∨ ) = 2(αj , αk )/(αk , αk ). Its diagonal is Cj,j = 2. Oﬀ-diagonal entries Cj,k are found from the number of edges between αj and αk and their

47

Topics in structure-preserving discretization

relative lengths:  0 if no edge,     − 1 if single edge, or multiple edge where α < α , j k Cj,k =  − 2 if double edge and αj > αk ,    − 3 if triple edge and αj > αk . The ratio of long and short roots is as follows. If αj > αk then αj = √ > = is a double edge and αj = √2αk in the cases where the division = 3αk when it is triple ≡ > ≡. It is convenient such √ to choose normalization ∨ that the longest roots have length αj = 2. This leads to αj = αj for the long roots. For short roots this implies αk∨ = 2αk in the double-line cases = > = and αk∨ = 3αk in the triple-edge case ≡ > ≡. Let D be the diagonal matrix Dk,k = αk . Then S = D−1 CD/2 is a symmetric matrix, and we compute bases for the root system Φ and the dual root system Φ∨ as the columns of the matrices6 Σ and Σ∨ given as Σ = RD, ∨

Σ = 2RD

(4.23) −1

,

(4.24)

where R is the Cholesky factorization of S, i.e., R is upper-triangular, with positive diagonal elements, such that R R = D−1 CD/2. A root system leads to the deﬁnition of Weyl groups and aﬃne Weyl groups. Let Φ be a d-dimensional root system with dual root system Φ∨ . For any root α ∈ Φ consider the reﬂection sα : V → V given by sα (t) = t −

2(t, α) α = t − (t, α∨ )α. (α, α)

For the dual roots α∨ ∈ Φ∨ we deﬁne translations τα∨ : V → V by τα∨ (t) = t + α∨ . The Weyl group of Φ is the ﬁnite group of isometries on V generated by the reﬂections sα for α ∈ Σ: W = {sα }α∈Σ . The dual root lattice L∨ is the lattice spanned by the translations τα∨ for α∨ ∈ Σ∨ . We identify this with the abelian group of translations on V generated by the dual roots L∨ = {τα∨ }α∨ ∈Σ∨ . ˜ is the inﬁnite crystallographic symmetry group The aﬃne Weyl group W of V generated by the reﬂections sα for α ∈ Σ and the translations τα∨ for 6

By abuse of notation we use Σ to denote both the basis for the root system and the matrix whose columns form the basis of the root system.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

α∨ ∈ Σ∨ , thus it is the semidirect product of the Weyl group W with the dual7 root lattice L∨ ˜ = {sα }α∈Σ , {τα∨ }α∨ ∈Σ∨ = W L∨ . W Let Λ denote the reciprocal lattice of L∨ , i.e., for all λ ∈ Λ and all α∨ ∈ Φ∨ we have (λ, α∨ ) ∈ Z. The lattice Λ is spanned by vectors {λj }dj=1 such that (λj , αk∨ ) = δj,k for all αk∨ ∈ Σ∨ . The vectors λj are called the fundamental dominant weights of the root system Φ and Λ is called the weights lattice. We use the weights λj and dual roots αk∨ to represent the Weyl group in matrix form as follows: wj,k := (λj , wαk∨ )

for all w ∈ W .

(4.25)

In particular, the reﬂections sαr for αr ∈ Σ are represented as (sαr )j,k = δj,k − Cr,k δr,j ⇒ sαr = I − er e r C, where C is the Cartan matrix and {er } is the standard basis on Rd . This shows that the Weyl group can be represented as integer matrices with respect to the basis Σ∨ and the dual basis {λ1 , . . . , λd } for V. The positive Weyl chamber C+ is deﬁned as the closed conic subset of V containing the points with non-negative coordinates with respect to the dual basis {λ1 , . . . , λd }, in other words C+ = {t ∈ V : (t, αj ) ≥ 0}. This is a fundamental domain for the Weyl group acting on V. The boundary of C+ consists of the hyperplanes perpendicular to {α1 , . . . , αd }. The aﬃne Weyl group contains reﬂection symmetries about aﬃne planes perpendicular to the roots, shifted a half-integer multiple of the length of a co-root away from the origin, i.e., for each α∨ ∈ Φ∨ and each k ∈ Z there is an aﬃne plane consisting of the points Pk,α∨ = {t ∈ V : 2(t, α∨ ) = k(α∨ , α∨ )} = {t ∈ V : (t, α) = k}, and this aﬃne plane is invariant under the aﬃne reﬂection τkα∨ · sα . A connected closed subset of V limited by such aﬃne planes is ˜. called an alcove, and is a fundamental domain for the aﬃne Weyl group W The situation is particularly simple for irreducible root systems, where the alcoves are always d-simplexes. Recall that any root α ∈ Φ can be expressed as α = dk=1 nk αk , where the nk = 2(α, λk )/(αk , αk ) are either all non-negative or all non-positive integers. A root α ˜ strictly dominates another root α, written α ˜ α, if n ˜ k ≥ nk for all k, with strict inequality for at least one k. Irreducible root systems have a unique dominant root 7

Since the Weyl groups of Φ and of Φ∨ are identical, it is no problem to instead deﬁne ˜ = W L as the semidirect product of the Weyl group with the primal root lattice. W We have, however, chosen here to follow the most common deﬁnition, which leads to a slightly simpler notation for the Fourier analysis.

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49

A × A root system 1 1 α 2

λ

2

λ1

α1

Figure 4.3. Fundamental domains of reducible root system A1 × A1 .

α ˜ ∈ Φ such that α ˜ α for all α = α ˜ . The dominant root α ˜ is the unique long root in the Weyl chamber (possibly on the boundary). Basic geometric properties of aﬃne Weyl groups are summarized by the following lemma. Lemma 4.11. (i) If Φ is irreducible with dominant root α ˜ then a fundamental domain ˜ is the simplex ⊂ V given by for W = {t ∈ V : (t, α) ˜ ≤ 1 and (t, αj ) ≥ 0 for all αj ∈ Σ}, where has corners in the origin and in the points λj /(λj , α ˜ ) for j = 1, . . . , d. (ii) The aﬃne Weyl group is generated by the aﬃne reﬂections8 about the boundary faces of the fundamental domain . For irreducible Φ these are ˜ = {sα }dj=1 , τα˜ · sα˜ . W j (iii) If Φ is reducible then a fundamental domain for the aﬃne Weyl group is given as the Cartesian product of the fundamental domains for each of its irreducible components. Example 4.12. The simplest rank d root system is the reducible system A1 × · · · × A1 , where the Dynkin diagram consists of d non-connected dots. Figure 4.3 shows A1 × A1 . The square outlined in black is the fundamental domain of the root lattice, and the small shaded square is the fundamental ˜ . Reducible root systems can be easily domain of the aﬃne Weyl group W understood as products of irreducible root systems. For instance, the multivariate Chebyshev polynomials corresponding to A1 ×· · ·×A1 are the tensor products of d univariate (classical) Chebyshev polynomials on a d-cube. 8

˜ is generated by reﬂections it is also a special case of a Coxeter group. Since W

50

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren α2 A2 root system α

2

λ

λ

2

1

2

λ2

α2

λ1

λ1 α1

α

α1

(a)

G root system

B2 root system

λ2

1

(b)

(c)

Figure 4.4. The 2D irreducible root systems.

Figure 4.5. Fundamental domains of A3 root system.

Example 4.13. The 2D and 3D irreducible root systems are A2 , B2 , G2 , A3 , B3 and C3 . The Cartan matrix, the matrix representation of generators for the Weyl group, and the size of the Weyl group are shown in Table 4.3. Figure 4.4 shows the 2D cases with roots α (large dots), dominant root α ˜ (larger dot), simple positive roots (α1 , α2 ), and their fundamental dominant weights√(λ1 , λ2 ). The roots are normalized such that the longest roots have length 2, thus for long roots α∨ = α. For short roots we have for B2 that α∨ = 2α2 and for G2 that α∨ = 3α. The fundamental domain of the dual root lattice (Voronoi region of L∨ ) is indicated by , and , and the fundamental domain for the aﬃne Weyl group is indicated by triangles. Figure 4.5 shows the A3 case (self-dual), where the fundamental domain (Voronoi region) of the root lattice is a rhombic dodecahedron, a convex polyhedron with 12 rhombic faces. Each of these faces (composed of two triangles) is part of a plane perpendicular to one of the 12 roots, half-way out to the root (roots are not drawn). The fundamental domain of the aﬃne Weyl group is the tetrahedron with an inscribed octahedron. The

51

Topics in structure-preserving discretization Table 4.3. The Cartan matrix, the matrix representation of generators for the Weyl group, and the size of the Weyl group. C A2 B2 G2  A3

B3

C3

2 −1 −1 2 2 −2 −1 2 2 −1 −3 2

2 −2 0  2 −1 0  2 −1 0

s1

 −1 0 2 −1 −1 2  −1 0 2 −2 −1 2  −1 0 2 −1 −2 2



−1 1 0 1 −1 2 0 1 −1 1 0 1

−1 0 0  −1 0 0  −1 0 0

s2

 1 0 1 0 0 1  1 0 1 0 0 1  1 0 1 0 0 1

1 1 1 1 1 3

|W |

s3

0 −1

6

0 −1

8

0 −1

12

   1 0 0 −1 0 1 −1 1  0 1 0 0 1 0 1    1 0 0 −1 0 1 −1 2  0 1 0 0 1 0 1    1 0 0 −1 0 1 −1 1  0 1 0 0 1 0 2

 0 0 −1  0 0 −1  0 0 −1

24

48

48

corners of this tetrahedron constitute the fundamental dominant weights λ1 , λ2 , λ3 . The regular octahedron drawn inside the Weyl chamber is important for application of multivariate Chebyshev polynomials, discussed in Section 4.2.2. Laplacian eigenfunctions on triangles, tetrahedra and simplexes. In this subsection we will consider real- or complex-valued functions on V-respecting symmetries of an aﬃne Weyl group. Consider ﬁrst L2 (T), the space of complex-valued L2 -integrable periodic functions on the torus T = V/L∨ i.e., functions f such that f (y + α∨ ) = f (y) for all y ∈ V and α∨ ∈ L∨ . Since the weights lattice Λ is reciprocal to L∨ , the Fourier basis for periodic functions is given as {exp(2πi(λ, t))}λ∈Λ and Fourier transforms are deﬁned for f ∈ L2 (T): 1 f (λ) = F(f )(λ) = f (t)e−2πi(λ,t) dt, (4.26) vol(T) T f (t) = F −1 (f)(t) = (4.27) f(λ)e2πi(λ,t) . λ∈Λ

We are interested in functions which are periodic under translations in L∨ ˜ , e.g., functions and which in addition respect the other symmetries in W

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

˜ (we will with odd or even symmetry with respect to the reﬂections in W see below that there are also other possibilities). Due to the semi-direct ˜ = W L∨ it follows that any w ˜ can be written product structure W ˜∈W ∨ ∨ ∨ w ˜ = w · τα , where w ∈ W and α ∈ L . Thus, on the space of periodic ˜ and the ﬁnite group W are identical. We functions L2 (T) the action of W deﬁne subspaces of symmetric and skew-symmetric periodic functions as follows: L2∨ (T) = {f ∈ L2 (T) : f (wt) = f (t) for all w ∈ W , t ∈ T }, L2∧ (T) = {f ∈ L2 (T) : f (wt) = (−1)|w| f (t) for all w ∈ W , t ∈ T }, where |w| denotes the length, deﬁned as |w| = , where w = sαj1 · · · sαj is written in the shortest possible way as a product of reﬂections about the simple positive roots αj ∈ Σ. Thus (−1)|w| = det(w) = ±1 depending on whether w is a product of an even or odd number of reﬂections. We deﬁne L2 -orthogonal projections π∨W and π∧W on these subspaces as 1 π∨W f (t) = f (wt), (4.28) |W | w∈W 1 W π∧ f (t) = (−1)|w| f (wt). (4.29) |W | w∈W

Orthogonal bases for these subspaces are obtained by projecting the exponentials, yielding the cosine- and sine-type basis functions cλ (t) := π∨W exp 2πi(λ, t), sλ (t) := π∧W exp 2πi(λ, t) as in (4.21)–(4.22). Note that these are not all distinct functions. Since they possess symmetries cλ (t) = cwλ (t), sλ (t) = (−1)|w| swλ (t), for every w ∈ W , we need only one λ from each orbit of W . The weights in the Weyl chamber, Λ+ = C+ ∩ Λ, form a natural index set of orbit representatives, and we ﬁnd L2 -orthogonal bases by taking the corresponding cλ (t) and sλ (t). Lemma 4.9 also holds in the following more general case. Lemma 4.14. Let φ be a rank d root system with weights lattice Λ. Let W be the Weyl group and let denote the fundamental domain of the aﬃne ˜ = W L∨ . The functions {cλ (t)}λ∈Λ and {sλ (t)}λ∈Λ Weyl group W + + form two distinct L2 -orthogonal bases for L2 (). These basis functions are eigenfunctions of the Laplacian ∇2 on satisfying homogeneous Neumann and Dirichlet boundary conditions, as in Lemma 4.9.

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53

Unfortunately, truncations of these bases do not form excellent approximation spaces for analytic functions on . Approximation of a function f , deﬁned on , in terms of {cλ (t)} is equivalent to Fourier approximation of the even extension of f in L2∨ (T), and in general we only observe quadratic convergence due to discontinuity of the gradient across the boundary of . A route to spectral convergence is by a change of variables, which turns the trigonometric polynomials {cλ (t)} and {sλ (t)} into multivariate Chebyshev polynomials of the ﬁrst and second kind. Before we dive into this topic, we remark on an alternative approach. Trigonometric polynomials with generalized symmetries. In 1D it is known that spectral (exponentially converging) approximations for analytic f deﬁned on can be obtained by employing the frame (i.e., a spanning set of vectors which is not linearly independent) generated by taking both the cosine and sine functions. In the frame we compute an approximating function by solving the following periodic extension problem. Find a smooth extension of f to L2 (T) and approximate this by the frame on . A detailed discussion of such techniques is found in Huybrechs (2010). One question is whether a frame consisting of only cλ (t) and sλ (t) might also be suﬃcient to obtain spectral convergence for higher-dimensional domains, or whether we have to include basis functions with other kinds of symmetries. Consider L2 (T, W), the space of functions periodic on T returning values in a Hilbert space W with inner product (·, ·)W . The inner product on L2 (T, W) is (f, g) = (f (t), g(t))W dt. T

Recall that a representation of the group W on W is a map ρ : W → Gl(W), deﬁning a linear action of W on W, i.e., for w ∈ W , ρ(w) is an invertible linear map on W such that ρ(ww) ˜ = ρ(w)ρ(w). ˜ The integer dρ = dim(W) is called the dimension of the representation ρ. We will (without loss of generality) always assume representations are unitary with respect to (·, ·)W . We say that f ∈ L2 (T, W) is ρ-symmetric if f (wt) = ρ(w)f (t)

for all w ∈ W , t ∈ T.

The space of such symmetric functions is denoted L2ρ (T, W). As an example, the functions in L2∨ (T) and L2∧ (T) are ρ-symmetric with respect to the trivial representation ρs (w) = 1 and the alternating representation ρa (w) = (−1)|w| acting on W = R. Since the Weyl group W is constructed as a group of linear transformations on V, we always also have the faithful identity representation ρI (w) = w, acting on W = V. Seen through a kaleidoscope,

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

a vector ﬁeld on appears to be ρI -symmetric on T, where the vectors are reﬂected in the mirrors. We deﬁne a projection πρW : L2 (T, W) → L2ρ (T, W) as πρW f (t) =

1 ρ(w−1 )f (wt). |W |

(4.30)

w∈W

Since ρ is unitary, it can be shown that πρW is an L2 -orthogonal projection. Note that scalar functions f : T → C can be symmetrized with respect to representations on any vector space W in a similar manner, i.e., 1 πρW f (t) = ρ(w−1 )f (wt), |W | w∈W

which yields a function πρW f (t) returning values in the space of all matrices over W. Using Theorem 4.6, we can recover f on T from its symmetrizations on . Proposition 4.15. Let f be a periodic scalar function f ∈ L2 (T). Given the symmetrizations fρ = πρW f ∈ L2ρ (T, W) for all ρ ∈ R, where R is a complete list of irreducible representations of W , we recover f on all of T as f (wt) = dρ trace (ρ(w)fρ (t)), for all t ∈ and w ∈ W . ρ∈R

It should be remarked that for the root system An the Weyl group is W Sn+1 , the symmetric group consisting of all permutations of n + 1 items. The representation theory of Sn+1 is well known, and there also exist fast algorithms for computing the corresponding transforms. Similarly to the symmetrization of f we can symmetrize the Fourier basis with respect to ρ and obtain matrix-valued symmetrized exponentials deﬁned as the dρ × dρ matrices expρ,λ (t) := πρW e2πi(λ,t) ,

(4.31)

where dρ = dim(W). Restricting to λ ∈ Λ+ and ρ ∈ R, where R is a complete list of irreducible representations of W , the components of {expρ,λ }λ∈Λ+ ,ρ∈R form an L2 -orthogonal basis for L2 (T). Restricted to they form a tight frame with frame bound |W |: see Huybrechs (2010) for a treatment of the classical case A1 . These generalized symmetric exponential functions are (almost) equivalent to the irreducible representations of the ˜ obtained by the technique of small groups (Serre 1977). aﬃne Weyl group W In Section 4.2.2 we will see that the odd and even symmetrized exponentials cλ (t) and sλ (t) transform to multivariate Chebyshev polynomials of the ﬁrst and second kind under a particular change of variables. Also, the

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55

more general ρ-symmetrized exponentials expρ,λ (t) are related to multivariate Chebyshev polynomials of ‘other kinds’. Approximation theory on the simplex using the frame spanned by {expρ,λ }, and the theory of multivariate Chebyshev polynomials of other kinds, is currently under investigation.9 It is known in the A2 case that the frame consisting only of cλ (t) and sλ (t) is not suﬃcient to guarantee spectral convergence of arbitrary analytic functions on . By including the last four basis functions arising from the irreducible two-dimensional representation ρI (w) = w we do obtain spectral convergence. However, practical use of these techniques in discretizations of PDEs has not yet been completed. 4.2.2. Multivariate Chebyshev polynomials Bivariate Chebyshev polynomials were constructed independently by Koornwinder (1974) and Lidl (1975) by folding exponential functions. Multidimensional generalizations (the A2 family) appeared ﬁrst in Eier and Lidl (1982). Hoﬀman and Withers (1988) presented a general folding construction. Characterization of such polynomials as eigenfunctions of diﬀerential operators is found in Beerends (1991) and Koornwinder (1974). Applications to the solution of diﬀerential equations are found in Munthe-Kaas (2006) and to triangle-based spectral element Clenshaw–Curtis-type quadratures in Ryland and Munthe-Kaas (2011). Recall that classical Chebyshev polynomials of the ﬁrst and second kind, Tk (x) and Uk (x), are obtained from cos(kθ) and sin(kθ) by a change of variable x = cos(θ), as Tk (x) = cos(kθ), sin((k + 1)θ) Uk (x) = . sin(θ) We want to understand this construction in the context of aﬃne Weyl groups. We recognize cos(kθ) and sin(kθ) as the symmetrized and skewsymmetrized exponentials. The cos(θ) used in the change of variables is the 2π-periodic function, which is symmetric, non-constant and has the longest wavelength (as such, uniquely deﬁned up to constant). In other words cos(θ) = π∨ exp(λ1 θ), where λ1 = 1 is the generator of the weights lattice. Any periodic band-limited even function f has a symmetric Fourier series of ﬁnite support on the weights lattice, and must hence be a polynomial in the variable x = cos(θ). The denominator sin(θ) is similarly the odd function of longest possible wavelength. Any periodic band-limited odd function f has a skew-symmetric Fourier series on the weights lattice. Dividing out sin(θ) results in a band-limited even function which again must 9

H. Munthe-Kaas and D. Huybrechs, work in progress.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

map to a polynomial under our change of variables. The denominator, which in our special case is sin(θ), we call the Weyl denominator. It plays an important role in the representation theory of compact Lie groups as the denominator in Weyl’s character formula, the jewel in the crown of representation theory. We will detail these constructions below. Notation. As in Section 4.2.1, we let Φ be a rank d root system on V = Rd , ˜ = W L∨ . with Weyl group W , co-root lattice L∨ and aﬃne Weyl group W ∨ Let T = V/L be the torus of periodicity and Λ = spanZ {λ1 , . . . , λd } the reciprocal lattice of L∨ . The lattice Λ is an abelian group known as the Pontryagin dual of the abelian group T. Speciﬁcally, each λ ∈ Λ corresponds uniquely to an exponential function exp(2πi(λ, t)), which is one of the irreducible one-dimensional representations on T. These form an abelian group under multiplication. In the following, it is convenient to write the group Λ in multiplicative form as follows. Let {eλ }λ∈Λ denote the elements of the multiplicative group,10 understood as formal symbols such that for λ, µ ∈ Λ we have eλ · eµ = eλ+µ . Let E = E(C) denote the free complex over the symbols eλ . vector space λ This consists of all formal sums a = λ∈Λ a(λ)e where the coeﬃcients a(λ) ∈ C and all but a ﬁnite number of these are non-zero. This is the complex group algebra over the multiplicative group Λ, with a commutative product induced from the group product a(λ) b(µ) = a(λ)b(µ) · eν . λ∈Λ

We deﬁne conjugation

µ∈Λ

ν∈Λ

λ+µ=ν

: E → E as a= a(λ)e−λ . λ∈Λ

An element a ∈ E is identiﬁed with a trigonometric polynomial f (t) = F −1 (a)(t) on the torus T (i.e., a band-limited periodic function) through the Fourier transforms given in (4.26)–(4.27). Multiplication and conjugation in E corresponds to pointwise multiplication and complex conjugation of the functions on T. Let E∨W ⊂ E denote the symmetric subalgebra of those elements that are invariant under the action of the Weyl group W on E. This consists of those a ∈ E where a(λ) = a(wλ) for all λ and all w ∈ W . Similarly, E∧W ⊂ E denotes those a ∈ E whose sign alternates under reﬂections sα , i.e., where the coeﬃcients satisfy a(λ) = (−1)|w| a(wλ). Here |w| denotes the length 10

Those who prefer can simply consider eλ to be shorthand for exp(2πi(λ, t)), as a function of t ∈ T.

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57

of w, the length of the shortest factorization of w into a product of simple reﬂections. Thus (−1)|w| = det(w). The alternating elements do not form a subalgebra, since the product of two alternating functions is symmetric. The subspaces E∨W and E∧W correspond to W -symmetric and W -alternating trigonometric polynomials on T. As a digression we remark that a parallel theory is the representation theory of compact connected Lie groups G: see Bump (2004) for an excellent exposition. In this setting T is the maximal torus of G, i.e., a maximal abelian subgroup. Any g ∈ G is conjugate to a point in T, i.e., there exist a k ∈ G such that kgk−1 ∈ T. The normalizer N (T) is composed of those elements of G which ﬁx T under conjugation, N (T) = {k ∈ GkTk −1 = T}. The Weyl group is W = N (T)/T, which is always ﬁnite. It can be shown that the action of W on T through conjugation can be represented in terms of a root system. This implies that class functions on G, i.e., functions such that f (g) = f (kgk −1 ) for all k, can be represented as W invariant functions on T, and that any function in E∨W can be interpreted as a class function on G. Of particular interest are the class functions which arise as traces of the irreducible representations on G. These are called the irreducible characters, and are known explicitly through the celebrated Weyl character formula. We will see that the irreducible characters and multivariate Chebyshev polynomials of the second kind are equivalent: see Beerends (1991). Multivariate Chebyshev polynomials. Deﬁne a projection π∧W : E → E∧W by its action on the coeﬃcients a → π∧W a: (π∧W a)(λ) =

1 (−1)|w| a(wλ). |W | w∈W

The projection π∨W : E → E∨W is deﬁned similarly, omitting (−1)|w| . The algebra E is generated by {eλj }dj=1 ∪ {e−λj }dj=1 where λj are the fundamental dominant weights. E∨W is the subalgebra generated by the symmetric generators {zj }dj=1 deﬁned by zj = π∨W eλj =

1 wλj 2 wλj e = e , |W | |W | + w∈W

w∈W

where W + denotes the even subgroup of W containing those w such that |w| is even. The latter identity follows from sαj λj = λj , so it is enough to consider w of even length. The action of W + on λj is free and eﬀective. It can be shown that E∨W is a unique factorization domain over the generators {zj }, i.e., any a ∈ E∨W can be expressed uniquely as a polynomial in {zj }dj=1 .

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

The skew subspace E∧W does not form an algebra, but this can be corrected by dividing out the Weyl denominator. Deﬁne the Weyl vector ρ ∈ Λ as ρ=

d

λj =

j=1

1 α. 2 + α∈Φ

We deﬁne the Weyl denominator D ∈ E∧W as (−1)|w| ewρ . D= w∈W

Proposition 4.16. Any a ∈ E∧W is divisible by D, i.e., there exists a unique b ∈ E∨W such that a = bD. Proof.

See Bump (2004, Proposition 25.2).

Any a ∈ E∨W can be written as a polynomial in z1 , . . . , zd , and hence the following polynomials are well-deﬁned. Definition 4.17. For λ ∈ Λ we deﬁne multivariate Chebyshev polynomials of the ﬁrst and second kind, Tλ and Uλ , as the unique polynomials that satisfy 1 wλ Tλ (z1 , . . . , zd ) = π∨W eλ = e , (4.32) |W | w∈W |w| w(ρ+λ) |W |π∧W eλ+ρ w∈W (−1) e Uλ (z1 , . . . , zd ) = . (4.33) = |w| wρ D w∈W (−1) e By a slight abuse of notation we will also consider zj as W -invariant functions on T as 2 2πi(λj ,t) e . (4.34) zj (t) = F −1 (zj )(t) = |W | + w∈W

We will study coordinates on T obtained from these. The functions zj (t) may be real or complex. If there exists an w ∈ W + such that wλj = λj , then zj = zj is real. Otherwise there must exist an index j = j and a w ∈ W + such that wλj = λj and we have zj = zj . In the latter case we can 1 (zj − zj ). replace these with d real coordinates xj = 12 (zj + zj ), xj = 2i ∨ d Using the basis {αj }, we identify T (R/Z) , and let tj ∈ [0, 1) denote standard coordinates on this unit-periodic torus. Let Jk,l (t) =

∂zk (t) ≡ (αl∨ · ∇t )zj (t) ∂αl∨

k, l ∈ {1, . . . , d}

be the Jacobian matrix of the map (t1 , . . . , td ) → (z1 , . . . , zd ).

59

Topics in structure-preserving discretization

Proposition 4.18. The Fourier transform of the Jacobian determinant, J := F(det(J)(t)), is the alternating function J = cD ∈ E∧W , where D is the Weyl denominator and the constant c =

4πi d |W |

.

Proof. Let sα be any reﬂection in W . Since z(t) = z(sα t) we have J(t) = J(sα t)sα . Hence det(J(t)) = − det(J(sα t)) and we conclude that J ∈ E∧W . By Proposition 4.16, D divides J. We need to conﬁrm that J/D is constant. We compute 4πi J α , wλk ewλk . k, = |W | + w∈W

+ Thus, J k, is supported on W λk and J is supported on the set w j λj . w1 ,...,wd ∈W +

The highest weight in this set is the Weyl vector ρ = dj=1 λj , which is reached if and only if w1 = w2 = · · · wd = I. Since the highest weight is ρ we conclude that J/D = c. The constant c is computed as follows: d c = J(ρ) = sign(σ) Jj,σj (ρ) j=1

σ∈Sd

=

sign(σ)

σ∈Sd

=

d j=1

4πi d

|W |

Jj,σj (λj ) =

σ∈Sd

sign(σ)

σ∈Sd

d j=1

δσj ,j =

d 4πi ασj , λj sign(σ) |W |

j=1

4πi |W |

d

.

Finally, in this paragraph, we want to remark that (4.33) is exactly the same formula as Weyl’s character formula, giving the trace of all the irreducible characters on a semi-simple Lie group (Bump 2004). These characters form an L2 -orthogonal basis for the space of class functions on the Lie group. Thus, expansions in terms of multivariate Chebyshev polynomials of the second kind is equivalent to expansions in terms of irreducible characters on a Lie group. Just as the basis given by the irreducible representations block-diagonalize equivariant linear operators on a Lie group, we can also use irreducible characters to obtain block diagonalizations: see James and Liebeck (2001). Thus, our software, which is primarily constructed to deal with spectral element discretizations of PDEs, may also have important applications to computations on Lie groups. This opens up a whole area

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

of possible applications of these approximations, such as in random matrix theory. We will not pursue these topics here. We will focus below on applications of Chebyshev polynomials of the ﬁrst kind in spectral element discretizations. Analytical properties. The polynomials Tλ (z) of the ﬁrst kind are by construction W-symmetric in λ, Tλ = Twλ for all w ∈ W . Thus the full space of multivariate polynomials is spanned by {Tλ }λ∈Λ+ , where Λ+ are the weights within the positive Weyl chamber. We will ﬁrst show that this is a family of multivariate orthogonal polynomials. ˜ . The change of variLet ⊂ T denote the fundamental domain of W ables t → z in (4.34) has a Jacobian determinant proportional to the Weyl denominator D. It is known from representation theory that D is always zero on the boundary of and non-zero in the interior. Thus the coordinate map t → z is invertible on , regular in the interior of and singular on the boundary. Let δ = z(). This is a domain which in the A2 case is a deltoid or three-cusp Steiner hypocycloid: see Figure 4.6(b). Note that D ∈ E∧W is an alternating function, thus DD ∈ E∨W is a symmetric function, hence it is a real multivariate polynomial in z. Proposition 4.19. The coordinate map t → z is invertible on δ = z(). The absolute value of the Jacobian determinant is = |c| DD, |J| is a real polynomial in z. The boundary of δ is given as an where DD algebraic ideal, DD = 0. In the A1 case we have J = sin(t), hence DD = sin2 (t) = 1 − z 2 . In the A2 case we have DD =

1 8 3 + (x − 3xy 2 ) − (x2 + y 2 )2 − 2(x2 + y 2 ), 3 3

where x = 12 (z + z) and y = 12 (z − z). Proposition 4.20. The polynomials {Tλ }λ∈Λ+ are orthogonal on δ with respect to the weighted inner product 1 dz. (f, g) = f (z)g(z) √ δ DD Proof. Under the change of variable z → t, we have that Tλ maps to π∨W exp(2πi(λ, t)), and these are L2 -orthogonal on . The weight follows from the formula for the Jacobian determinant.

Topics in structure-preserving discretization

(a)

(b)

61

(c)

Figure 4.6. The equilateral domain ∆ in (a) maps to the deltoid δ in (b) under coordinate map t → z, and to a triangle in (c) under a straightening map.

From the convolution product in E∨W one ﬁnds that the Tλ satisfy the recurrence relations T0 = 1,

(4.35)

Tλj = zj ,

(4.36)

Tλ = Twλ for w ∈ W, T−λ = Tλ , Tλ T µ =

1 |W |

(4.37) (4.38)

Tλ+wµ .

(4.39)

w∈W

These reduce to classical three-term recurrences for A1 and four-term recurrences for A2 : see Munthe-Kaas (2006). The recurrences provide a practical way of computing values of Tk (z) at arbitrary points z. For special collocation points we will see below that FFT-based computations are much more eﬃcient. Discrete properties. Pulled back to t-coordinates, polynomials in z become band-limited symmetric functions f ∈ L2∨ (T). Due to the Shannon sampling theorem, any band-limited function f ∈ L2 (T) can be exactly reconstructed from sampling on a suﬃciently ﬁne lattice S ⊂ T. Due to periodicity, the reciprocal lattice S ∗ must be a sublattice of Λ. The condition for perfect reconstruction is that the only point of the reciprocal lattice S ∗ within the support of f in Λ is the origin. In addition to this condition, we also require the sampling lattice to be W -invariant, W S = S.

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There are several ways to construct such a lattice. A convenient way is to let S be a down-scaled version of the co-root lattice, S = L∨ /M,

(4.40)

where M is an integer suﬃciently large to ensure the Shannon condition for perfect reconstruction. On the lattice S the sampled function fS becomes a symmetric function on a ﬁnite abelian group. Thus we can employ FFTs to ﬁnd f ∈ E∨W . Special group-theoretic versions of symmetrized FFTs can also be used, which are more eﬃcient, but harder to program than the standard FFT: see Munthe-Kaas (1989). The lattice S maps to collocation points z(S) for the multivariate polynomials, as shown in Figure 4.6 for the A2 case, M = 12. The nodal points z(S) ⊂ δ form the two-dimensional analogue of Chebyshev extremal points. The straight lines in Figure 4.6(a), which are in the direction of a root, map to straight lines in Figure 4.6(b). Along these lines inside δ, the nodal points distribute like 1D classical Chebyshev collocation points (either as Chebyshev zeros, or as Chebyshev extremal points). Similarly, the multivariate Chebyshev points in higher dimensions contain ﬂat hyperplanes of lower dimension, and under restriction to these one restricts to lower-dimensional Chebyshev nodes. This is an important feature of these polynomials, which simpliﬁes restriction of polynomials to (certain speciﬁc) linear subspaces. In particular, we note that the hexagon in Figure 4.6(a) maps to an equilateral triangle embedded nicely inside δ (Figure 4.6(b)). The fact that the collocation nodes distribute as 1D Chebyshev nodes on the boundary of this triangle allows us to express in closed form conditions for continuity of polynomials patched together along these triangle boundaries. For high-order polynomial collocation it is extremely important that the Lebesgue number of the collocation points grows slowly in M . It is well known that for classical Chebyshev polynomials the Lebesgue number grows at the optimal rate O(log(M )). From properties of the Dirichlet kernel, one can show a similar result for multivariate Chebyshev polynomials. Proposition 4.21. For the collocation points z(S), where S = L∨ /M ⊂ Rd , the Lebesgue number grows as O(logd (M )). 4.2.3. Spectral element methods based on triangles and simplexes Triangle-based spectral element methods were ﬁrst considered in Dubiner (1991), where a basis was constructed by a ‘warping’ of a 2D tensor product to a triangle. More recent work is Giraldo and Warburton (2005), Hesthaven (1998) and Warburton (2006), based on interpolation of polynomials in good interpolation nodes on a triangle. Our approach diﬀers from these by basing the construction on the multivariate Chebyshev polynomials, which allows for fast FFT-based computations of all the basic operations on the polynomials. It is well known

Topics in structure-preserving discretization

63

that a logarithmic growth of the Lebesgue number implies exponentially fast (spectral) convergence of polynomial approximations to analytic functions (Canuto et al. 2006). This result, together with the availability of FFT-based algorithms is, just as in the univariate case, among the most important properties of the multivariate Chebyshev polynomials as a practical tool in computational approximation theory and solution of PDEs. Thus, as a conclusion, the coordinate map t → z has a dramatic positive eﬀect on the convergence rate of ﬁnite approximations using multivariate Chebyshev polynomials, compared to expansions in the symmetric exponentials π∧W exp(2πi(λ, t). A practical diﬃculty is, however, the fact that under this change of coordinates the simplex becomes a signiﬁcantly more complicated domain with cusps in the corners. This has to be dealt with in an eﬃcient manner if we want to construct practical spectral element methods. We have been working with three possible solutions to this problem. • Work with the domain ⊂ T. Use the tight frame provided by the ρ-symmetrized exponentials for ρ ∈ R, and solve the periodic extension problem. This approach has not been developed in detail yet. • Straighten δ to a nearby simplex by a straightening map (MuntheKaas 2006). We have had reasonably good experiences with this approach, but the straightening maps must be singular in the cusps. In numerical experiments with this approach we have seen quite good Lebesgue numbers, and reasonably good convergence rates: see MuntheKaas (2006). However, we have not been able to obtain spectral convergence, probably due to the corner singularities of the straightening map. • Patch together deltoids with overlap, so that the equilateral triangles that are inscribed in the deltoid form a simplicial subdivision of the total space. This approach is working successfully, and will be discussed below. One important question is whether or not this inscribed triangle is a particular feature for the A2 case, or if the higher-dimensional cases also have a similar structure with a nicely inscribed simplex in δ. This is indeed the case. We can prove this for the An family (which is the most important case), and we conjecture that a similar property also holds for the other inﬁnite families of Dynkin diagrams. In Figure 4.7 we illustrate the δ domain in the A3 case. The 3D deltoid-shaped domain contains an inscribed tetrahedron. This tetrahedron is the image of the regular octahedron inscribed in ⊂ T as seen in Figure 4.5. On the faces of this tetrahedron, the nodal points distribute as the Chebyshev points for the A2 case.

64

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

Figure 4.7. The deltoid domain δ in the A3 case. The inscribed tetrahedron is the image of the regular octahedron inscribed in the Weyl chamber of : see Figure 4.5.

Numerical experiments. A library of routines has been implemented for the A2 case, both in MATLAB and in C++. All basic algorithms such as products of functions, derivation and integration is implemented using FFTbased techniques; see Ryland and Munthe-Kaas (2011) for more details. Algorithms exist for exactly integrating the polynomials both over δ and also over the inscribed triangle, with respect to weight function 1 or the 1 weight function (DD)− 2 . In Ryland and Munthe-Kaas (2011) we demonstrate the use of these bases in spectrally converging Clenshaw–Curtis-type cubatures based on triangular subdivisions. The software allows for triangles mapped from the equilateral reference triangle both with linear and with non-linear non-singular coordinate maps. The Jacobian of these maps can be supplied analytically, or computed numerically from the sampling values. One problem with this type of quadrature is that, since the deltoid stretches outside its inscribed triangle, one has to evaluate functions at points outside the domain. If this is not possible, one could use the collocation points of the straightened triangle, Figure 4.6(c), for the triangles on the boundary. This, however, reduces the convergence rate. We are currently exploring spectral element discretizations for PDEs based on similar subdivisions. In the experiments, we have explicitly imposed continuity across triangle boundaries. The numerical experiments conﬁrm the exponential convergence rate for these approximations.

65

Topics in structure-preserving discretization solution N=48

100 80 60 40 20 0 −20 50 40

50 30

40 30

20

20

10

10 0

0

Figure 4.8. The numerical experiment shows spectral element solution of ∇2 u = −1 on an L-shaped domain composed of three squares, with Dirichlet boundary conditions. Each square is divided into two isosceles triangles, and all these 6 subdomains are patched together by explicitly imposing conditions for continuity across common boundaries of two triangles. Each triangle has a Chebyshev A2 family of Chebyshev polynomials, in this ﬁgure using the root lattice down-scaled with the factor 48 as collocation lattice.

5. Finite element systems of diﬀerential forms Many partial diﬀerential equations (PDEs) can be naturally thought of as expressing that a certain ﬁeld, say a scalar ﬁeld or a vector ﬁeld, is a critical point of a certain functional. Often this functional will be of the form S(u) = L(x, u(x), ∇u(x), . . .) dx. S

u is a section of a vector bundle over S and L is Here S is a domain in a Lagrangian involving x ∈ S as well as values of u and its derivatives. In this context the functional S is called the action. Criticality of the action can be written as Rd ,

∀u

DS(u)u = 0.

More generally, a PDE written in the form F(u) = 0 can be given a variational formulation: ∀u

F(u), u = 0.

(5.1)

The operator F is continuous from a Banach space X to another one Y , and ·, · denotes a duality product on Y × Y .The unknown u is sought in

66

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

X and the equation is tested with u in Y . The duality product is most often a generalization of the L2 (S) scalar product, such as a duality between Sobolev spaces. In the case of least action principles it is of interest to ﬁnd X such that one can take Y = X (and Y = X). This point of view leads quite naturally to discretizations. We construct a trial space Xn ⊆ X and a test space Yn ⊆ Y , and solve u ∈ Xn , ∀u ∈ Yn

F(u), u = 0.

(5.2)

In theory we have sequences indexed by n ∈ N and establish convergence properties of the method as n → ∞. This discretization technique is called the Galerkin method when Xn = Yn and the Petrov–Galerkin method when Xn and Yn are diﬀerent. The ﬁnite element (FE) method consists in constructing ﬁnite-dimensional trial and test spaces adapted to classes of PDEs. Mixed FE spaces have been constructed to behave well for the diﬀerential operators grad, curl and div. They provide a versatile tool-box for discretizing PDEs expressed in these terms. Thus Raviart–Thomas div-conforming ﬁnite elements (Raviart and Thomas 1977) are popular for PDEs in ﬂuid dynamics and N´ed´elec’s curl-conforming ﬁnite elements (N´ed´elec 1980) have imposed themselves in electromagnetics. For reviews and references see Brezzi and Fortin (1991), Roberts and Thomas (1991), Hiptmair (2002) and Monk (2003). These spaces ﬁt the deﬁnition of a ﬁnite element given by Ciarlet (1978). In particular they are equipped with unisolvent degrees of freedom, which determine inter-element continuity and provide interpolation operators: projections In onto Xn which are deﬁned at least for smooth ﬁelds. In applications one often needs a pair of spaces Xna × Xnb , and these spaces should be compatible in the sense of satisfying a Brezzi inf-sup condition (Brezzi 1974). The spaces are linked by a diﬀerential operator d : Xna → Xnb (one among grad, curl, div), and the proof of compatibility would follow from a commuting diagram: Xa

d

Ina

Xna

d

/ Xb

Inb

/ Xb

n

A technical problem is that one would like the interpolators to be deﬁned and bounded on the Banach spaces X a , X b , whereas the natural degrees of freedom are not bounded on these. For instance X a is usually of the form: X a = {u ∈ L2 (S) ⊗ E a : du ∈ L2 (S) ⊗ E b }, for some ﬁnite-dimensional ﬁbres E a , E b , but this continuity is usually insufﬁcient for degrees of freedom, such as line integrals, to be well-deﬁned. The convergence of eigenvalue problems requires even more boundedness prop-

Topics in structure-preserving discretization

67

erties of the interpolation operators, something more akin to boundedness in L2 (S) (see Christiansen and Winther (2010) for precise statements). Mixed ﬁnite elements have been constructed to contain polynomials of arbitrarily high degree. Applying the Bramble–Hilbert lemma to the interpolators gives high orders of approximation, and under stability conditions this gives a high order of convergence for numerical methods. But here too the intention is hampered by the lack of continuity of interpolators. As remarked by Bossavit (1988), lowest-order mixed ﬁnite elements, when translated into the language of diﬀerential forms, correspond to constructs from diﬀerential topology called Whitney forms (Weil 1952, Whitney 1957). The above-mentioned diﬀerential operators can all be interpreted as instances of the exterior derivative. From this point of view it becomes natural to arrange the spaces in full sequences linked by operators forming commuting diagrams: X0

d

In0

Xn0

d

/ X1

d

/ ...

d

In1

/ X1

n

d

/ . . .

d

/ Xd

Ind

/ X d. n

An important property is that the interpolators should induce isomorphisms on cohomology groups. This is essentially de Rham’s theorem, when the top row consists of spaces of smooth diﬀerential forms and the bottom row consists of Whitney forms. In particular, on domains homeomorphic to balls, the sequences of FE spaces should be exact. This applies in particular to single elements such as cubes and simplexes. High-order FE spaces of diﬀerential forms naturally generalizing Raviart– Thomas–N´ed´elec elements were presented by Hiptmair (1999). For a comprehensive review, encompassing Brezzi–Douglas–Marini elements (Brezzi, Douglas and Marini 1985), see Arnold, Falk and Winther (2006b). Elements are constructed using the Koszul operator, or equivalently the Poincar´e homotopy operator, to ensure local sequence exactness, that is, exactness of the discrete sequence attached to a single element. Mixed FE spaces have thus been deﬁned for simplexes. Tensor product constructions yield spaces on Cartesian products. Given spaces deﬁned on some mesh, it would sometimes be useful to have spaces constructed on the dual mesh, matching the spaces on the primal mesh in some sense. Since the dual mesh of a simplicial mesh is not simplicial, this motivates the construction of FE spaces on meshes consisting of general polytopes. On these, it seems unlikely that good FE spaces can be constructed with only polynomials: one should at least allow for piecewise polynomials. In some situations, such as convection-dominated ﬂuid ﬂow, stability requires some form of upwinding. This could be achieved by a

68

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

Petrov–Galerkin method, by including upwinded basis functions in either the trial space Xn or the test space Yn . For a recent review of this topic see Morton (2010). This provides another motivation for constructing a framework for ﬁnite elements that includes non-polynomial functions. Discretization of PDEs expressed in terms of grad, curl and div on polyhedral meshes has long been pursued in the framework of mimetic ﬁnite diﬀerences, reviewed in Bochev and Hyman (2006). The convergence of such methods has been analysed in terms of related mixed ﬁnite elements (Brezzi, Lipnikov and Shashkov 2005). For connections with ﬁnite volume methods see Droniou, Eymard, Gallou¨et and Herbin (2010). While recent developments tend to exhibit similarities and even equivalences between all these methods, the FE method, as it is understood here, could be singled out by its emphasis on deﬁning ﬁelds inside cells, and ensuring continuity properties between them, in such a way that the main discrete diﬀerential operators acting on discrete ﬁelds are obtained simply by restriction of the continuous ones. Ciarlet’s deﬁnition of a ﬁnite element does not capture the fact that mixed ﬁnite elements behave well with respect to restriction to faces of elements. From the opposite point of view, once this is realized, it becomes natural to prove properties of mixed ﬁnite elements by induction on the dimension of the cell. The goal is then to construct a framework for FE spaces of diﬀerential forms on cellular complexes accommodating arbitrary functions. One requires stability of the ansatz spaces under restriction to subcells and under the exterior derivative. For the good properties of standard spaces to be preserved, one must impose additional conditions on the ansatz spaces, essentially surjectivity of the restriction from the cell to the boundary, and sequence exactness under the exterior derivative on each cell. As it turns out, these conditions, which we refer to as compatibility, imply the existence of interpolation operators commuting with the exterior derivative. Degrees of freedom are not part of the deﬁnition of compatible ﬁnite element systems but are rather deduced, and it becomes more natural to compare various degrees of freedom for a given system. The local properties of surjectivity and sequence exactness also ensure global topological properties by the general methods of algebraic topology. Interpolation operators still lack desired continuity properties. But combining them with a smoothing technique yields commuting projection operators that are stable in L2 (S). Stable commuting projections were ﬁrst proposed in Sch¨oberl (2008). Smoothing was achieved by taking averages over perturbations of the grid. Another construction using cut-oﬀ and smoothing by convolution on reference macro-elements was introduced in Christiansen (2007). While commutativity was lost, the lack of it was controlled by an auxiliary parameter. As observed in Arnold et al. (2006b), for quasi-uniform

Topics in structure-preserving discretization

69

meshes one can simplify these constructions and use smoothing by convolution on the physical domain. If, in the method of Sch¨ oberl (2008), one can say that the nodes of the mesh are shaken independently, smoothing by convolution consists in shaking them in parallel. Much earlier, in Dodziuk and Patodi (1976), convergence for the eigenvalue problem for the Hodge– Laplacian, discretized with Whitney forms, was proved using smoothing by the heat kernel. For scalar functions, smoothing by convolution in the FE method has been used at least as far back as Strang (1972) and Hilbert (1973), but Cl´ement interpolation (Cl´ement 1975) seems to have supplanted it. Christiansen and Winther (2008) introduced a space-dependent smoothing operator, commuting with the exterior derivative, allowing for general shape-regular meshes. These constructions also require a commuting extension operator, extending diﬀerential forms outside the physical domain. This section is organized as follows. In Section 5.2, cellular complexes and the associated framework of ﬁnite element systems are introduced. Basic examples are included, as well as some constructions like tensor products. Section 5.2 serves to introduce degrees of freedom and interpolation operators on FE systems. In Section 5.3, we construct smoothers and extensions which commute with the exterior derivative and preserve polynomials locally. When combined with interpolators they yield Lq (S)-stable commuting projections for scale-invariant FE systems. In Section 5.4 we apply these constructions to prove discrete Sobolev injections and a translation estimate. The framework of FE systems was implicit in Christiansen (2008a) and made explicit in Christiansen (2009), but we have improved some of the proofs. Upwinding in this context is new, as well as the discussion of interpolation and degrees of freedom. The construction of smoothers and extensions improves that of Christiansen and Winther (2008) by having the additional property of preserving polynomials locally, up to any given maximal degree. The analysis is also extended from L2 to Lq estimates, for all ﬁnite q. This is used in the proof of the Sobolev injection and translation estimate, which are also new (improving Christiansen and Scheid (2011) and Karlsen and Karper (2010)). 5.1. Finite element systems Cellular complexes. For any natural number k ≥ 1, let Bk be the closed unit ball of Rk and Sk−1 its boundary. For instance S0 = {−1, 1}. We also put B0 = {0}. Let S denote a compact metric space. A k-dimensional cell in S is a closed subset T of M for which there is a Lipschitz bijection Bk → T with a Lipschitz inverse. If a cell T is both k- and l-dimensional then k = l. For k ≥ 1, we denote by ∂T its boundary, the image of Sk−1 by the chosen

70

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

bi-Lipschitz map. Diﬀerent such maps give the same boundary. The interior of T is by deﬁnition T \ ∂T (it is open in T but not necessarily in S). Definition 5.1. A cellular complex is a pair (S, T ) where S is a compact metric space and T is a ﬁnite set of cells in S, such that the following conditions hold. • Distinct cells in T have disjoint interiors. • The boundary of any cell in T is a union of cells in T . • The union of all cells in T is S. In this situation we also say that T is a cellular complex on S. We ﬁrst make the following remarks. Proposition 5.2. The intersection of two cells in T is a union of cells in T . Proof. Let T, U be two cells in T and suppose x ∈ T ∩ U . Choose a cell T included in T of minimal dimension such that x ∈ T . Choose also a cell U included in U of minimal dimension such that x ∈ U . Suppose neither of the cells T and U are points. Then x belongs to the interiors of both T and U , so that T = U . Therefore there exists a cell included in both T and U to which x belongs. This conclusion also trivially holds if T or U is a point. In fact, if (S, T ) is a cellular complex, S can be recovered from T as follows. Proposition 5.3.

Let (S, T ) be a cellular complex.

• S is recovered as a set from T as its union. • The topology of S is determined by the property that a subset U of S is closed if and only if, for any cell T in T , of dimension k, the image of U ∩ T under the chosen bi-Lipschitz map is closed in Bk . • A compatible metric on S can be recovered from metrics dT on each cell T in T inherited from Bk , for instance by (max ∅ = +∞): d(x, y) = min{1, max{dT (x, y) : x, y ∈ T and T ∈ T }}. A simplicial complex is a cellular complex in which the intersection of any two cells is a cell (not just a union of cells) and such that the boundary of any cell is split into subcells in the same way as the boundary of a reference simplex is split into sub-simplexes. The reference simplex ∆k of dimension k is k+1 (x0 , . . . , xk ) ∈ R : xi = 1 and ∀i xi ≥ 0 . i

Topics in structure-preserving discretization

71

Its sub-simplexes are parametrized by subsets J ⊆ {0, · · · , k} as the solution spaces {x ∈ ∆k : ∀i ∈ J xi = 0}. For each k ∈ N the set of k-dimensional subsets of T is denoted by T k = {T ∈ T : dim T = k}. The k-skeleton of T is the cellular complex consisting of cells of dimension at most k: T (k) = T 0 ∪ · · · ∪ T k . The boundary ∂T of any cell T of T can be naturally equipped with a cellular complex, namely {T ∈ T : T ⊆ T and T = T }. We use the same notation for the boundary of a cell and the cellular complex it carries. A reﬁnement of a cellular complex T on S is a cellular complex T on S such that each element of T is the union of elements of T . We will be particularly interested in simplicial reﬁnements of cellular complexes. A cellular subcomplex of a cellular complex T on S is a cellular complex T on some closed part S of S such that T ⊆ T . For instance, if T ∈ T is a cell, its subcells form a subcomplex of T , which we denote by T˜. We have seen that the boundary of any cell T ∈ T can be equipped with a cellular complex which is a subcomplex of T˜. Fix a cellular complex (S, T ). In the following we suppose that for each T ∈ T of dimension ≥ 1, the manifold T has been oriented. The relative orientation of two cells T and T in T , also called the incidence number , is denoted o(T, T ) and deﬁned as follows. For any edge e ∈ T 1 its vertices are ordered, from say e˙ to e¨. Deﬁne o(e, e) ˙ = −1 and o(e, e¨) = 1. As for higher-dimensional cells, ﬁx k ≥ 1. Given T ∈ T k+1 and T ∈ T k such that T ⊆ T , we deﬁne o(T, T ) = 1 if T is outward-oriented compared with T and o(T, T ) = −1 if it is inward-oriented. For all T, T ∈ T not covered by these deﬁnitions we put o(T, T ) = 0. For each k, let C k (T ) denote the set of maps c : T k → R. Such maps associate a real number with each k-dimensional cell and are called k-cochains. The coboundary operator δ : C k (T ) → C k+1 (T ) is deﬁned by (δc)T = o(T, T )cT . T ∈T k

The space of k-cochains has a canonical basis indexed by T k . The coboundary operator is the operator whose canonical matrix is the incidence matrix o, indexed by T k+1 × T k . We remark that the coeﬃcients in the sum can be non-zero only when T ∈ ∂T ∩ T k .

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Lemma 5.4. Proof.

We have that δδ = 0 as a map C k (T ) → C k+2 (T ).

See, e.g., Christiansen (2009, Lemma 3.6).

In other words the family C • (T ) is a complex, called the cochain complex and represented by 0 → C 0 (T ) → C 1 (T ) → C 2 (T ) → · · · . When S is a smooth manifold we denote by Ωk (S) the space of smooth diﬀerential k-forms on S. Diﬀerential forms can be mapped to cochains as follows. Let S be a manifold and T a cellular complex on S. For each k we denote by ρk : Ωk (S) → C k (T ) the de Rham map, which is deﬁned by k u . ρ : u → T ∈T k

T

Proposition 5.5.

For each k the following diagram commutes:

ρk

ρk+1

/ C k+1 (T )

δ

C k (T ) Proof.

/ Ωk+1 (S)

d

Ωk (S)

This is an application of Stokes’ theorem.

Suppose T is a cellular complex equipped with an orientation (of the cells) and T is a cellular reﬁnement also equipped with an orientation (for instance the same complex but with diﬀerent orientations). For each cell T ∈ T k and each T ∈ T k deﬁne ι(T, T ) = ±1 if T ⊆ T and they have the same/diﬀerent orientation, and ι(T, T ) = 0 in all other cases. Proposition 5.6.

The map ι : C • (T ) → C • (T ), deﬁned by ι(T, T )uT , (ιu)T = T ∈T

is a morphism of complexes, meaning that ι and δ commute. Proof.

See, e.g., Christiansen (2009, Proposition 3.5).

We also remark that for a manifold S we have the following. Proposition 5.7.

The following diagram commutes: Ω(S)

w ww ww w w {w w ρ

C(T )

ι

GG GG ρ GG GG # / C(T )

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Element systems. For any cell T , we denote by Ωks,q (T ) the space of diﬀerential k-forms on T with Ws,q (T ) Sobolev regularity, and put Ωkq (T ) = Ωk0,q (T ). Fix q ∈ [1, +∞[ and deﬁne X k (T ) = {u ∈ Ωkq (T ) : du ∈ Ωkq (T )}. When i : T → T is an inclusion of cells and u is a smooth enough form on T we denote by u|T = i u the pullback of u to T . Thus we restrict to the subcell and forget about the action of u on vectors not tangent to it. In the topology (5.1), restrictions to subcells T of codimension one are well-deﬁned, for instance as elements of Ωk−1,q (T ). When T is a cell in a given cellular complex T we may therefore set ˆ k (T ) = {u ∈ X k (T ) : ∀T ∈ T X

T ⊆ T ⇒ u|T ∈ X k (T )}.

Definition 5.8. Suppose T is a cellular complex. For each k ∈ N and each ˆ k (T ) called a diﬀerential T ∈ T we suppose we are given a space Ak (T ) ⊆ X k-element on T . We suppose that the exterior derivative induces maps d : Ak (T ) → Ak+1 (T ) and that if i : T ⊆ T is an inclusion of cells, pullback induces a map i : Ak (T ) → Ak (T ). Such a family of elements is called an element system. A diﬀerential element is said to be ﬁnite if it is ﬁnite-dimensional. A ﬁnite element system is an element system in which all the elements are ﬁnite. ˆ • (•) themselves deﬁne an element system. It Example 5.9. The spaces X is far from ﬁnite. Example 5.10. Let U be an open subset of a vector space V . We denote by Pp (U ) the space of real polynomials of degree at most p on U . For k ≥ 1 the space of alternating maps V k → R is denoted Ak (V ). The space of diﬀerential k-forms on U , which are polynomial of degree at most p, is denoted PAkp (U ). We identify PAkp (U ) = Pp (U ) ⊗ Ak (V )

and

PA0p (U ) = Pp (U ).

Choose a cellular complex where all cells are ﬂat. Choose a function π : T × N → N and deﬁne Ak (T ) = PAkπ(T,k) (T ). One gets a ﬁnite element system when the following conditions are satisﬁed: T ⊆ T ⇒ π(T , k) ≥ π(T, k)

and

π(T, k + 1) ≥ π(T, k) − 1.

Example 5.11. Denote the Koszul operator on vector spaces by κ. It is the contraction of diﬀerential forms by the identity, considered as a vector ﬁeld: (κu)x (ξ1 , . . . , ξk ) = ux (x, ξ1 , . . . , ξk ).

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Alternatively one can use the Poincar´e operator associated with the canonical homotopy from the identity to the null-map. Let T be a simplicial complex. Deﬁne, for non-zero p ∈ N, k+1 k Λkp (T ) = {u ∈ PAkp (T ) : κu ∈ PAk−1 p (T )} = PAp−1 (T ) + κPAp−1 (T ).

For ﬁxed p we call this the trimmed polynomial ﬁnite element system of order p. The case p = 1 corresponds to constructs in Weil (1952) and Whitney (1957). Arbitrary order elements were introduced in N´ed´elec (1980) for vector ﬁelds in R3 . In Hiptmair (1999) these spaces were extended to diﬀerential forms. The correspondence between lowest-order mixed ﬁnite elements and Whitney forms was pointed out in Bossavit (1988). See Arnold et al. (2006b) for a comprehensive review. It was usual to start the indexing at p = 0 but, as remarked in the preprint of Christiansen (2007), the advantage of letting the lowest order be p = 1 is that the wedge product induces maps: +k1 ∧ : Λkp00 (T ) × Λkp11 (T ) → Λkp00 +p (T ). 1

See also Arnold et al. (2006b, p. 34). In words, the wedge product respects the grading in k and the ﬁltering in p. This observation was useful in the implementation of a scheme for the Yang–Mills equation (Christiansen and Winther 2006). The ﬁrst example of a ﬁnite element system yields quite useless Galerkin spaces in general, whereas the second one yields good ones. We shall elaborate on this in what follows, starting by deﬁning what the Galerkin space associated with a ﬁnite element system is. For any subcomplex T of T we deﬁne Ak (T ) as follows: k k A (T ) = u ∈ A (T ) : ∀T, T ∈ T T ⊆ T ⇒ uT |T = uT . T ∈T

Elements of Ak (T ) may be regarded as diﬀerential forms deﬁned piecewise, which are continuous across interfaces between cells, in the sense of equal pullbacks. For a cell T its collection of subcells is the cellular complex T˜. Applied to this case, the above deﬁnition gives a space canonically isomorphic to Ak (T ). We can identify Ak (T˜) = Ak (T ). An FE system over a cellular complex is an inverse system of complexes: for an inclusion of cells there is a corresponding restriction operator. The space A• (T ) deﬁned above is an inverse limit of this system and is determined by this property up to unique isomorphism. We also point out that this kind of construction, involving glueing of polynomial diﬀerential forms on cellular complexes, has been used to study homotopy theory (Griﬃths and Morgan 1981).

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Of particular importance is the application of the above construction to the boundary ∂T of a cell T , considered as a cellular complex consisting of all subcells of T except T itself. Considering ∂T as a cellular complex (not only a subset of T ), we denote the constructed space by Ak (∂T ). If i : ∂T → T denotes the inclusion map, the pullback by i deﬁnes a map i : Ak (T ) → Ak (∂T ), which we denote by ∂ and call restriction. Conventions. In the following, the arrows starting or ending at 0 are the only possible ones. Arrows starting at R are, unless otherwise speciﬁed, the maps taking a value to the corresponding constant function. Arrows ending at R are integration of forms of maximal degree. Other unspeciﬁed arrows are instances of the exterior derivative. Consider now the following conditions on an element system A on a cellular complex T . • Extensions. For each T ∈ T and k ∈ N, restriction ∂ : Ak (T ) → Ak (∂T ) is onto. • Exactness. The following sequence is exact for each T : 0 → R → A0 (T ) → A1 (T ) → · · · → Adim T (T ) → 0. The ﬁrst condition can be written symbolically as

∂Ak (T )

=

(5.3)

Ak (∂T ).

Definition 5.12. We will say that an element system admits extensions if the ﬁrst property holds, is locally exact if the second condition holds and is compatible if both hold. Given a ﬁnite element system A, we say that its points carry reals if, for each point T ∈ T 0 , A0 (T ) contains the constant maps T → R (so that A0 (T ) = RT ≈ R). Proposition 5.13. If A admits extensions and its points carry reals, then for each cell T , Adim T (T ) contains a form with non-zero integral. Proof.

By induction on the dimension of the cell, using Stokes’ theorem.

Notation. We let Ak0 (T ) denote the kernel of ∂ : Ak (T ) → Ak (∂T ). Proposition 5.14.

We have dim Ak (T ) ≤

dim Ak0 (T ),

T ∈T

with equality when the ﬁnite element system admits extensions. Proof. For a given m ≥ 0, let T (m) be the m-skeleton of T . We have a sequence: 0→ Ak0 (T ) → Ak (T (m) ) → Ak (T (m−1) ) → 0. T ∈T m

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The second arrow is bijective onto the kernel of the third. If all cells of dimension m admit extensions, the whole sequence is exact. The proposition follows from applying these remarks for all m. Combining Propositions 5.13 and 5.14, we get the following. Corollary 5.15. If A admits extensions and its points carry reals, then Ak (T ) has dimension at least the number of k-dimensional subcells of T . Proposition 5.16. When the element system is compatible, the de Rham map ρ• : A• (T ) → C • (T ) induces isomorphisms in cohomology. Proof. We use induction on the dimension of T . In dimension 0 it is clear. Suppose now that m ≥ 1, and that we have proved the theorem when dim T < m. Suppose that T has dimension m. Note that the de Rham map gives isomorphisms in cohomology on cells, A• (T ) → C • (T˜), since both complexes are acyclic. Denote by U the (m − 1)-skeleton of T . Consider the diagram: • • A (U ) A (T ) A• (∂T ) • / / / A (T ) 0 T ∈T m

0

/ C (T ) •

• / C (U )

T ∈T m

T ∈T m

C • (T˜)

C • (∂T ) / T ∈T m

/0

/0

The vertical maps are de Rham maps. The second horizontal arrow consists of restricting to the summands whereas the third one consists of restricting and comparing, as in the Mayer–Vietoris sequence. Both rows are exact sequences of complexes, the diagram commutes, and the last two vertical arrows induce isomorphisms in cohomology by the induction hypothesis. Write the long exact sequences of cohomology groups associated with both rows (e.g., Christiansen (2009, Theorem 3.1)) and connect them with the induced morphisms. Applying the ﬁve lemma (e.g., Christiansen (2009, Lemma 3.2)) gives the result for T . Proposition 5.17. For an element system with extensions the exactness of (5.3) on each T ∈ T is equivalent to the combination of the following two conditions. • For each T ∈ T , A0 (T ) contains the constant functions. • For each T ∈ T , the following sequence (with boundary condition) is exact: 0 → A00 (T ) → A10 (T ) → · · · → Adim T (T ) → R → 0. (5.4)

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Proof. When the extension property is satisﬁed on all cells, both versions (with and without boundary condition) of the cohomological condition guarantee that for each T , Adim T (T ) contains a form with non-zero integral, by Proposition 5.13. When the spaces A0 (T ) all contain the constant functions, we may consider the following diagrams, where T is a cell of dimension m and Am − (T ) denotes the space of forms with zero integral: 0

/R O

/ A0 (∂T ) O

/ ··· O

/ Am−1 (∂T ) O

/R O

/0

0

/R O

/ A0 (T ) O

/ ··· O

/ Am−1 (T ) O

/ Am (T ) O

/0

0

/ A0 (T )

/ ···

/ Am−1 (T )

/ Am (T ) −

/0

0

0

(5.5) The columns, extended by 0, are exact if and only if the extension property holds on T and Am (T ) contains a form of non-zero integral. In this case, if one row is exact, the two other rows are either both exact or both inexact. We now prove the stated equivalence. (i) If compatibility holds then in (5.5) the extended columns are exact, as well as the ﬁrst and second row, so also the third. Hence (5.4) is exact. (ii) Suppose now exactness of (5.4) holds for each T . Choose m ≥ 1 and suppose that we have proved exactness of (5.3) for cells of dimension up to m − 1. Let T be a cell of dimension m. In (5.5) the extended columns are exact. Apply Proposition 5.16 to the boundary of T to get exactness of the ﬁrst row. The third row is exact by hypothesis, and we deduce exactness of the second. The induction, whose initialization is trivial, is complete. ˆ is a compatible element system. Corollary 5.18. X Tensor products. Suppose we have two manifolds M and N , equipped with cellular complexes U and V. We suppose we have diﬀerential elements Ak (U ) for U ∈ U and B k (V ) for V ∈ V, both forming systems as deﬁned above. Let U × V denote the product cellular complex on M × N , whose cells are all those of the form U × V for U ∈ U and V ∈ V. Recall that the tensor product of diﬀerential forms u on U and v on V , is the form on U × V deﬁned as the wedge product of their pullbacks by the respective canonical projections pU : U × V → U and pV : U × V → V . In symbols we can write u ⊗ v = (pU u) ∧ (pV v). We equip U × V with elements C • (U × V ) = A• (U ) ⊗ B • (V ).

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Explicitly we put C k (U × V ) =

Al (U ) ⊗ B k−l (V ).

l

This deﬁnes an element system C, called the tensor product of A and B. Proposition 5.19.

We have C0• (U × V ) = A•0 (U ) ⊗ B0• (V ).

Proof.

See Christiansen (2009, Lemma 3.10).

Proposition 5.20.

When A and B admit extensions we have C • (U × V) = A• (U) ⊗ B • (V).

Proof. The right-hand side is included in the left-hand side. Moreover, by Proposition 5.14, dim C(U × V) ≤ U,V dim C0 (U × V ). On the other hand Proposition 5.19 gives U,V dim C0 (U × V ) = U,V dim A0 (U ) ⊗ B0 (V ) = U,V dim A0 (U ) dim B0 (V ) = U dim A0 (U ) V dim B0 (V ) = dim A(U ) dim B(V). This completes the proof. Proposition 5.21. product. Proof.

If A and B admit extensions, so does their tensor

Consider cells U ∈ U and V ∈ V. Note that (∂U × V ) ∪ (U × ∂V ) = ∂(U × V ), (∂U × V ) ∩ (U × ∂V ) = ∂U × ∂V.

The Mayer–Vietoris principle gives an exact sequence, 0 → C(∂(U × V )) → C(∂U × V ) ⊕ C(U × ∂V ) → C(∂U × ∂V ) → 0, where the second and third mappings are w → w|∂U ×V ⊕ w|U ×∂V , u ⊕ v → u|∂U ×∂V − v|∂U ×∂V . It follows that dim C(∂(U × V )) = dim C(∂U × V ) + dim C(U × ∂V ) − dim C(∂U × ∂V ).

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Applying Proposition 5.20 three times, we get dim C(∂(U × V )) = dim A(∂U ) dim B(V ) + dim A(U ) dim B(∂V ) − dim A(∂U ) dim B(∂V ) = dim A(U ) dim B(V ) − dim A0 (U ) dim B0 (V ) = dim C(U × V ) − dim C0 (U × V ) = dim ∂C(U × V ). Therefore we obtain ∂C(U × V ) = C(∂(U × V )), as claimed. Proposition 5.22. product.

If A and B are locally exact, then so is their tensor

Proof. This follows from the Kunneth theorem (e.g., Christiansen (2009, Theorem 3.2)). Nesting. Suppose T is a cellular complex and that (Ξ, #) is an ordered set. Suppose that, for each parameter ξ ∈ Ξ, an FE system A[ξ] on T has been chosen. We assume that if ξ # ξ then A[ξ]k (T ) ⊆ A[ξ ]k (T ). Choose now a parameter function π : T → Ξ, which is order-preserving in the sense that if T ⊆ T then π(T ) # π(T ). Deﬁne an FE system A[π] by A[π]k (T ) = {u ∈ A[π(T )]k (T ) : ∀T ∈ T

T ⊆ T ⇒ u|T ∈ A[π(T )]k (T )}.

Proposition 5.23. If, for each ξ ∈ Ξ, the system A[ξ] is compatible, then the constructed system A[π] is too. Proof. Note that if u ∈ A[π]k (∂T ) then u ∈ A[ξ(T )]k (∂T ) so it can be extended to an element u ∈ A[ξ(T )]k (T ). This element is in A[π]k (T ). Thus A[π]k (T ) admits extensions. We also have A[π]k0 (T ) = A[π(T )]k0 (T ), which gives local exactness thanks to Proposition 5.17. Example 5.24. For a simplicial complex T , let A[p] denote the trimmed ﬁnite element system of order p. We model variable orders of approximation by a function π : T → N∗ such that when T ⊆ T we have π(T ) ≤ π(T ). The above construction deﬁnes an FE system A[π], of the type used for hp-methods (Demkowicz, Kurtz, Pardo, Paszy´ nski, Rachowicz and Zdunek 2008). We will check later that the trimmed systems for ﬁxed order p are compatible, and then the above result gives compatibility of the variableorder system deﬁned by π. Locally harmonic forms. Let T be a cell where, for each k, Ak (T ) is equipped with a scalar product a. Orthogonality with respect to a will be denoted

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by ⊥. We say that a k-form u on T is A-harmonic if du ⊥ dAk0 (T )

and

u ⊥ dAk−1 0 (T ).

(5.6)

One can, for instance, take a to be the L2 scalar product on diﬀerential forms, associated with some Riemannian metric. Denote by d the formal adjoint of d with respect to this scalar product. The continuous analogue of the above condition (5.6) is d du = 0

and

d u = 0.

(5.7)

From the other point of view, (5.6) is the Galerkin variant of (5.7). Proposition 5.25. Let A be a ﬁnite element system where each Ak (T ) is equipped with a scalar product a. Suppose T is a cell such that (5.4) is exact. Put m = dim T . m • For

each α ∈ R, there is a unique A-harmonic u ∈ A (T ) such that T u = α.

• For k < m, any u ∈ Ak (∂T ) admitting an extension in Ak (T ), has a unique A-harmonic extension in Ak (T ). Let A be a ﬁnite element system on T . Deﬁne a ﬁnite element system ˚ A by ˚k (T ) = {u ∈ Ak (T ) : ∀T ∈ T A

T ⊆ T ⇒ u|T is A-harmonic}.

˚ is the subsystem of locally harmonic forms. We say that A ˚ is a compatProposition 5.26. If A is a compatible FE system then A k k ˚ ible FE system such that the de Rham map ρ : A (T ) → C k (T ) is an isomorphism. Proof.

This was essentially proved in Christiansen (2008a).

This construction generalizes Kuznetsov and Repin (2005), in which divconforming ﬁnite elements are deﬁned on polyhedra in R3 . Since C k (T ) has a canonical basis, the de Rham map determines a corresponding canonical ˚k (T ). Its elements can be constructed by recursive harmonic basis of A extension. Example 5.27. On simplexes, lowest-order trimmed polynomial diﬀerential forms are locally harmonic in the sense of (5.7), with respect to the L2 -product associated with any piecewise constant Euclidean metric (Christiansen 2008a). Example 5.28. Locally harmonic forms can be used to deﬁne ﬁnite element spaces on the dual cellular complex of a given simplicial one (Buﬀa and Christiansen 2007). Choose a simplicial reﬁnement of the dual mesh,

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for instance the barycentric reﬁnement of the primal mesh. Consider Whitney forms on this reﬁnement as a compatible ﬁnite element system on the dual mesh. Then take the subsystem of locally harmonic forms. In space dimension d, this provides a space of k-forms with the same dimension as the space of (d−k)-forms on the primal mesh. Duality in the sense of an inf-sup condition was proved in Buﬀa and Christiansen (2007) with applications to the preconditioning of integral operators appearing in electromagnetics. Duality methods are quite common in ﬁnite volume settings; for a recent development see Andreianov, Bendahmane and Karlsen (2010). Example 5.29. For a given ﬁne mesh one can agglomerate elements into a coarser cellular mesh. The ﬁnite element system on the ﬁne mesh then provides a ﬁnite element system on the coarse one, with identical function spaces. If the former system is compatible so is the latter. Associated with the latter, we can consider the subsystem of locally harmonic forms. This procedure can be applied recursively: at each level one can consider the locally harmonic forms of the ﬁner level. This yields a multilevel analysis which can be used for multigrid preconditioning (Pasciak and Vassilevski 2008). Here is a third application, again involving the dual ﬁnite elements. Example 5.30. written as

Recall that the incompressible Euler equation can be

u˙ + div(u ⊗ u) + grad p = 0

and

div u = 0.

Here, u is a vector ﬁeld with time derivative u. ˙ One uses div-conforming Raviart–Thomas elements for u: let Xh denote this space. One uses a weak formulation with locally harmonic curl-conforming elements on the dual grid as test functions: let Yh denote this space. The L2 duality and its extensions to Sobolev spaces are denoted ·, ·. It is invertible on Xh × Yh , at least in dimension 2, as proved in Buﬀa and Christiansen (2007). The semi-discrete problem reads as follows. Find a time-dependent uh ∈ Xh such that, for all vh ∈ Yh which are orthogonal to the subspace of Yn of curl-free elements, we have u˙h , vh + div(uh ⊗ uh ), vh = 0. To see that the second bracket is well-deﬁned, note that div(uh ⊗ uh ) and div div(uh ⊗ uh ) both have W−1,q Sobolev regularity for all q < 2. On the other hand vh and curl vh have W0,q -regularity, for q > 2. Analogous spaces, with regularity expressed by a second-order operator, have been used for Regge calculus (Christiansen 2008b) and elasticity (Sch¨oberl and Sinwel 2007).

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In computational ﬂuid dynamics it is often important to include some form of upwinding in the numerical method to obtain stability. A model problem for this situation is the equation ∆u + V · grad u = f,

(5.8)

where the ﬁeld u ∈ H1 (S) satisﬁes homogeneous Dirichlet boundary condition, f is a given forcing term, whereas V is a given ﬂow ﬁeld, which we take to be divergence-free. One is interested in the asymptotic behaviour for small positive (viscosity). Such problems can for instance be solved with Petrov–Galerkin methods. Consider a cellular complex and a large compatible ﬁnite element system on it, obtained for instance by reﬁnement. We let one space (say the trial space) consist of locally harmonic forms for the standard L2 -product. For the other space (say the test space) we use locally harmonic forms for a weighted L2 -product. To motivate our choice of weight we introduce some more notions of diﬀerential geometry. Given a 1-form α we deﬁne the covariant exterior derivative: dα : u → du + α ∧ u, These operators do not form a complex, but we have dα dα u = (dα) ∧ u. In gauge theory the term dα is called curvature. Supposing that α = dβ for a function β, we have dα = exp(−β) d exp(β)u. One says that u → exp(β)u is a gauge transformation. We suppose the domain S is equipped with a Riemannian metric. It provides in particular an L2 scalar product on diﬀerential forms. We denote by dα the formal adjoint of d. When α = exp(−β), we have dα u = exp(β)d exp(−β). A natural generalization of (5.7) is dα du = 0

and

dα u = 0,

d exp(−β)du = 0

and

d exp(−β)u = 0.

which can be written as

Example 5.31. To address (5.8) deﬁne the 1-form α by α(ξ) = −−1 V · ξ, and note that (5.8) can be rewritten as dα du = −−1 f.

(5.9)

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Given a cellular complex T on S, with ﬂat cells, and a large compatible FE system A, we construct two spaces of locally harmonic forms, distinguished by the choice of scalar product a. For one (the trial space) take a to be the L2 scalar product. For the other (the test space), we choose for each T a constant approximation αT of the pullback of α to T . Let βT be the aﬃne function with zero mean on T such that dβT = αT . For the trial spaces, use the scalar product deﬁned on a cell T by a(u, v) = exp(−βT )u · v (5.10) T

to deﬁne the locally harmonic functions. If v is a constant diﬀerential form, u = exp(βT )v satisﬁes the equations (5.9). The canonical basis of the test space will then be upwinded or downwinded (depending on the sign in front of βT in (5.10)), compared with the canonical basis of the trial space. Example 5.32. equation:

Similar notions can be used to address the Helmholtz ∆u + k 2 u = 0.

(5.11)

We wish to construct a compatible FE system over C, which contains a certain number of plane waves: uξ : x → exp(iξ · x). To contain just one of them we remark that for any (ﬂat) cell T , if ξT is the tangent component of ξ on T we have, on T , ∆T uξ |T − iξT · gradT uξ |T = 0. Connecting with the previous example, on a cell T we let βT be the aﬃne real function with zero mean and gradient ξT on T . Deﬁne a(u, v) = exp(−iβT )u · v. T

Here, extra care must be taken because this bilinear form is not positive deﬁnite and will in fact be degenerate at interior resonances of the cell. Away from them, the locally harmonic forms for the inﬁnite-dimensional ˆ (with q = 2) are well behaved, in the sense of satisfying element system X Proposition 5.26, and contain the plane wave uξ . More generally, suppose one wants to construct an FE system containing a good approximation to a particular solution of (5.11). To the extent that the solution can be locally approximated by a plane wave, the ﬁnite element system will contain a good approximation of it, for a choice of a family of functions βT , one for each cell T , to be determined (maybe adaptively).

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5.2. Interpolators Mirrors and interpolators. The notion of a mirror system formalizes that of degrees of freedom, with particular emphasis on their geometric location. Definition 5.33. A mirror system is a choice, for each k and T , of a ˆ k (T ) , called a k-mirror on T . subspace Z k (T ) of X ˆ k (T ) then gives a linear form ·, u on Z k (T ), which Any k-form u in X we call the mirror image of u. For a global k-form u the mirror images can be collected into a single object. We deﬁne Φk u = ·, u|T T ∈T ∈ Z k (T ) , where, for any subcomplex T of T , we deﬁne a (global) mirror Z k (T ) by Z k (T ) = Z k (T ). T ∈T

We say that a mirror system is faithful to an element system A if, for any subcomplex T , restricting Φ to T determines an isomorphism: Φk (T ) : Ak (T ) → Z k (T ) . Example 5.34. The canonical mirror system for the trimmed polynomial FE system of order p is the following, where dim T = m:

Z k (T ) = {u → T v ∧ u : v ∈ PAm−k p−m+k−1 (T )}. It follows from results in Arnold et al. (2006b) that it is faithful. Proposition 5.35. When A admits extensions, a given mirror system Z is faithful if and only if the duality product on Z k (T ) × Ak0 (T ) is invertible for each k and T . Proof. (i) Suppose Z is faithful. By induction on dimension, dim Z k (T ) = dim Ak0 (T ). Moreover, Φk (T ) induces an injection Ak0 (T ) → Z k (T ) . Thus duality on Z k (T ) × Ak0 (T ) is invertible. (ii) Suppose duality on Z k (T ) × Ak0 (T ) is invertible for all k and T . Then dim Ak (T ) = T ∈T dim Ak0 (T ) = dim Z k (T ) . Moreover, Φk (T ) is injective. Indeed, if Φk u = 0 then for T ∈ T , u|T is proved to be 0 by starting with cells T of dimension k, and incrementing cell dimension inductively using u|∂T = 0. Definition 5.36. For a ﬁnite element system A, an interpolator is a colˆ k (T ) → Ak (T ), one for each k ∈ N lection of projection operators I k (T ) : X and T ∈ T , which commute with restrictions to subcells.

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One can then denote it simply with I • and extend it unambiguously to any subcomplex T of T . Any faithful mirror system deﬁnes an interpolator by Φk I k u = Φk u. We call this the interpolator associated with the mirror system. Proposition 5.37.

The following are equivalent.

• A admits extensions. • A has a faithful mirror system. • A can be equipped with an interpolator. Proof. (i) Suppose A admits extensions. For each k and T , choose a closed ˆ k (T ) and let Z k (T ) be its annihilator. The supplementary of Ak0 (T ) in X duality product on Z k (T ) × Ak0 (T ) is then invertible for each k and T , so that Z is faithful to A. (ii) As already stated, any faithful mirror system deﬁnes an interpolator. ˆ k (T ) and (iii) Suppose A has an interpolator. If u ∈ Ak (∂T ) extend it in X k interpolate it to get an extension in A (T ). Example 5.38. In particular, the trimmed polynomial FE system of order p is compatible. As remarked in Christiansen (2010), it is minimal among compatible ﬁnite element systems containing polynomial diﬀerential forms of order p − 1. For a given element system A, admitting extensions, it will be useful to construct extension operators Ak (∂T ) → Ak (T ), i.e., linear left inverses of the restriction. One also remarks that a faithful mirror system determines a particular extension. Namely, to u ∈ Ak (∂T ) one associates the unique v ∈ Ak (T ) extending u and such that, for all l ∈ Z k (T ), l(v) = 0. Proposition 5.39. Let A be an element system on T admitting extensions. Let M denote the set of mirror systems that are faithful to A, let I be the set of interpolators onto A and let E be the set of extensions in A. The natural map M → I × E is bijective. Proof. For a given interpolator I ∈ I and E ∈ M, deﬁne a mirror system z(I, E) as follows. For k ∈ N and T ∈ T consider the map ˆ k (T ) → X ˆ k (T ). QkT = (id −E∂) ◦ I : X It is a projector with range Ak0 (T ). Let z(I, E)k (T ) denote the annihilator of its kernel. In other words l ∈ z(I, E)k (T ) if and only if ˆ k (T ) ∀u ∈ X

Iu = EI∂u ⇒ l(u) = 0.

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We claim that z inverts the given map a : M → I × E. (i) Given (I, E), we check that the interpolator and extension deﬁned by z(I, E) are I and E. ˆ k (T ). We have I(u − Iu) = 0 and EI∂(u − Iu) = 0, hence • Pick u ∈ X l(u − Iu) = 0. The interpolator deduced from z(I, E) is thus I. • Pick u ∈ Ak (∂T ). We have IEu = EI∂Eu, hence for all l ∈ z(I, E)k (T ) we have l(u) = 0. The extension deduced from z(I, E) is thus E. (ii) Given a faithful mirror system Z with associated interpolators I and E, we check that z(I, E) = Z. ˆ k (T ) is such that Iu = EI∂u then l(u) = Pick l ∈ Z k (T ). If u ∈ X l(Iu) = l(EI∂u) = 0. Hence l ∈ z(I, E)k (T ). On the other hand z(I, E) is also faithful to A by Proposition 5.35. The inclusion Z k (T ) ⊆ z(I, E)k (T ) then implies equality. ˆ k (T ) Example 5.40. Let A be a ﬁnite element system. Equip each X k with a continuous bilinear form a which is non-degenerate on A0 (T ) (e.g., ˆ k (T ) is continuously embedded in a Hilbert space). One can deﬁne a X mirror system by Z k (T ) = {a(·, v) : v ∈ Ak0 (T )}. When the FE system admits extensions, the associated interpolator can be interpreted as a recursive a-projection, starting from cells of minimal dimension, and continuing by incrementing dimension at each step, projecting with respect to a with given boundary conditions. Commuting interpolators. It is of interest to construct interpolators that commute with the exterior derivative. When T ⊆ T , restriction maps ˆ k (T ) → X ˆ k (T ), so that a k-mirror Z k (T ) on T can also be considered X as a k-mirror on T . If the mirror system is faithful it must be in direct sum with Z k (T ). Proposition 5.41. An interpolator commutes with the exterior derivative if and only if its mirror system satisﬁes ∀T ∈ T ∀l ∈ Z k (T )

l ◦ d ∈ Z k−1 (T˜).

(5.12)

ˆ k−1 (T ). Suppose ﬁrst that Φk−1 u = 0. Then, Proof. (If ) Pick u ∈ X k ˜ for all l ∈ Z (T ), l(du) = 0, hence Φk (du) = 0. In the general case, since Φk−1 (u − Φk−1 u) = 0, we deduce Φk (du − dΦk−1 u) = 0, so that Φk du = dΦk−1 u. ˆ k−1 (T ) such (Only if ) Suppose l ∈ Z k (T ) and l ◦ d ∈ Z k−1 (T˜). Pick u ∈ X k−1 that l(du) = 0, but for all l ∈ Z (T˜), l (u) = 0. Then I k du = 0 but I k−1 u = 0 (so dI k−1 u = 0).

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Example 5.42. The canonical mirror system of trimmed polynomials of order p yields a commuting interpolator. Suppose that Z is a mirror system on T such that (5.12) holds. Then we ˆ : l → l ◦ d from Z k (T ) to Z k−1 (T ). Denote by δ have a well-deﬁned map d its adjoint, which maps from Z k−1 (T ) to Z k (T ) . Remark 5.43. The following diagram commutes: ˆ k−1 (T ) X

Φk−1

Z k−1 (T ) Proof.

d

δ

/X ˆ k (T )

Φk

/ Z k (T )

ˆ k−1 (T ). We have Pick l ∈ Z k (T ) and u ∈ X ˆ = (Φu)(l ◦ d) = l(du) = (Φdu)(l). (δΦu)(l) = (Φu)(dl)

This concludes the proof. ˆ k (T ) with a continuous scalar product Proposition 5.44. Equip each X a. For any compatible ﬁnite element system A, the following is a faithful mirror system yielding a commuting interpolator. For k = dim T ,

Z k (T ) = {a(·, v) : v ∈ dAk−1 0 (T )} + {R ·}. For k < dim T , k Z k (T ) = {a(·, v) : v ∈ dAk−1 0 (T )} + {a(d·, v) : v ∈ dA0 (T )}.

This is the natural generalization, to the adopted setting, of projectionbased interpolation, as deﬁned in Demkowicz and Babuˇska (2003) and Demkowicz and Buﬀa (2005). When the scalar products a are all the L2 -product on forms, we call it harmonic interpolation. Suppose U is a cellular complex on M and V is a cellular complex on N giving rise to the product complex U × V on M × N . If Z is a mirror system on U and Y is a mirror system on V, then Z ⊗ Y is a mirror system on U × V deﬁned by (Z ⊗ Y)k (U × V ) = Z l (U ) ⊗ Y(V )k−l . l

Proposition 5.45. faithful.

The tensor product of two faithful mirror systems is

Proof. We have three FE systems, A, B and A ⊗ B, all admitting extensions. Moreover, by Proposition 5.19, Al0 (U ) ⊗ B0k−l (V ). (A ⊗ B)k0 (U × V ) = l

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Now apply Proposition 5.35. Extension–projection interpolators. It will also be of interest to construct extension operators which commute with the exterior derivative. More precisely, for a cell of dimension m a commuting extension operator is a family of operators E k : Ak (∂T ) → Ak (T ) for 0 ≤ k ≤ m − 1, together with a map E m : R → Am (T ) such that the following diagram commutes: 0

0

/R

/ A0 (∂T )

/R

/ ···

/ Am−1 (∂T )

/ ···

E0

/ A0 (T )

E m−1

/ Am−1 (T )

/R

/0

Em

/ Am (T )

/0

(5.13) When has an element with non-zero integral an such that the above diagram commutes, is uniquely determined by E m−1 and exists if and only if E m−1 sends elements with zero integral to closed forms. Am−1 (∂T )

Em

Proposition 5.46. Let A be a ﬁnite element system where each Ak (T ) is equipped with a scalar product a. Suppose T is a cell such that (5.4) is exact. Put m = dim T . If T admits extensions, the harmonic extension operators deﬁned by Proposition 5.25 commute in the sense of diagram (5.13). We suppose that for each cell T , extension operators E : A• (∂T ) → A (T ) and projections P : X • (T ) → A• (T ) have been deﬁned. Then E and P uniquely determine an interpolator J as follows. One constructs J inductively. The initialization on cells of dimension 0 (points) is trivial. Now let T be a cell and suppose that J has been constructed for all cells on ˆ k (T ) with k < dim T , its boundary. On T we deﬁne, for u ∈ X •

Ju = EJ∂u + (id −E∂)P u = P u + E(J∂u − ∂P u), and for k = dim T we simply put Ju = P u. The only thing to check is that J commutes with restriction from T to cells on the boundary, which is trivial. We call J the associated extension– projection (EP) interpolator. Let T be cell of dimension m. endomorphism F of X m (T )

We say that an m preserves integrals if F u = u for all u ∈ X (T ). An interpolator is said to preserve integrals if it preserves integrals on all cells. For an interpolator I, this is equivalent to commutation of the following diagram, involving

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de Rham maps: / A• (T ) vv vv v vv ρ zv v

I

ˆ • (T ) X

II II II ρ III $

C • (T )

Furthermore, we would like the interpolator to commute with the exterior derivative. Proposition 5.47. Suppose that the projectors P : X • (T ) → A• (T ) commute with the exterior derivative, that the extensions E commute in the sense of diagram (5.13) and that P preserves integrals. Then the associated EP interpolator commutes with the exterior derivative and preserves integrals. Proof. We use induction on the dimension of cells. Let T be a cell of dimension m. ˆ k (T ) with k ≤ m − 2, the commutation dJu = Jdu follows If u ∈ X immediately from the commutation of P , E and ∂, as well as J on ∂T (which follows from the induction hypothesis). ˆ m−1 (T ) we have For u ∈ X dJu = dP u + dE(J∂u − ∂P u). But we have

(J∂u − ∂P u) =

∂T

∂T

∂u −

∂T

∂u −

dP u = T

∂u −

=

∂T

P du, T

du = 0. T

The ﬁrst thing we used is that J preserves integrals on the boundary (which is an induction hypothesis). Hence, by (5.13), dE(J∂u − ∂P u) = 0. Thus dJu = dP u = P du = Jdu. That J preserves integrals simply from the fact that, for

on T follows

u ∈ X m (T ), we have Ju = P u = u. Compared with the use of mirror systems, the advantage of deﬁning an interpolator from extensions and projections is that approximation properties of the interpolator follow directly from estimates on the extensions and projections. In the following we denote the Lq (U )-norm simply by · U .

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Proposition 5.48. Let (Tn ) be a sequence of cellular complexes, each ˜ k (T ) equipped with a compatible FE system. Suppose that for each T , X is equipped with a densely deﬁned seminorm |·|T . We also suppose that we have functions λn : Tn → R∗+ and τn : Tn → R∗+ and extensions and projections satisfying u − Pn uT # τn (T )|u|T , u − Pn u∂T # τn (T )λn (T )−1 |u|T , En uT # λn (T )u∂T . We suppose that for T, T ∈ Tn , if T ⊆ T then λn (T ) λn (T ) and τn (T ) τn (T ). Then the associated EP interpolator satisﬁes u − In uT # τn (T ) λn (T )dim T −dim T |u|T , T ⊆T

where we sum over subcells T of T in Tn . We shall express this bound as order optimality. Proof. We use induction. Suppose T is a cell and that In is order-optimal ˆ k (T ) with k < dim T , we then have on its boundary. For u ∈ X u − In uT # u − Pn uT + λn (T )In ∂u − ∂Pn u∂T # u − Pn uT + λn (T )u − Pn u∂T + λn (T )u − In u∂T # τn (T )|u| + λn (T )u − In u∂T . This completes the proof. This proposition was designed with the p-version of the ﬁnite element method in mind. One can think of τn (T ) = p−1 and λn (T ) = p−1/q . The seminorms involved would correspond to Sobolev spaces, possibly weighted. Thus one would require extension operators whose Lq (∂T ) → Lq (T )-norm is of order p−1/q . The construction used to prove Proposition 3.3 in the preprint of Christiansen (2007) might be useful here. 5.3. Quasi-interpolators Let S be a domain in Rd . Let (Tn ) be a sequence of cellular complexes on S, each equipped with a compatible FE system A[n]. We deﬁne Xnk = A[n]k (Tn ). Definition 5.49.

Consider a sequence of maps Qkn : Lq (S) → Xnk .

• We say that they are stable if, for u ∈ Lq (S), Qkn uLq (S) # uLq (S) .

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• We say that they are order-optimal if, for u ∈ W,q (S), u − Qkn uLq (S) # τn uW,q (S) , where τn is the order of best approximation on Xnk in the W,q (S) → Lq (S)-norm. • We say that they are quasi-projections if, for some α < 1, we have for all u ∈ Xnk u − Qkn uLq (S) ≤ αuLq (S) . • We say that they commute if they commute with the exterior derivative. We shall construct stable quasi-interpolators of the form Q = IRE, where I is an interpolator, R is a regularization (smoother) approximating the identity and E is an extension operator (usually R requires values outside S). The following technique will be referred to as scaling. Lemma 5.50. For a cell T of diameter hT and barycentre bT , consider the scaling map σT : x → hT x + bT and let Tˆ be the pre-image of T by σT , called the reference cell. Let m be the dimension of T (0 ≤ m ≤ d). Let u be a k-multilinear form on T and let u ˆ = σT u be the pullback of u to Tˆ. Then we have −k+m/q

uLq (T ) = hT

ˆ uLq (Tˆ) .

We adopt the h-setting, in that the diﬀerential elements A[n]k (T ), when pulled back to reference domains Tˆ, belong to compact families: see Arnold et al. (2006b, Remark p. 64). This hypothesis excludes, for instance, methods where the polynomial degree is unbounded as n → ∞, and normally requires the cells to be shape-regular. Proposition 5.51. Suppose the ﬁnite elements A[n]k (T ) contain polynomials of degree − 1. A sequence of commuting interpolators In can be constructed to satisfy +(dim T −dim T )/q u − In uLq (T ) # hT ∇ uLq (T ) , (5.14) T ⊆T

where we sum over subcells T of T in Tn . We shall express this bound as order optimality.

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Proof. Choose commuting interpolators In , which when pulled back to the reference cell satisfy u − Iˆn uLq (Tˆ) # ∇ uLq (Tˆ ) . T ⊆T

Estimate (5.14) follows by scaling. Such interpolators can be constructed from mirror systems (e.g., harmonic interpolation), or from extensions and projections as in Proposition 5.48. Regularizer. We consider regularizing operators constructed as follows. We require a function ψ supported in the unit ball Bd and a function Φ:

Bd × Rd → Rd , (y, x) → Φy (x).

For any given y we denote by Φy the pullback by Φy : Rd → Rd . That is, for any k-multilinear form u, (Φy u)[x](ξ1 , . . . , ξk ) = u[Φy (x)](Dx Φy (x)ξ1 , . . . , Dx Φy (x)ξk ). We deﬁne

Ru = Bd

ψ(y)Φy u dy.

(5.15)

For the function ψ we choose one which is smooth, rotationally invariant, non-negative, with support in the unit ball and with integral 1. Later on, we will require convolution by ψ to preserve polynomials up to a certain degree (see (5.24)), but until then this hypothesis will be irrelevant. In what follows, when we integrate with respect to the variable y it will always be on the unit ball Bd , so we omit this from expressions such as (5.15). Given such a ψ we are interested in how properties of Φ reﬂect upon R. For this purpose we call R deﬁned by (5.15) the regularizer associated with Φ. To emphasize its dependence on Φ we sometimes denote the regularizer by R[Φ]. Formally, R[Φ] will commute with the exterior derivative, since pullbacks do. The minimal regularity we assume of Φ is to have continuous second derivatives. We also suppose that, for given x, y → Φy (x) is a diﬀeomorphism from Bd to its range. This is enough for the commutation to hold. In what follows we shall be a bit more careful about the regularizing eﬀects of R[Φ] for given Φ. Let B(x, δ) denote the ball with centre x and radius δ. For any subset U of Rd , its δ-neighbourhood is deﬁned by V δ (U ) = ∪{B(x, δ) : x ∈ U }. As a ﬁrst result we state the following.

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Proposition 5.52. For any δ > 0 and C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, and Dy Φy (x)−1 , Dx Φy (x) ≤ C, the associated regularizer satisﬁes (R[Φ]u)(x) ≤ C uL1 (V δ (x)) . Proof.

(5.16)

With x ﬁxed, the Jacobian of the map Bd % y → Φy (x) ∈ V δ (x)

has an inverse bounded by C. We are interested in estimates on derivatives of R[Φ]u when u is a k-form. We have (∇R[Φ]u)(x) = ψ(y)Dx (Φy u)[x] dy, (5.17) where we can substitute the expression Dx Φy u[x](ξ0 , . . . , ξk )

(5.18)

= Du[Φy (x)](Dx Φy (x)ξ0 , . . . , Dx Φy (x)ξk ) + ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2xx Φy (x)(ξ0 , ξi ), . . . , Dx Φy (x)ξk ). The purpose of the following lemma is to get an integral expression for ∇R[Φ]u not involving any derivatives of u. Essentially, in (5.17), derivatives acting on u are transferred to other terms under the integral sign, using integration by parts. Without the integral sign, this corresponds to identifying the derivatives of u as a ‘total’ divergence, up to expressions involving no derivatives of u. Lemma 5.53.

We have

Dx Φy u[x](ξ0 , . . . , ξk ) = ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2xx Φy (x)(ξ0 , ξi ), . . . , Dx Φy (x)ξk ) − ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2yx Φy (x)(Ξy (x)ξ0 , ξi ), . . . , Dx Φy (x)ξk ) + Dy Φy u[x](Ξy (x)ξ0 , ξ1 , . . . , ξk ), with Ξy (x) deﬁned by Ξy (x)ξ = Dy Φy (x)−1 Dx Φy (x)ξ.

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We also have ψ(y)Dy Φy u[x](Ξy (x)ξ0 , ξ1 , . . . , ξk ) = divy ψ(y)Ξy (x)ξ0 Φy u[x](ξ1 , . . . , ξk ) − divy ψ(y)Ξy (x) ξ0 Φy u[x](ξ1 , . . . , ξk ). The following proposition shows that the regularizer maps forms of L1 regularity to continuously diﬀerentiable forms, while being careful about the operator norm. Proposition 5.54. For any δ > 0 and C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, and

Dy Φy (x)−1 , Dx Φy (x), D2xx Φy (x)

!

D2yx Φy (x), Dy (Dy Φy (x)−1 )

≤ C,

the associated regularizer satisﬁes (∇R[Φ]u)(x) ≤ C uL1 (V δ (x)) .

(5.19)

Proof. The preceding lemma gives an expression for (∇R[Φ]u)(x), from which the claim follows. In the estimate (5.19) one has one order more of diﬀerentiation on the left-hand side. In the following proposition we consider an equal amount of diﬀerentiation on both sides. Proposition 5.55. Pick an integer ≥ 0, and δ > 0. For any C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, −1

Dy Φy (x)

≤ C,

(5.20) (5.21)

and Dx Φy (x), D2xx Φy (x), . . . , D+1 x...x Φy (x) ≤ C,

(5.22)

the associated regularizer satisﬁes (∇ R[Φ]u)(x) ≤ C uW,1 (V δ (x)) . Proof. For = 0 one uses the change of variable formula in the deﬁnition (5.15), the Jacobian being taken care of by estimate (5.21). The case = 1 follows from expression (5.18). Diﬀerentiating this expression several times gives the claimed result for general (an expression for the diﬀerential of order of a pullback will be given later).

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From now on we suppose Φ has the form Φy (x) = x + φ(x)y,

(5.23)

for some function φ : S → R. Most of the discussion would work with matrix-valued maps φ : S → Rd×d , which could be important for anisotropic meshes. But for simplicity we do not consider anisotropic meshes here, and thus scalar φ are suﬃcient. We also assume that ψ satisﬁes ψ(y)f (y) dy = f (0), (5.24) for all polynomials f of degree at most p + d, for some integer p ≥ 0. The purpose of these hypotheses is to make the regularizer preserve polynomials of degree up to p. To see this, ﬁrst note that property (5.24) guarantees that convolution by ψ preserves polynomials of degree p + d. We state the following. Proposition 5.56. Suppose |φ(x)| ≤ δ for some δ > 0. If u is also a polynomial of degree at most p on V δ (x), then ∇ Ru(x) = ∇ u(x) for all . Proof.

We have Dx Φy (x)ξ = ξ + (Dφ(x)ξ)y.

Suppose u is a k-form which is a polynomial of degree at most p. Recall that (Φy u)[x](ξ1 , . . . , ξk ) = u[Φy (x)](Dx Φy (x)ξ1 , . . . , Dx Φy (x)ξk ). As a function of y this is a polynomial of degree at most p + k ≤ p + d. Its value at y = 0 is u[x](ξ1 , . . . , ξk ). This gives the case = 0 of the proposition. For = 1 one uses expression (5.18). Greater are obtained by further diﬀerentiation of this expression. To see to what extent Propositions 5.52, 5.54 and 5.55 can be applied, we remark that Φy (x) − x = φ(x)y, Dy Φy (x)ξ = φ(x)ξ, Dy (Dy Φy (x)−1 ) = 0, Dx Φy (x)ξ = (Dφ(x)ξ)y, D2yx Φy (x)(ξ, ξ ) = (Dφ(x)ξ)ξ , Dx...x Φy (x)(ξ1 , . . . , ξ ) = D φ(x)(ξ1 , . . . , ξ ) y.

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Proposition 5.57. Pick ≤ p + 1. For any δ > 0 and any C > 0, there exists C > 0 such that, for all φ satisfying, at some point x, |φ(x)| ≤ δ, |φ(x)−1 | ≤ C, and Dx φ(x), D2xx φ(x), . . . , D+1 x...x φ(x) ≤ C, the associated regularizer satisﬁes (∇ Ru)(x) ≤ C ∇ uL1 (V δ (x)) . Proof. By the Deny–Lions lemma there exists C > 0 such that, for all u ∈ W,1 (V δ (x)), inf u − f W,1 (V δ (x)) # ∇ uL1 (V δ (x)) .

f ∈P−1

The regularizer R preserves the space P−1 of polynomials of degree up to − 1. Combining this with Proposition 5.55 gives the claimed result. The regularizer is adapted to the mesh Tn as follows. We choose φn such that, for all x ∈ T ∈ Tn , φn (x) hT , D

1+r

φn (x) #

h−r T ,

(5.25) for 0 ≤ r ≤ .

(5.26)

We introduce a parameter > 0 and consider the regularizations Rn = R[Φ] associated with the maps Φy (x) = x + φn (x)y. We deﬁne Vn (T ) = ∪{B(x, φn (x)) : x ∈ T }. We choose ﬁxed but small. Proposition 5.58.

Fix > 0. For any T ∈ Tn we have an estimate

1+(d−dim T )/q ∇Rn uLq (T ) hT (d−dim T )/q ∇ Rn uLq (T ) hT

# uLq (Vn (T )) ,

(5.27)

# ∇ uLq (Vn (T )) .

(5.28)

For T of maximal dimension we also have u − Rn uLq (T ) # hT ∇ uLq (Vn (T )) .

(5.29)

Proof. Pick T ∈ Tn . Its diameter is hT and its barycentre bT . Consider the scaling map σT : x → hT x + bT and let Tˆ be the pre-image of T by σT , called the reference cell.

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Note that, quite generally, the regularization R transforms as follows under pullback by a diﬀeomorphism σ: R[Φ] = (σ )−1 R[σ −1 ◦ Φ• ◦ σ]σ , where σ −1 ◦ Φ• ◦ σ : (y, x) → σ −1 (Φy (σ(x))). In our case the operator Rn [σT−1 ◦ Φ• ◦ σT ] is regularizing on the reference ˆ . We have cell Tˆ and we denote it by R n

(σT−1

◦ Φy ◦ σT )(x) = x + h−1 T φn (hT x + bT )y.

The conditions (5.25), (5.26) put us in a position to conclude from Propositions 5.54 and 5.55 that ˆ n u ∞ ˆ # u 1 −1 ∇R L (σ V (T )) , L (T ) n

T

ˆ n u ∞ ˆ # ∇ u 1 −1 ∇ R L (σ V (T )) . L (T ) n

T

From this we deduce ˆ u q ˆ # u q −1 ∇R n L (σ V (T )) , L (T ) T

n

ˆ n u q ˆ # ∇ u q −1 ∇ R L (σ V (T )) . L (T )

T

n

Then the estimates (5.27) and (5.28) follow from scaling. Let T ∈ Tn have dimension d. From Proposition 5.52, we get ˆ Ru Lq (Tˆ) # uLq (σ −1 V (T )) . T

n

Preservation of polynomials and the Deny–Lions lemma then gives ˆ u − Ru Lq (Tˆ) # ∇ uLq (σ −1 V (T )) . T

n

Scaling then gives (5.29). For a cell T ∈ T we denote by Mn (T ) the macro-element surrounding T in Tn , that is, the union of the cells T ∈ Tn touching T : Mn (T ) = ∪{T ∈ Tn : T ∩ T = ∅}. Choose so small that, for all n and all T ∈ Tn , Vn (T ) ∩ S ⊆ Mn (T ). Proposition 5.59.

For u deﬁned on Vn (S) we have estimates In Rn uLq (S) # uLq (Vn (S)

and, for ≤ p + 1, u − In Rn uLq (S) # hn uLq (Vn (S) .

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Extension. We shall deﬁne extension operators which extend diﬀerential forms on S to some neighbourhood of S, preserve polynomials up to a certain degree p, commute with the exterior derivative and are continuous in W,q -norms for ≤ p + 1. For this purpose we will use maps Φs depending on a parameter s, deﬁned outside S with values in S, and pull back by these maps. By taking judiciously chosen linear combinations of such pullbacks we meet the requirements of continuity and polynomial preservation. First we derive some formulas for the derivative of order of the pullback of a diﬀerential form. Antisymmetry in the variables will not be important for these considerations, so we consider multilinear rather than diﬀerential forms. Consider then a smooth map Φ : Rd → Rd and u, a k-multilinear form on Rd , with k ≥ 1. We want to give an expression for (∇ Φ u)[x](ξ1 , . . . , ξk+ ), as a linear combination of terms of the form (∇r u)[Φ(x)] Dm1 Φ(x)(ζ1 , . . . , ζm1 ), Dm2 Φ(x)(ζm1 +1 , . . . , ζm1 +m2 ), . . . , (5.30) where m1 + · · · + mk+r = k + and (ζ1 , . . . , ζk+ ) = (ξσ(1) , . . . , ξσ(k+) ), for any permutation σ of the indexes {1, . . . , k + }. To handle the combinatorics behind this problem, we let D be the set of pairs (σ, m) where m = (m1 , . . . , mv(m) ) is a multi-index of valence v(m) ≥ 1 and weight w(m) = m1 +· · ·+mv(p) , such that mi ≥ 1 for each 1 ≤ i ≤ v(m), and σ is a permutation of the indexes (1, . . . , w(m)). We deﬁne the partial weights |m|0 = 0 and |m|i = m1 + · · · + mi for 1 ≤ i ≤ v(m). We deﬁne (σ, m)◦ [Φ, x](ξ1 , . . . , ξw(m) ) to be the v(m)-multivector: v(m)

⊗i=1 Dmi Φ(x)(ζ|m|i−1 +1 , . . . , ζ|m|i ), with (ζ1 , . . . , ζw(m) ) = (ξσ(1) , . . . , ξσ(w(m)) ). Thus (σ, m)◦ [Φ, x] ∈ L(⊗w(m) Rd , ⊗v(m) Rd ).

(5.31)

Let D be the free group generated by D. We deﬁne some operations in D, which correspond to diﬀerentiating (5.31) with respect to x. For a given (σ, m) we deﬁne, for j = 1, . . . , v(m), ∂j (σ, m) to consist of

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the multi-index (m1 , . . . , mj + 1, . . . , mv(m) ) and the permutation σ(i) if σ(i) ≤ |m|j , σ(i) + 1 if σ(i) > |m|j ,

1 ≤ i ≤ w(m) →

w(m) + 1 → |m|j + 1. Deﬁning

v(m)

D(σ, m) =

∂j (σ, m),

j=1

we have

D((σ, m)◦ [Φ, x]) = (D(σ, m))◦ [Φ, x].

On the left, D is ordinary diﬀerentiation with respect to x, and on the right, D is an operation in D. If u is a k-multilinear form and ζ = ζ1 ⊗ . . . ⊗ ζk ∈ ⊗k Rd , we deﬁne the contraction u : ζ = u(ζ1 , . . . , ζk ). Contraction is bilinear. Given a k-multilinear form u, we want to diﬀerentiate, with respect to x, expressions of the form (∇r u)[Φ(x)] : (σ, m)◦ [Φ, x], where v(m) = k+r. This corresponds to (5.30). Since contraction is bilinear and we know how to diﬀerentiate the right-hand side, it remains to diﬀerentiate the left-hand side. For this purpose we introduce one more operation in D. Deﬁne e such that e(σ, m) consists of the multi-index (m1 , . . . , mv(m) , 1) and the permutation 1 ≤ i ≤ w(m) → σ(i), w(m) + 1 → w(m) + 1. Then we have ∇ (∇r u)[Φ(x)] : (σ, m)◦ [Φ, x] = (∇

r+1

(5.32) ◦

r

◦

u)[Φ(x)] : (e(σ, m)) [Φ, x] + (∇ u)[Φ(x)] : (D(σ, m)) [Φ, x].

In the free group D we now deﬁne Γr [k] for ≥ 0 and 0 ≤ r ≤ recursively. We initialize by Γ00 [k] = (id, (1, . . . , 1)), with the identity permutation and k terms in the multi-index. Then we deﬁne for ≥ 0 Γ+1 0 [k] = eΓ0 [k], Γ+1 i [k] = eΓi [k] + DΓi−1 [k], for 1 ≤ i ≤ , Γ+1 +1 [k] = DΓ [k].

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We check that Γ0 [k] = Γ00 [k + ], Γ [k] = D Γ00 [k]. Proposition 5.60.

We have, for a given k-form u,

∇ (Φ u)[x] =

(∇−i u)[Φ(x)] : Γi [k]◦ [Φ, x].

i=0

The expression k + . Proof.

Γi [k]

∈ D is a sum of terms with valence k + − i and weight

We use induction on , using (5.32).

For x outside S, let δ(x) denote the distance from x to S. Given a polynomial degree p, we will construct an extension operator E as a linear combination of pullbacks: E= ψs Φs , with Φs : V (S) \ S → S, s∈I

subject to the following conditions. • The index set I is a ﬁnite subset of the interval [2, 3] and the coeﬃcients (ψs )s∈I are chosen such that, for any polynomial f of degree at most p + d, ψs f (s) = f (0). s∈I

• There is a function φ : V (S) \ S → Rd such that, for all s and x, Φs (x) = x + sφ(x),

(5.33)

and, moreover, φ(x) δ(x), D1+r φ(x) # δ(x)−r ,

for 0 ≤ r ≤ p.

• Finally, for s ∈ [2, 3] we should have DΦs (x)−1 # 1, and the Φs should determine diﬀeomorphisms from V (S) \ S for some > 0, to an interior neighbourhood of ∂S. In order to show that the above list of conditions can be met, we need some results of a general nature that are given in the Appendix. Proposition 5.61.

The conditions listed above can be met.

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Proof. Let I consist of p + d + 1 points in the interval [2, 3]. Numbers ψs are then determined by solving a linear system with a Vandermonde matrix. If ∂S had been smooth, the orthogonal projection ℘ onto ∂S would be well-deﬁned and smooth on a neighbourhood. Then we could have taken φ(x) = ℘(x) − x. In the following we modify this construction to allow for Lipschitz boundaries. Choose a smooth vector ﬁeld ν according to Proposition A.3, pointing outwards on ∂S, so that for some > 0 the following map is a Lipschitz isomorphism onto its open range: g:

∂S×] − , [ (z, t)

→ Rd , → z + tν(z).

Deﬁne f on V (S) \ S by f (g(z, t)) = z − g(z, t) = −tν(z). The problem with f , to serve as φ in (5.33), is its lack of regularity. From Theorem 2 in Stein (1970, p. 171) we get a regularized distance function δ˜ deﬁned outside S, such that ˜ δ(x) δ(x), and for all r ≥ 1 ˜ Dr δ(x) # δ(x)1−r . We regularize f by a variant of (5.15), (5.23). We put ˜ φ(x) = ψ(y)f (x − δ(x)y) dy, ˜ where the parameter is chosen to satisfy δ(x) ≤ 1/2δ(x), so that f is evaluated far enough from S. For an illustration we refer to Figure 5.1. The conditions are then met. In the following we choose an integer ≤ p. Proposition 5.62.

We have

∇ Eu = ∇ u[Φ2 (x)]

+

ψs (∇ u[Φs (x)] − ∇ u[Φ2 (x)]) : ⊗k+ (id +sDφ(x))

s

+

(5.34a)

1

ψs 0

s i=1

∇ u[tΦs (x) + (1 − t)Φ2 (x)] :

⊗i (s − 2)φ(x) ⊗ Γi [k]◦ [Φs , x]

(1 − t)i−1 dt. (i − 1)!

(5.34b) (5.34c)

(5.34d)

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n ν Φ2 (x) Φ3 (x)

x s=0

s=1

n

s=2 s=3 S

Figure 5.1. Deﬁnition of Φs (x), without smoothing.

Proof.

We have ∇

(Φs u)[x]

= (∇−i u)[Φs (x)] : Γi [k]◦ [Φs , x].

(5.35)

i=0

Concerning Γi [k]◦ [Φs , x], note that DΦs (x) = id +sDφ(x), Dr Φs (x) = sDr φ(x) for r ≥ 2. Therefore Γi [k]◦ [Φs , x] is a polynomial in s of degree at most k + − i. For i = 0 the value in s = 0 is Γ0 [k]◦ [Φs , x]|s=0 = ⊗k+ (id +sDφ(x))|s=0 = ⊗k+ id. For i ≥ 1 we have a sum of terms that are products where at least one derivative of order at least two appears, so the value at s = 0 is 0. In (5.35) the i = 0 term gives rise to ψs ∇ u[Φs (x)] : Γ0 [k]◦ [Φs , x] = ∇ u[Φ2 (x)] s

+

ψs (∇ u[Φs (x)] − ∇ u[Φ2 (x)]) : ⊗k+ (id +sDφ(x)).

s

This corresponds to (5.34a,b).

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For i ≥ 1, Taylor’s formula with integral remainder gives i−1

(∇−i u)[Φs (x)] =

∇−i+j u[Φ2 (x)] : ⊗j (s − 2)φ(x)

j=0

1

+ 0

1 j!

∇ u[tΦs (x) + (1 − t)Φ2 (x)] : ⊗i (s − 2)φ(x)

(5.36)

(1 − t)i−1 dt. (i − 1)!

When this expression is contracted with Γi [k]◦ [Φs , x], the sum (5.36) consists of polynomials in s with degree j + k + − i ≤ + d, with value 0 at s = 0. Therefore ψs (∇−i u)[Φs (x)] : Γi [k]◦ [Φs , x] s

=

1

ψs 0

s

∇ u[tΦs (x) + (1 − t)Φ2 (x)] :

⊗i (s − 2)φ(x) ⊗ Γi [k]◦ [Φs , x]

(1 − t)i−1 dt. (i − 1)!

Summing over 1 ≤ i ≤ , this corresponds to (5.34c,d). From this formula several conclusions can be drawn. Proposition 5.63. • If u is of class C (S), Eu is of class C (V (S)). • If u is a polynomial of degree at most p, Eu is too. More precisely, if T is a cell touching S, and u is a polynomial on its macro-element, of degree at most p, then Eu is a polynomial of degree at most p on V (T ). • E is bounded W,q (S) → W,q (V (S)). For cells T ∈ Tn touching ∂S, ∇ EuLq (Vn (T )) # ∇ uLq (Mn (T )) . Proof.

For (σ, m) ∈ D, (σ, m)◦ [Φs , x] # δ(x)v(m)−w(m) ,

from which it follows that Γi [k]◦ [Φs , x] # δ(x)−i . This gives ⊗i φ(x) ⊗ Γi [k]◦ [Φs , x] # 1.

(5.37)

We now check continuity properties on ∂S. Suppose x0 ∈ ∂S and x → x0 . Then we have ∇ u[Φs (x)] − ∇ u[Φ2 (x)] → 0,

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so that (5.34b) converges to 0. If the integrals (5.34c) had been evaluated at ∇ u[Φ2 (x)], their sum over s ∈ I would be 0. An argument similar to the above, combined with (5.37), then shows that (5.34c,d) also converges to 0. We are left with the term on (5.34a), which converges to ∇ u(x0 ). Boundedness properties of E follow from (5.37) and the assumption that the Φs determine diﬀeomorphisms with uniformly bounded Jacobian determinants. Quasi-interpolator. Putting together the pieces, we get the following. Theorem 5.64. For any > 0, the operators Q n = In Rn E satisfy local estimates, for T ∈ Tnd , Q n uLq (T ) # uLq (Mn (T )) , u − Q n uLq (T ) # hT ∇ uLq (Mn (T )) , as the corresponding global ones: Q n uLq (S) # uLq (S) , u − Q n uLq (S) # h ∇ uLq (S) . Moreover, for any , choosing small enough will yield, for u ∈ Xnk , u − Q n uLq (T ) ≤ uLq (Mn (T )) , u − Q n uLq (S) # uLq (S) . Finally, Q n commutes with the exterior derivative (when it is in Lq (S)). When is chosen so small that (id −Q n )|Xnk Lq (S)→Lq (S) ≤ 1/2, Q n |Xnk is invertible with norm less than 2. We deﬁne operators: Pn = (Q n |Xnk )−1 Q n . Proposition 5.65. The operators Pn are Lq (S)-stable projections onto Xnk which commute with the exterior derivative. The case q = 2 leads to eigenvalue convergence for the operator d d discretized by the Galerkin method on Xnk and therefore for the Hodge– Laplacian in mixed form. For a discussion of eigenvalue convergence we refer to Boﬃ (2010), Arnold et al. (2010) and Christiansen and Winther (2010). 5.4. Sobolev injection and translation estimate In this subsection we prove a Sobolev injection theorem generalizing the one we introduced in Christiansen and Scheid (2011). The proof technique is slightly diﬀerent, and we generalize to diﬀerential forms in all dimensions. We also prove a translation estimate of the type introduced in Karlsen and

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Karper (2010, Theorem A.1). Compared with that paper we get an optimal bound and a generalization to all known mixed ﬁnite elements in the hversion. Given a cellular complex T , we deﬁne the broken H1 -seminorm |·| by 2 |u|2 = ∇u2L2 (T ) + h−1 (5.38) T [u]T L2 (T ) . T ∈T d

T ∈T d−1

Given a simplex T ∈ T d−1 , [u]T denotes the jump of u across T . The scaling factor in front of the jump terms is chosen so that the two terms that are summed scale in the same way: see Lemma 5.50. We assume we have a sequence of cellular complexes (Tn ) and that each Tn is equipped with a compatible FE system A[n]. We also assume that they are of the type discussed in Section 5.3 so that those constructions apply. We deﬁne Xnk = A[n]k (Tn ). The broken seminorm (5.38) will be deﬁned relative to one of the Tn and, since the particular Tn should be clear from the context, we omit dependence on Tn from our notation. In the L2 (S) case deﬁne X k = {u ∈ L2 (S) : du ∈ L2 (S)}, W k = {u ∈ X k : du = 0}, V k = {u ∈ X k : ∀w ∈ W k

u · w = 0}.

Deﬁne Wnk = {u ∈ Xnk : du = 0}, Vnk = {u ∈ Xnk : ∀w ∈ Wnk

u · w = 0}. k

Denote by H the L2 -orthogonal projection onto V , the completion of V k in L2 (S). This operator realizes a Hodge decomposition of u in the form u = (u − Hu) + Hu. The following well-known trick is also useful in the proof of eigenvalue convergence. Proposition 5.66.

We have, for u ∈ Vnk ,

u − HuL2 (S) ≤ Hu − Pn HuL2 (S) . Proof.

We have u − Pn Hu = Pn (u − Hu) ∈ Wnk .

Now write Hu − Pn Hu = (u − Pn Hu) − (u − Hu), and note that the two terms on the left-hand side are orthogonal.

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Proposition 5.67.

For u ∈ Xnk , |u| ∇Rn EuL2 (S) ,

u − Proof.

Rn EuL2 (S)

# h|u|.

(5.39) (5.40)

The parameter is chosen so small that, for all u ∈ Xnk , |u − Rn Eu| ≤ 1/2|u|.

Then we have |u| ≤ 2|Rn Eu|, |Rn Eu| ≤ 3/2|u|. Since Rn Eu is smooth, we have |Rn Eu| = ∇Rn EuL2 (S) . This gives (5.39). The other estimate is proved locally by scaling from a reference macroelement. Proposition 5.68.

For u ∈ H1 (S), |Pn u| # ∇uL2 (S) .

Proof.

By restriction to reference simplexes and scaling, |Qn u| # ∇uL2 (S) .

Choosing small enough, we also have, for u ∈ Xnk , |u − Qn u| ≤ 1/2|u|. Combining the two, we get the proposition. Proposition 5.69. Suppose S is convex and that the meshes are quasiuniform. Then, for all u ∈ Vnk , |u| # duL2 (S) . Proof.

We have, for u ∈ Vnk , |u| = |Pn u| ≤ |Pn (u − Hu)| + |Pn Hu| ≤ h−1 Pn (u − Hu)L2 + ∇HuL2 # h−1 u − HuL2 + ∇HuL2 # ∇HuL2 .

From this (5.41) follows.

(5.41)

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Proposition 5.70. Let q be the relevant Sobolev exponent, so that H1 (S) is contained in Lq (S). For all u ∈ Xnk , uLq (S) # |u| + uL2 (S) . Proof.

We have uLq (S) # Rn EuLq (S) # ∇Rn EuL2 (S) + Rn EuL2 (S) # |u| + uL2 (S) ,

as claimed. Denote by τy the translation by the vector y, so that when u is deﬁned in x − y we have (τy u)(x) = u(x − y). Proposition 5.71.

For all u ∈ Xnk ,

u − τy EuL2 (S) # (|y| + h1/2 |y|1/2 )|u|. On a reference simplex Tˆ we can write for u ∈ Xnk pulled back: ˆ2L2 (Tˆ) # |ˆ y |2 ∇ˆ u2L2 (T ) + |ˆ y| [ˆ u]T 2L2 (T ) . ˆ u − τyˆu

Proof.

T ∈Mn (Tˆ)d

Scaling back to T of size h, we get, with y = hˆ y, ∇u2L2 (T ) + |y| u − τy u2L2 (T ) # |y|2 T ∈Mn (T )d

T ∈Mn (Tˆ)d−1

[u]T 2L2 (T ) ,

T ∈Mn (T )d−1

so that u − τy u2L2 (T ) # (|y|2 + h|y|)|u|2Mn (T ) . This estimate comes with a restriction of the type |y| ≤ h/C, ensuring that one does not translate T out of its associated macro-element. For |y| ≥ h/C we can write u − τy Eu = (u − Rn Eu) + (Rn Eu − τy ERn Eu) + τy E(Rn Eu − u). From this we deduce u − τy EuL2 # u − Rn EuL2 + Rn Eu − τy ERn EuL2 # h|u| + |y|∇Rn EuL2 # |y| |u|. This concludes the proof.

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Acknowledgements First author. On ﬁnite elements, I have beneﬁted from the insights of, in particular, Annalisa Buﬀa, Jean-Claude N´ed´elec and Ragnar Winther. On algebraic topology I am grateful for the help of John Rognes and Bjørn Jahren, particularly with Propositions 5.16 and 5.21. I am also grateful to Martin Costabel for a helpful introduction to Stein’s construction of universal extension operators. Example 5.30 is from a discussion with Thomas Dubos (shallow water on the sphere) in June 2009. Trygve Karper’s remarks on upwinding were also helpful. Second author. I would, in particular, like to thank Brett Ryland for his important contribution to the development of the theory of generalized Chebyshev polynomials and also for doing a major part of the computer implementations. Furthermore, I thank Morten Nome for his eﬀorts in realizing the spectral element implementations. I am also grateful to Daan Huybrechs for important contributions to the understanding of generalized symmetries and Chebyshev polynomials of ‘other kinds’. Finally, thanks to Krister ˚ Ahlander for his contributions to developing both the theory and the computer codes for numerical treatment of equivariance. Third author. Morten Dahlby and Takaharu Yaguchi have contributed substantially to the section on integral-preserving methods. Elena Celledoni has proofread parts of this paper and made numerous suggestions for improvement. This work, conducted as part of the award ‘Numerical Analysis and Simulations of Geometric Wave Equations’, made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program. The work is also supported by the Norwegian Research Council through the project ‘Structure Preserving Algorithms for Diﬀerential Equations: Applications, Computation and Education’ (SpadeACE).

Appendix Lemma A.1. In a Banach space E, let U be an open set and let f : U → E be a contraction mapping, i.e., for some δ < 1, f (x) − f (y) ≤ δx − y.

(A.1)

Then the map g : U → E, x → x+f (x) has an open range V and determines a Lipschitz bijection U → V , with a Lipschitz inverse. Proof. Suppose that g(x0 ) = y0 . For y − y0 ≤ ﬁnd the solution x of g(x) = y as a ﬁxed point of the map z → y − f (z). More precisely, construct

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109

a sequence starting at x0 and deﬁned by xn+1 = y − f (xn ). Then, as long as it is deﬁned (in U ), we have xn+1 − xn ≤ δxn − xn−1 ≤ δ n−1 . If is chosen so small that the closed ball with centre x0 and radius (1−δ)−1 is included in U , the sequence is deﬁned for all n, and converges to a limit x ∈ U solving g(x) = y. It follows that g is an open mapping. In particular, the range V is open. Moreover, g(x) − g(y) ≥ (1 − δ)x − y. This gives injectivity, so that g : U → V is bijective. The inverse is Lipschitz with constant no worse than (1 − δ)−1 . Lemma A.2. In some Euclidean space E, let S be a bounded domain whose boundary ∂S is locally the graph of a Lipschitz function. Let n be the outward-pointing normal on ∂S. Suppose m is a unit vector, that x0 ∈ ∂S, and that for x in a neighbourhood of x0 in ∂S we have n(x) · m ≥ , for some > 0. Then there is a neighbourhood of x0 in ∂S which is a Lipschitz graph above the plane orthogonal to m. Proof. We know that for a certain outward-pointing unit vector m0 , a neighbourhood U0 of x0 is a Lipschitz graph above an open ball B0 in m⊥ 0. Choose θ ∈ [0, π/2[ such that, for x ∈ U0 , n(x) · m0 ≥ cos θ, and moreover ≥ cos θ. Let f : B0 → R be the function such that U1 is the range of B0 y

→ E, → y + f (y)m0 .

Since, for y ∈ B0 , n(y + f (y)m0 ) = (m0 − grad f (y))/(1 + | grad f (y)|2 )1/2 , we get | grad f (x)| ≤ tan θ. Choose y, y ∈ B0 and put x = y + f (y)m0 and x = y + f (y )m0 . For s, s ∈ R we have |(x + sm0 ) − (x + s m0 )|2 = |y − y |2 + (f (y) − f (y ))2 + 2(f (y) − f (y ))(s − s ) + (s − s )2 .

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Then note that, for M > 0, 2(f (y) − f (y ))(s − s ) ≤ (1 + M −2 )(f (y) − f (y ))2 + (1 + M −2 )−1 (s − s )2 ≤ (f (y) − f (y ))2 + M −2 tan2 θ|y − y |2 + (1 + M −2 )−1 (s − s )2 . In particular, with M = tan θ we get |(x + sm0 ) − (x + s m0 )|2 ≥ (1 − (1 + tan−2 θ)−1 )(s − s )2 , which simpliﬁes to |(x + sm0 ) − (x + s m0 )| ≥ cos θ|s − s |. Deﬁne the function g0 :

U0 × R x

→ E, → y + sm0 .

(A.2)

Its range is B0 + Rm0 , and it is bijective onto it. Consider now a unit vector m1 such that n(x) · m1 ≥ cos θ for x ∈ U0 . Deﬁne g1 as in (A.2), replacing m0 by m1 . We shall show that g1 is open and bi-Lipschitz, when |m1 − m0 | < cos θ. Deﬁne f1 on B0 + Rm0 , by f1 (x) = g1 ◦ g0−1 − x. With the preceding notation, we have f1 (x + sm0 ) − f1 (x + s m0 ) = (s − s )(m1 − m0 ), and hence |f1 (x + sm0 ) − f1 (x + s m0 )| ≤ |m1 − m0 |/ cos θ|(x + sm0 ) − (x + s m0 )|. We then apply Lemma A.1 and deduce that g1 is open and bi-Lipschitz. It follows that there is a ball B1 in m⊥ 1 above which a neighbourhood U1 ⊆ U0 of x0 is a graph. We may repeat the above considerations to construct a sequence of such vectors m1 , m2 , . . . , mk reaching m in a ﬁnite number of steps. Proposition A.3. In some Euclidean space E, let S be a bounded domain whose boundary ∂S is locally the graph of a Lipschitz function. Then there exists a smooth vector ﬁeld ν on E, of unit length and outward-pointing on ∂S, such that, for some > 0, the map ∂S×] − , [ (x, s)

→ E, → x + sν(x)

has open range and determines a Lipschitz isomorphism onto it.

(A.3)

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111

Proof. Cover ∂S with a ﬁnite number of orthogonal cylinders Ci directed along a unit vector ni pointing out of S, and with a base Ui , such that above Ui , ∂S is the graph of a Lipschitz function. Let n denote the outwardpointing normal on ∂S. For some θ ∈ [0, π/2[, we have for all i and all x ∈ ∂S ∩ Ci ni · n(x) ≥ cos(θ). (A.4) Choose smooth functions αi on E whose restrictions to ∂S form a partition of unity. Deﬁne ν˜(x) = αi (x)ni , i

and normalize by putting ν(x) = ν˜(x)/|ν(x)|. We have ν(x) · n(x) ≥ cos(θ), for all x ∈ ∂S. Denote by g the function g:

∂S × R (x, s)

→ E, → x + sν(x).

Pick x0 ∈ ∂S and put m0 = ν(x0 ). By Lemma A.2, ∂S is locally a Lipschitz graph above the plane orthogonal to m0 . Let U (x0 ) be the corresponding neighbourhood of x0 in ∂S. Denote by g0 the function g0 :

∂S ∩ U (x0 ) × R (x, s)

→ E, → x + sm0 .

It is a Lipschitz isomorphism onto its range, which is open in E. Deﬁne f by f (x) = g ◦ g0−1 (x) − x. We have f (x + sm0 ) = s(ν(x) − m0 ), so that f (x + sm0 ) − f (y + tm0 ) = (s − t)(ν(x) − m0 ) + t(ν(x) − ν(y)). It follows that, for a small enough and possibly reducing U(x0 ), f is short on g0 (U(x0 )×] − , [). By Lemma A.1 it follows that g ◦ g0−1 restricted to g0 (U (x0 )×] − , [), for some small enough , called (x0 ), has open range and determines a Lipschitz isomorphism onto it. Hence g restricted to U (x0 )×] − (x0 ), (x0 )[ has open range and determines a Lipschitz isomorphism onto it. In particular, g is an open mapping. The open subsets U(x0 ) associated with each x0 ∈ ∂S cover ∂S. Choose a ﬁnite subset F of ∂S such that the sets U (x) for x ∈ F cover ∂S. Pick µ > 0 such that if |x − x | ≤ µ they belong to a common U (x) for x ∈ F. In (A.3), choose smaller than each (x) for x ∈ F, and also smaller than

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µ/3. Now pick two points x, x ∈ ∂S, and s, s in ] − , [. If |x − x | ≤ µ, they belong to a common U (x), x ∈ F. If, on the other hand, |x − x | ≥ µ, we have |g(x, s) − g(x , s )| ≥ |x − x | − |s| − |s |, ≥ µ/3. Based on these two cases we may conclude that, for some global m, |g(x, s) − g(x , s )| ≥ m(|x − x |2 + |s − s |2 ). The lemma follows.

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Acta Numerica (2011), pp. 121–209 doi:10.1017/S0962492911000031

c Cambridge University Press, 2011 Printed in the United Kingdom

Mathematical and computational methods for semiclassical Schr¨ odinger equations∗ Shi Jin† Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: [email protected]

Peter Markowich‡ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected]

Christof Sparber§ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA E-mail: [email protected] We consider time-dependent (linear and nonlinear) Schr¨ odinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of eﬃcient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schr¨ odinger equation in the semiclassical regime. ∗ †

‡ §

Colour online available at journals.cambridge.org/anu. Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285, a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the University of Wisconsin–Madison. Supported by a Royal Society Wolfson Research Merit Award and by KAUST through a Investigator Award KUK-I1-007-43. Partially supported by the Royal Society through a University Research Fellowship.

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CONTENTS 1 Introduction 2 WKB analysis for semiclassical Schr¨ odinger equations 3 Wigner transforms and Wigner measures 4 Finite diﬀerence methods for semiclassical Schr¨odinger equations 5 Time-splitting spectral methods for semiclassical Schr¨odinger equations 6 Moment closure methods 7 Level set methods 8 Gaussian beam methods: Lagrangian approach 9 Gaussian beam methods: Eulerian approach 10 Asymptotic methods for discontinuous potentials 11 Schr¨odinger equations with matrix-valued potentials and surface hopping 12 Schr¨ odinger equations with periodic potentials 13 Numerical methods for Schr¨odinger equations with periodic potentials 14 Schr¨odinger equations with random potentials 15 Nonlinear Schr¨odinger equations in the semiclassical regime References

122 125 130 134 140 146 151 155 160 167 173 178 183 190 193 200

1. Introduction This goal of this article is to give an overview of the currently available numerical methods used in the study of highly oscillatory partial diﬀerential equations (PDEs) of Schr¨ odinger type. This type of equation forms a canonical class of (nonlinear) dispersive PDEs, i.e., equations in which waves of diﬀerent frequency travel with diﬀerent speed. The accurate and eﬃcient numerical computation of such equations usually requires a lot of analytical insight, and this applies in particular to the regime of high frequencies. The following equation can be seen as a paradigm for the PDEs under consideration: iε∂t uε = −

ε2 ∆uε + V (x)uε , 2

uε (0, x) = uεin (x),

(1.1)

for (t, x) ∈ R × Rd , with d ∈ N denoting the space dimension. In addition, ε ∈ (0, 1] denotes the small semiclassical parameter (the scaled Planck’s constant), describing the microscopic/macroscopic scale ratio. Here, we have already rescaled all physical parameters, such that only one dimensionless

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parameter ε 1 remains. The unknown uε = uε (t, x) ∈ C is the quantum mechanical wave function whose dynamics is governed by a static potential function V = V (x) ∈ R (time-dependent potentials V (t, x) can usually also be taken into account without requiring too much extra work, but for the sake of simplicity we shall not do so here). In this article, several diﬀerent classes of potentials, e.g., smooth, discontinuous, periodic, random, will be discussed, each of which requires a diﬀerent numerical strategy. In addition, possible nonlinear eﬀects can be taken into account (as we shall do in Section 15) by considering nonlinear potentials V = f (|uε |2 ). In the absence of V (x) a particular solution to the Schr¨odinger equation is given by a single plane wave, i t 2 ε , u (t, x) = exp ξ · x − |ξ| ε 2 for any given wave vector ξ ∈ Rd . We see that uε features oscillations with frequency 1/ε in space and time, which inhibit strong convergence of the wave function in the classical limit ε → 0+ . In addition, these oscillations pose a huge challenge for numerical computations of (2.1); in particular, they strain computational resources when run-of-the-mill numerical techniques are applied in order to numerically solve (1.1) in the semiclassical regime ε 1. For the linear Schr¨odinger equation, classical numerical analysis methods (such as the stability–consistency concept) are suﬃcient to derive meshing strategies for discretizations (say, of ﬁnite diﬀerence, ﬁnite element or even time-splitting spectral type) which guarantee (locally) strong convergence of the discrete wave functions when the semiclassical parameter ε is ﬁxed: see Chan, Lee and Shen (1986), Chan and Shen (1987), Wu (1996) and D¨orﬂer (1998); extensions to nonlinear Schr¨odinger equations can be found in, e.g., Delfour, Fortin and Payre (1981), Taha and Ablowitz (1984) and Pathria and Morris (1990). However, the classical numerical analysis strategies cannot be employed to investigate uniform in ε properties of discretization schemes in the semiclassical limit regime. As we shall detail in Section 4, even seemingly reasonable, i.e., stable and consistent, discretization schemes, which are heavily used in many practical application areas of Schr¨odinger-type equations, require huge computational resources in order to give accurate physical observables for ε 1. The situation gets even worse when an accurate resolution of uε itself is required. To this end, we remark that time-splitting spectral methods tend to behave better than ﬁnite diﬀerence/ﬁnite element methods, as we shall see in more detail in Section 5. In summary, there is clearly a big risk in using classical discretization techniques for Schr¨ odinger calculations in the semiclassical regime. Certain schemes produce completely incorrect observables under seemingly reasonable meshing strategies, i.e., an asymptotic resolution of the oscillation is

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not always enough. Even worse, in these cases there is no warning from the scheme (such as destabilization) that something has gone wrong in the computation (since local error control is computationally not feasible in the semiclassical regime). The only safety anchor here lies in asymptotic mathematical analysis, such as WKB analysis, and/or a physical insight into the problem. They typically yield a (rigorous) asymptotic description of uε for small ε 1, which can consequently be implemented on a numerical level, providing an asymptotic numerical scheme for the problem at hand. In this work, we shall discuss several asymptotic schemes, depending on the particular type of potentials V considered. While one can not expect to be able to pass to the classical limit directly in the solution uε of (1.1), one should note that densities of physical observables, which are the quantities most interesting in practical applications, are typically better behaved as ε → 0, since they are quadratic in the wave function (see Section 2.1 below). However, weak convergence of uε as ε → 0 is not suﬃcient for passing to the limit in the observable densities (since weak convergence does not commute with nonlinear operations). This makes the analysis of the semiclassical limit a mathematically highly complex issue. Recently, much progress has been made in this area, particularly by using tools from micro-local analysis, such as H-measures (Tartar 1990) and Wigner measures (Lions and Paul 1993, Markowich and Mauser 1993, G´erard, Markowich, Mauser and Poupaud 1997). These techniques go far beyond classical WKB methods, since the latter suﬀers from the appearance of caustics: see, e.g., Sparber, Markowich and Mauser (2003) for a recent comparison of the two methods. In contrast, Wigner measure techniques reveal a kinetic equation in phase space, whose solution, the so-called Wigner measure associated to uε , does not exhibit caustics (see Section 3 for more details). A word of caution is in order. First, a reconstruction of the asymptotic description for uε itself (for ε 1) is in general not straightforward, since, typically, some phase information is lost when passing to the Wigner picture. Second, phase space techniques have proved to be very powerful in the linear case and in certain weakly nonlinear regimes, but they have not yet shown much strength when applied to nonlinear Schr¨odinger equations in the regime of supercritical geometric optics (see Section 15.2). There, classical WKB analysis (and in some special cases techniques for fully integrable systems) still prevails. The main mathematical reason for this is that the initial value problem for the linear Schr¨odinger equation propagates only one ε-scale of oscillations, provided the initial datum in itself is ε-oscillatory (as is always assumed in WKB analysis). New (spatial) frequencies ξ may be generated during the time evolution (typically at caustics) but no new scales of oscillations will arise in the linear case. For nonlinear Schr¨odinger problems this is diﬀerent, as new oscillation scales may be generated through the

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nonlinear interaction of the solution with itself. Further, one should note that this important analytical distinction, i.e., no generation of new scales but possible generation of new frequencies (in the linear case), may not be relevant on the numerical level, since, say, 100ε is analytically just a new frequency, but numerically a new scale. Aside from semiclassical situations, modern research in the numerical solution of Schr¨odinger-type equations goes in a variety of directions, of which the most important are as follows. (i) Stationary problems stemming from, e.g., material sciences. We mention band diagram computations (to be touched upon below in Sections 12 and 13) and density functional theory for approximating the full microscopic Hamiltonian (not to be discussed in this paper). The main diﬀerence between stationary and time-dependent semiclassical problems is given by the fact that in the former situation the spatial frequency is ﬁxed, whereas in the latter (as mentioned earlier) new frequencies may arise over the course of time. (ii) Large space dimensions d 1, arising, for example, when the number of particles N 1, since the quantum mechanical Hilbert space for N indistinguishable particles (without spin) is given by L2 (R3N ). This is extremely important in quantum chemistry simulations of atomistic/molecular applications. Totally diﬀerent analytical and numerical techniques need to be used and we shall not elaborate on these issues in this paper. We only remark that if some of the particles are very heavy and can thus be treated classically (invoking the so-called Born–Oppenheimer approximation: see Section 11), a combination of numerical methods for both d 1 and ε 1 has to be used.

2. WKB analysis for semiclassical Schr¨ odinger equations 2.1. Basic existence results and physical observables We recall the basic existence theory for linear Schr¨ odinger equations of the form ε2 iε∂t uε = − ∆uε + V (x)uε , uε (0, x) = uεin (x). (2.1) 2 For the sake of simplicity we assume the (real-valued) potential V = V (x) to be continuous and bounded, i.e., V ∈ C(Rd ; R) :

|V (x)| ≤ K.

The Kato–Rellich theorem then ensures that the Hamiltonian operator H ε := −

ε2 ∆ + V (x) 2

(2.2)

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is essentially self-adjoint on D(−∆) = C0∞ ⊂ L2 (Rd ; C) and bounded from below by −K: see, e.g., Reed and Simon (1975). Its unique self-adjoint extension (to be denoted by the same symbol) therefore generates a strongly ε continuous semi-group U ε (t) = e− itH /ε on L2 (Rd ), which ensures the global existence of a unique (mild) solution uε (t) = U ε (t)uin of the Schr¨odinger equation (2.1). Moreover, since U ε (t) is unitary, we have uε (t, ·)2L2 = uεin 2L2 ,

∀ t ∈ R.

In quantum mechanics this is interpreted as conservation of mass. In addition, we also have conservation of the total energy ε2 ε ε 2 |∇u (t, x)| dx + V (x)|uε (t, x)|2 dx, (2.3) E[u (t)] = 2 Rd d R which is the sum of the kinetic and the potential energies. In general, expectation values of physical observables are computed via quadratic functionals of uε . To this end, denote by aW (x, εDx ) the operator corresponding to a classical (phase space) observable a ∈ Cb∞ (Rd × Rd ), obtained via Weyl quantization, x+y 1 W , εξ f (y) e i(x−y)·ξ dξ dy, (2.4) a a (x, εDx )f (x) := (2π)m 2 Rd ×Rd where εDx := −iε∂x . Then the expectation value of a in the state uε at time t ∈ R is given by a[uε (t)] = uε (t), aW (x, εDx )uε (t) L2 ,

(2.5)

where ·, · L2 denotes the usual scalar product on L2 (Rd ; C). Remark 2.1. The convenience of the Weyl calculus lies in the fact that an (essentially) self-adjoint Weyl operator aW (x, εDx ) has a real-valued symbol a(x, ξ): see H¨ormander (1985). The quantum mechanical wave function uε can therefore be considered only an auxiliary quantity, whereas (real-valued) quadratic quantities of uε yield probability densities for the respective physical observables. The most basic quadratic quantities are the particle density ρε (t, x) := |uε (t, x)|2 ,

(2.6)

and the current density

j ε (t, x) := ε Im uε (t, x)∇uε (t, x) .

(2.7)

It is easily seen that if uε solves (2.1), then the following conservation law holds: ∂t ρε + div j ε = 0.

(2.8)

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In view of (2.3) we can also deﬁne the energy density 1 eε (t, x) := |ε∇uε (t, x)|2 + V (x)ρε (t, x). (2.9) 2 As will be seen (see Section 5), computing these observable densities numerically is usually less cumbersome than computing the actual wave function uε accurately. From the analytical point of view, however, we face the problem that the classical limit ε → 0 can only be regarded as a weak limit (in a suitable topology), due to the oscillatory nature of uε . Quadratic operations deﬁning densities of physical observables do not, in general, commute with weak limits, and hence it remains a challenging task to identify the (weak) limits of certain physical observables, or densities, respectively. 2.2. Asymptotic description of high frequencies In order to gain a better understanding of the oscillatory structure of uε we invoke the following WKB approximation (see Carles (2008) and the references given therein): ε→0

uε (t, x) ∼ aε (t, x) e iS(t,x)/ε ,

(2.10)

with real-valued phase S and (possibly) complex-valued amplitude aε , satisfying the asymptotic expansion ε→0

aε ∼ a + εa1 + ε2 a2 + · · · .

(2.11)

Plugging the ansatz (2.10) into (2.1), one can determine an approximate solution to (2.1) by subsequently solving the equations obtained in each order of ε. To leading order, i.e., terms of order O(1), one obtains a Hamilton–Jacobi equation for the phase function S: 1 ∂t S + |∇S|2 + V (x) = 0, S(0, x) = Sin (x). (2.12) 2 This equation can be solved by the method of characteristics, provided V (x) is suﬃciently smooth, say V ∈ C 2 (Rd ). The characteristic ﬂow is given by the following Hamiltonian system of ordinary diﬀerential equations: x(t, ˙ y) = ξ(t, y), ˙ y) = −∇x V (x(t, y)), ξ(t,

x(0, y) = y, ξ(0, y) = ∇Sin (y).

(2.13)

Remark 2.2. The characteristic trajectories y → x(t, y) obtained via (2.13) are usually interpreted as the rays of geometric optics. The WKB approximation considered here is therefore also regarded as the geometric optics limit of the wave ﬁeld uε .

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Figure 2.1. Caustics generated from initial data ∂x Sin (x) = − sin(πx)| sin(πx)|p−1 : (a) p = 1, and the solution becomes triple-valued; (b) p = 2, and we exhibit single-, triple- and quintuple-valued solutions.

By the Cauchy–Lipschitz theorem, this system of ordinary diﬀerential equations can be solved at least locally in time, and consequently yields the phase function t 1 |∇S(τ, y(t, x))|2 − V (τ, y(t, x)) dτ, S(t, x) = S(0, x) + 0 2 where y(τ, x) denotes the inversion of the characteristic ﬂow Xt : y → x(t, y). This yields a smooth phase function S ∈ C ∞ ([−T, T ] × Rd ) up to some time T > 0 but possibly very small. The latter is due to the fact that, in general, characteristics will cross at some ﬁnite time |T | < ∞, in which case the ﬂow map Xt : Rd → Rd is no longer one-to-one. The set of points at which Xt ceases to be a diﬀeomorphism is usually called a caustic set. See Figure 2.1 (taken from Gosse, Jin and Li (2003)) for examples of caustic formulation. Ignoring the problem of caustics for a moment, one can proceed with our asymptotic expansion and obtain at order O(ε) the following transport equation for the leading-order amplitude: a ∂t a + ∇S · ∇a + ∆S = 0, a(0, x) = ain (x). (2.14) 2 In terms of the leading-order particle density ρ := |a|2 , this reads ∂t ρ + div(ρ∇S) = 0,

(2.15)

which is reminiscent of the conservation law (2.8). The transport equation (2.14) is again solved by the methods of characteristics (as long as S is smooth, i.e., before caustics) and yields ain (y(t, x)) a(t, x) = , Jt (y(t, x))

|t| ≤ T,

(2.16)

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where Jt (y) := det ∇y x(t, u) denotes the Jacobi determinant of the Hamiltonian ﬂow. All higher-order amplitudes an are then found to be solutions of inhomogeneous transport equations of the form an ∂t an + ∇S · ∇an + ∆S = ∆an−1 . (2.17) 2 These equations are consequently solved by the method of characteristics. At least locally in time (before caustics), this yields an approximate solution of WKB type, uεapp (t, x) = e iS(t,x)/ε a(t, x) + εa1 (t, x) + ε2 a2 (t, x) + · · · , including amplitudes (an )N n=1 up to some order N ∈ N. It is then straightforward to prove the following stability result. Theorem 2.3. form,

Assume that the initial data of (2.1) are given in WKB uεin (x) = ain (x) e iSin (x)/ε ,

C ∞ (Rd ),

(2.18)

S(Rd ),

and let ain ∈ i.e., smooth and rapidly decaying. with Sin ∈ Then, for any closed time interval I ⊂ T , before caustic onset, there exists a C > 0, independent of ε ∈ (0, 1], such that sup uε (t) − uεapp (t)L2 ∩L∞ ≤ CεN . t∈I

The ﬁrst rigorous result of this type goes back to Lax (1957). Its main drawback is the fact that the WKB solution breaks down at caustics, where S develops singularities. In addition, the leading-order amplitude a blows up in L∞ (Rd ), in view of (2.16) and the fact that limt→T Jt (y) = 0. Of course, these problems are not present in the exact solution uε but are merely an artifact of the WKB ansatz (2.10). Caustics therefore indicate the appearance of new ε-oscillatory scales within uε , which are not captured by the simple oscillatory ansatz (2.10). 2.3. Beyond caustics At least locally away from caustics, though, the solution can always be described by a superposition of WKB waves. This can be seen rather easily in the case of free dynamics where V (x) = 0. The corresponding solution of the Schr¨odinger equation (2.1) with WKB initial data is then explicitly given by 1 uε (t, x) = ain (y) e iϕ(x,y,ξ,t)/ε dy dξ, (2.19) (2πε)d d d R ×R with phase function t ϕ(x, y, ξ, t) := (x − y) · ξ + |ξ|2 + Sin (y). 2

(2.20)

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The representation formula (2.19) comprises an oscillatory integral, whose main contributions stem from stationary phase points at which ∂y,ξ ϕ(x, t) = 0. In view of (2.20) this yields ξ = ∇S,

y = x − tξ.

The corresponding map y → x(t, y) is the characteristic ﬂow of the free Hamilton–Jacobi equation 1 ∂t S + |∇S|2 = 0, S(0, x) = Sin (x). 2 Inverting the relation y → x(t, y) yields the required stationary phase points {yj (t, x)}j∈N ∈ Rd for the integral (2.19). Assuming for simplicity that there are only ﬁnitely many such points, then 1 ε ain (y) eiϕ(x,y,ξ,t)/ε dy dξ u (t, x) = (2πε)d d d R ×R (2.21) J ain (yj (t, x)) iS(yj (t,x))/ε+ iπmj /4 ε→0 e ∼ , Jt (yj (t, x)) j=1 with constant phase shifts mj = sgn D2 S(yj (t, x))) ∈ N (usually referred to as the Keller–Maslov index). The right-hand side of this expression is usually referred to as multiphase WKB approximation. The latter can be interpreted as an asymptotic description of interfering wave trains in uε . Remark 2.4. The case of non-vanishing V (x), although similar in spirit, is much more involved in general. In order to determine asymptotic description of uε beyond caustics, one needs to invoke the theory of Fourier integral operators: see, e.g., Duistermaat (1996). In particular, it is in general very hard to determine the precise form and number of caustics appearing throughout the time evolution of S(t, x). Numerical schemes for capturing caustics have been developed in, e.g., Benamou and Solliec (2000), or Benamou, Laﬁtte, Sentis and Solliec (2003) and the references therein.

3. Wigner transforms and Wigner measures 3.1. The Wigner-transformed picture of quantum mechanics Whereas WKB-type methods aim for approximate solutions of uε , the goal of this section is to directly identify the weak limits of physical observable densities as ε → 0. To this end, one deﬁnes the so-called Wigner transform of uε , as given in Wigner (1932): 1 ε ε ε ε ε ε u x + η u x − η eiξ·η dη. (3.1) w [u ](x, ξ) := 2 2 (2π)d Rd

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Plancherel’s theorem together with a simple change of variables yields wε L2 (R2d ) = ε−d (2π)−d/2 uε 2L2 (Rd ) . The real-valued Wigner transform wε ∈ L2 (Rdx × Rdξ ) can be interpreted as a phase space description of the quantum state uε . In contrast to classical phase space distributions, wε in general also takes negative values (except for Gaussian wave functions). Applying this transformation to the Schr¨odinger equation (2.1), the timedependent Wigner function wε (t, x, ξ) ≡ wε [uε (t)](x, ξ) is easily seen to satisfy ∂t wε + ξ · ∇x wε − Θε [V ]wε = 0,

ε wε (0, x, ξ) = win (x, ξ),

(3.2)

where Θε [V ] is a pseudo-diﬀerential operator, taking into account the inﬂuence of V (x). Explicitly, it is given by i ε δV ε (x, y)f (x, ξ ) eiη(ξ−ξ ) dη dξ , (3.3) Θ [V ]f (x, ξ) := d (2π) Rd ×Rd where the symbol δV ε reads 1 ε ε δV ε := V x− y −V x+ y . ε 2 2 Note that in the free case where V (x) = 0, the Wigner equation becomes the free transport equation of classical kinetic theory. Moreover, if V ∈ C 1 (Rd ) we obviously have that ε→0

δV ε −→ y · ∇x V, in which case the ε → 0 limit of (3.2) formally becomes the classical Liouville equation in phase space: see (3.7) below. The most important feature of the Wigner transform is that it allows for a simple computation of quantum mechanical expectation values of physical observables. Namely, ε W ε u (t), a (x, εD)u (t) L2 = a(x, ξ)wε (t, x, ξ) dx dξ, (3.4) Rd ×Rd

where a(x, ξ) is the classical symbol of the operator aW (x, εDx ). In addition, at least formally (since wε ∈ L1 (Rd × Rd ) in general), the particle density (2.6) can be computed via wε (t, x, ξ) dξ, ρε (t, x) = Rd

and the current density (2.7) is given by ε ξwε (t, x, ξ) dξ. j (t, x) = Rd

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Similarly, the energy density (2.9) is ε H(x, ξ)wε (t, x, ξ) dξ, e (t, x) = Rd

where the classical (phase space) Hamiltonian function is denoted by 1 H(x, ξ) = |ξ|2 + V (x). (3.5) 2 Remark 3.1. It can be proved that the Fourier transform of wε with respect to ξ satisﬁes w ε ∈ C0 (Rdy ; L1 (Rdx )) and likewise for the Fourier transformation of wε with respect to x ∈ Rd . This allows us to deﬁne the integrals of wε via a limiting process after convolving wε with Gaussians; see Lions and Paul (1993) for more details. 3.2. Classical limit of Wigner transforms The main point in the formulae given above is that the right-hand side of (3.4) involves only linear operations of wε , which is compatible with weak limits. To this end, we recall the main result proved in Lions and Paul (1993) and G´erard et al. (1997). Theorem 3.2. ε, that is,

Let uε (t) be uniformly bounded in L2 (Rd ) with respect to sup uε (t)L2 < +∞,

0<ε≤1

∀ t ∈ R.

Then, the set of Wigner functions {wε (t)}0<ε≤1 ⊂ S (Rdx × Rdξ ) is weak-∗ compact and thus, up to extraction of subsequences, ε→0

wε [uε ] −→ w0 ≡ w

in L∞ ([0, T ]; S (Rdx × Rdξ )) weak-∗,

where the limit w(t) ∈ M+ (Rdx × Rdξ ) is called the Wigner measure. If, in addition ∀ t : (ε∇u(t)) ∈ L2 (Rd ) uniformly with respect to ε, then we also have ε→0

ρε (t, x) −→ ρ(t, x) = ε→0

j ε (t, x) −→ j(t, x) =

w(t, x, dξ), R

d

ξw(t, x, dξ). Rd

Note that although wε (t) in general also takes negative values, its weak limit w(t) is indeed a non-negative measure in phase space. Remark 3.3. The limiting phase space measures w(t) ∈ M+ (Rdx × Rdp ) are also often referred to as semiclassical measures and are closely related to the so-called H-measures used in homogenization theory (Tartar 1990). The fact that their weak limits are non-negative can be seen by considering

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the corresponding Husimi transformation, i.e., ε ε [u ] := wε [uε ] ∗x Gε ∗ξ Gε , wH

where we denote 2 /ε

Gε ( · ) := (πε)−d/4 e−| · |

.

ε is non-negative a.e. and has the same limit points The Husimi transform wH as the Wigner function wε ; see Markowich and Mauser (1993).

This result allows us to exchange limit and integration on the limit on the right-hand side of (3.4) to obtain ε→0 ε W ε u , a (x, εDx )u L2 −→ a(x, ξ)w(t, x, ξ) dx dξ. Rd ×Rd

The Wigner transformation and its associated Wigner measure are therefore highly useful tools for computing the classical limit of the expectation values of physical observables. In addition, it is proved in Lions and Paul (1993) and G´erard et al. (1997) that w(t, x, ξ) is the push-forward under the ﬂow corresponding to the classical Hamiltonian H(x, ξ), i.e., w(t, x, ξ) = win (F−t (x, ξ)), where win is the initial Wigner measure and Ft : R2d → R2d is the phase space ﬂow given by x˙ = ∇ξ H(x, ξ), ξ˙ = −∇x H(x, ξ),

x(0, y, ζ) = y, ξ(0, y, ζ) = ζ.

(3.6)

In other words w(t, x, ξ) is a distributional solution of the classical Liouville equation in phase space, i.e., ∂t w + {H, w} = 0,

(3.7)

where {a, b} := ∇ξ a · ∇x b − ∇x a · ∇ξ b denotes the Poisson bracket. Note that in the case where H(x, ξ) is given by (3.5), this yields ∂t w + ξ · ∇ξ w − ∇x V (x) · ∇ξ w = 0,

(3.8)

with characteristic equations given by the Newton trajectories: x˙ = ξ, ξ˙ = −∇x V (x). Strictly speaking we require V ∈ Cb1 (Rd ), in order to deﬁne w as a distributional solution of (3.8). Note, however, in contrast to WKB techniques, the equation for the limiting Wigner measure (3.8) is globally well-posed, i.e., one does not experience problems of caustics. This is due to the fact that the Wigner measure w(t, x, ξ) lives in phase space.

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3.3. Connection between Wigner measures and WKB analysis A particularly interesting situation occurs for uεin given in WKB form (2.18). The corresponding Wigner measure is found to be ε→0

wε [uεin ] −→ win = |ain (x)|2 δ(ξ − ∇Sin (x)),

(3.9)

i.e., a mono-kinetic measure concentrated on the initial velocity v = ∇Sin . In this case the phase space ﬂow Ft is projected onto physical space Rd , yielding Xt , the characteristic ﬂow of the Hamilton–Jacobi equation (2.12). More precisely, the following result has been proved in Sparber et al. (2003). Theorem 3.4. Let w(t, x, ξ) be the Wigner measure of the exact solution uε to (2.1) with WKB initial data. Then w(t, x, ξ) = |a(t, x)|2 δ(ξ − ∇S(t, x)) if and only if ρ = |a|2 and v = ∇S are smooth solutions of the leading-order WKB system given by (2.12) and (2.15). This theorem links the theory of Wigner measures with the WKB approximation before caustics. After caustics, the Wigner measure is in general no longer mono-kinetic. However, it can be shown (Sparber et al. 2003, Jin and Li 2003) that for generic initial data uεin and locally away from caustics, the Wigner measure can be decomposed as w(t, x, ξ) =

J

|aj (t, x)|2 δ(ξ − vj (t, x)),

(3.10)

j=1

which is consistent with the multiphase WKB approximation given in (2.21).

4. Finite diﬀerence methods for semiclassical Schr¨ odinger equations 4.1. Basic setting A basic numerical scheme for solving linear partial diﬀerential equations is the well-known ﬁnite diﬀerence method (FD), to be discussed in this section; see, e.g., Strikwerda (1989) for a general introduction. In the following, we shall be mainly interested in its performance as ε → 0. To this end, we shall allow for more general Schr¨ odinger-type PDEs in the form (Markowich and Poupaud 1999) iε∂t uε = H W (x, εDx )uε ,

uε (0, x) = uεin (x),

(4.1)

where H W denotes the Weyl quantization of a classical real-valued phase space Hamiltonian H(x, ξ) ∈ C ∞ (Rd × Rd ), which is assumed to grow at

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most quadratically in x and ξ. For the following we assume that the symbol is a polynomial of order K ∈ N in ξ with C ∞ -coeﬃcients Hk (x), i.e., Hk (x)ξ k , H(x, ξ) = |k|≤K

where k = (k1 , . . . , kd ) ∈ Nd denotes a multi-index with |k| := k1 + · · · + kd . The diﬀerential operator H(x, εDx )W can now be written as x+y ϕ(y) . H(x, εDx )W ϕ(x) = ε|k| Dyk Hk (4.2) 2 y=x |k|≤K

In addition, assume that H(x, εDx )W is essentially self-adjoint on L2 (Rd ),

(A1)

and, for simplicity, ∀ k, α ∈ Nd with |k| ≤ K ∃ Ck,α > 0 : |∂xα Hk (x)| ≤ Ck,α ∀x ∈ Rd .

(A2)

Under these conditions H(x, εDx )W can be shown (Kitada 1980) to generate W a unitary (strongly continuous) semi-group of operators U ε (t) = e−itH /ε , which provides a unique global-in-time solution uε = uε (t) ∈ L2 (Rd ). Next, let

Γ := γ = 1 r1 + · · · + m rm : j ∈ Z for 1 ≤ j ≤ d ⊆ Rd be the lattice generated by the linearly independent vectors r1 , . . . , rd ∈ Rd . For a multi-index k ∈ Nd we construct a discretization of order N of the operator ∂xk as follows: ∂xk ϕ(x) ≈

1 aγ,k ϕ(x + hγ). h|k| γ∈Γ

(4.3)

k

Here ∆x = h ∈ (0, h0 ] is the mesh size, Γk ⊆ Γ is the ﬁnite set of discretization points and aγ,k ∈ R are coeﬃcients satisfying aγ,k γ = k!δ,k , 0 ≤ | | ≤ N + |k| − 1, (B1) γ∈Γk

where δ,k = 1 if = k, and zero otherwise. It is an easy exercise to show that the local discretization error of (4.3) is O(hN ) for all smooth functions if (B1) holds. For a detailed discussion of the linear problem (B1) (i.e., possible choices of the coeﬃcients aγ,k ) we refer to Markowich and Poupaud (1999).

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4.2. Spatial discretization We shall now deﬁne the corresponding ﬁnite diﬀerence discretization of H(x, εDx )W by applying (4.3) directly to (4.2). To this end, we let Hh,ε (x, ξ) = |k| (−i)|k| aγ,k Hk (x) e iγ·ξ/ , (4.4) |k|≤K

γ∈Γk

with = hε being the ratio between the small semiclassical parameter ε and the mesh size h. Then we obtain the ﬁnite diﬀerence discretization of (4.2) in the form H(x, εDx )W ϕ(x) ≈ Hh,ε (x, εDx )W ϕ(x) hγ |k| |k| = (−i) aγ,k Hk x + ϕ(x + hγ). 2 |k|≤K

γ∈Γk

In view of (4.4), the discretization Hh,ε (x, εDx )W is seen to be a bounded operator on L2 (Rd ) and self-adjoint if i|k| aγ,k e iγ·ξ is real-valued for 0 ≤ |k| ≤ K. (B2) γ∈Γk

We shall now collect several properties of such ﬁnite diﬀerence approximations, proved in Markowich, Pietra and Pohl (1999). We start with a spatial consistency result. Lemma 4.1. =

ε ε,h→0 h −→

Let (A1), (B1), (B2) hold and ϕ ∈ S(Rdx × Rdξ ). Then, for

∞, ε,h→0

Hh,ε ϕ −→ Hϕ

in S(Rdx × Rdξ ).

(4.5)

For a given ε > 0, choosing h such that = hε → ∞ corresponds to asymptotically resolving the oscillations of wavelength O(ε) in the solution uε (t, x) to the Schr¨ odinger-type equation (4.1). In the case = const., i.e., putting a ﬁxed number of grid points per oscillation, the symbol Hh,ε (x, ξ) ≡ H (x, ξ) is independent of h and ε, i.e., H (x, ξ) = |k| aγ,k (−i)|k| Hk (x) e iγ·ξ/ . (4.6) |k|≤K

γ∈Γk

ε,h→0

In the case −→ 0, which corresponds to a scheme ignoring the εoscillations, we ﬁnd γ·ξ h,ε→0 H0 (x), Hh,ε ∼ aγ,0 cos γ∈Γ0

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and hence Hh,ε (x, εDx )W does not approximate H(x, εDx )W . We thus cannot expect reasonable numerical results in this case (which will not be investigated further). 4.3. Temporal discretization and violation of gauge invariance For the temporal discretization one can employ the Crank–Nicolson scheme with time step ∆t > 0. This is a widely used time discretization scheme for the Schr¨odinger equation, featuring some desirable properties (see below). We shall comment on the temporal discretizations below. The scheme reads uσn+1 − uσn 1 σ W 1 σ ε + u = 0, n = 0, 1, 2, . . . , (4.7) + iHh,ε (x, εDx ) u ∆t 2 n+1 2 n subject to initial data uσin = uεin (x), where from now on, we shall denote the vector of small parameters by σ = (ε, h, ∆t). Note that the self-adjointness of Hh,ε (x, εDx )W implies that the operator 1 + isHh,ε (x, εDx )W is invertible on L2 (Rd ) for all s ∈ R. Therefore the scheme (4.7) gives welldeﬁned approximations uσn for n = 1, 2, . . . if uεin ∈ L2 (Rd ). Moreover, we remark that it is suﬃcient to evaluate (4.7) at x ∈ hΓ in order to obtain discrete equations for uσn (hγ), γ ∈ Γ. Remark 4.2. For practical computations, one needs to impose artiﬁcial ‘far-ﬁeld’ boundary conditions. Their impact, however, will not be taken into account in the subsequent analysis. By taking the L2 scalar product of (4.7) with 12 uσn+1 + 12 uσn , one can directly infer the following stability result. Lemma 4.3.

The solution of (4.7) satisﬁes uσn L2 = uεin L2 ,

n = 0, 1, 2, . . . .

In other words, the physically important property of mass-conservation also holds on the discrete level. On the other hand, the scheme can be seen to violate the gauge invariance of (4.1). More precisely, one should note that expectation values of physical observables, as deﬁned in (2.5), are invariant under the substitution (gauge transformation) v ε (t, x) = uε (t, x) e iωt/ε ,

ω ∈ R.

In other words, the average value of the observable in the state uε is equal to its average value in the state v ε .

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Remark 4.4. Note that in view of (3.1), the Wigner function is seen to also be invariant under this substitution, i.e., ∀ω ∈ R : wε [uε (t)] = wε [uε (t) e iωt/ε ] ≡ wε [v ε (t)]. On the other hand, using this gauge transformation the Schr¨odinger equation (4.1) transforms to (4.8) iε∂t v ε = H(x, εDx )W + ω v ε , v ε (0, x) = uεin (x), which implies that the zeroth-order term H0 (x) in (4.2) is replaced by H0 (x) + ω, while the other coeﬃcients Hk (x), k = 0, remain unchanged. In physical terms, H0 (x) corresponds to a scalar (static) potential V (x). The corresponding force ﬁeld obtained via F (x) = ∇H0 (x) = ∇(H0 (x) + ω) is unchanged by the gauge transformation and thus (4.8) can be considered (physically) equivalent to (4.1). The described situation, however, is completely diﬀerent for the diﬀerence scheme outlined above. Indeed, a simple calculation shows that the discrete gauge transformation vnσ = uσn e iωtn /ε does not commute with the discretization (4.7), up to adding a real constant to the potential. Thus, the discrete approximations of average values of observables depend on the gauging of the potential. In other words, the discretization method is not time-transverse-invariant. 4.4. Stability–consistency analysis for FD in the semiclassical limit The consistency–stability concept of classical numerical analysis provides a framework for the convergence analysis of ﬁnite diﬀerence discretizations of linear partial diﬀerential equations. Thus, for ε > 0 ﬁxed it is easy to prove that the scheme (4.7) is convergent of order N in space and order 2 in time if the exact solution uε (t, x) is suﬃciently smooth. Therefore, again for ﬁxed ε > 0, we conclude convergence of the same order for average values of physical observables provided a(x, ξ) is smooth. However, due to the oscillatory nature solutions to (4.1) the local discretization error of the ﬁnite diﬀerence schemes and, consequently, also the global discretization error, in general tend to inﬁnity as ε → 0. Thus, the classical consistency–stability theory does not provide uniform results in the classical limit. Indeed, under the reasonable assumption that, for all multi-indices j1 and j2 ∈ Nd , ∂ |j1 |+|j2 | ε ε→0 u (t, x) ∼ ε−|j1 |−j2 ∂tj1 ∂xj2

in L2 (Rd ),

locally uniformly in t ∈ R, the classical stability–consistency analysis gives

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the following bound for the global L2 -discretization error: N (∆t)2 h O + O . ε3 εN +1 The situation is further complicated by the fact that for any ﬁxed t ∈ R, the solution uε (t, ·) of (4.1) and its discrete counterpart uσn in general converge only weakly in L2 (Rd ) as ε → 0, respectively, σ → 0. Thus, the limit processes ε → 0, σ → 0 do not commute with the quadratically nonlinear operation (2.5), needed to compute the expectation value of physical observables a[uε (t)]. In practice, one is therefore interested in ﬁnding conditions on the mesh size h and the time step ∆t, depending on ε in such a way that the expectation values of physical observables in discrete form approximate a[uε (t)] uniformly as ε → 0. To this end, let tn = n∆t, n ∈ N, and denote aσ (tn ) := a(·, εDx )W uσn , uσn . The function aσ (t), t ∈ R, is consequently deﬁned by piecewise linear interpolation of the values aσ (tn ). We seek conditions on h, k such that, for all m a ∈ S(Rm x × Rξ ), lim (aσ (t) − a[uε (t)]) = 0 uniformly in ε ∈ (0, ε0 ],

h,∆t→0

(4.9)

and locally uniformly in t ∈ R. A rigorous study of this problem will be given by using the theory of Wigner measures applied in a discrete setting. Denoting the Wigner transformation (on the scale ε) of the ﬁnite diﬀerence solution uσn by wσ (tn ) := wε [uσn ], and deﬁning, as before, wσ (t), for any t ∈ R, by the piecewise linear interpolation of wσ (tn ), we conclude that (4.9) is equivalent to proving, locally uniformly in t, lim (wσ (t) − wε (t)) = 0 in S (Rdx × Rdξ ), uniformly in ε ∈ (0, ε0 ], (4.10)

h,∆t→0

where wε (t) is the Wigner transform of the solution uε (t) of (4.1). We shall now compute the accumulation points of the sequence {wσ (t)}σ as σ → 0. We shall see that for any given subsequence {σn }n∈N , the set of Wigner measures of the diﬀerence schemes µ(t) := lim wσn (t) n→∞

depends decisively on the relative sizes of ε, h and ∆t. Clearly, in those cases in which µ = w, where w denotes the Wigner measure of the exact solution uε (t), the desired property (4.10) follows. On the other hand (4.10) does not hold if the measures µ and w are diﬀerent. Such a Wigner-measure-based

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study of ﬁnite diﬀerence schemes has been conducted in Markowich et al. (1999) and Markowich and Poupaud (1999). The main result given there is as follows. Theorem 4.5. Fix a scale ε > 0 and denote by µ the Wigner measure of the discretization (4.7) as σ → 0. Then we have the following results. Case 1. Let h/ε → 0 (or, equivalently, → ∞). (i) If ∆t/ε → 0, then µ satisﬁes ∂t µ + {H, µ} = 0,

µ(0, x, ξ) = win (x, ξ).

R+ ,

(ii) If ∆t/ε → ω ∈ then µ solves ∂ 2 ω µ+ arctan H , µ = 0, ∂t ω 2

µ(0, x, ξ) = win (x, ξ).

(iii) If ∆t/ε → ∞ and if in addition there exists C > 0 such that |H(x, ξ)| ≥ C, ∀x, ξ ∈ Rd , then µ is constant in time, i.e., µ(t, x, ξ) ≡ µin (x, ξ),

∀ t ∈ R.

Case 2. If h/ε → 1/ ∈ R+ , then the assertions (i)–(iii) hold true, with H replaced by H deﬁned in (4.6). The proof of this result proceeds similarly to the derivation of the phase space Liouville equation (3.7), in the continuous setting. Note that Theorem 4.5 implies that, as ε → 0, expectation values for physical observables in the state uε (t), computed via the Crank–Nicolson ﬁnite diﬀerence scheme, are asymptotically correct only if both spatial and temporal oscillations of wavelength ε are accurately resolved. Remark 4.6. Time-irreversible ﬁnite diﬀerence schemes, such as the explicit (or implicit) Euler scheme, behave even less well, as they require ∆t = o(ε2 ) in order to guarantee asymptotically correct numerically computed observables; see Markowich et al. (1999).

5. Time-splitting spectral methods for semiclassical Schr¨ odinger equations 5.1. Basic setting, ﬁrst- and second-order splittings As has been discussed before, ﬁnite diﬀerence methods do not perform well in computing the solution to semiclassical Schr¨odinger equations. An alternative is given by time-splitting trigonometric spectral methods, which will be discussed in this subsection; see also McLachlan and Quispel (2002) for a broad introduction to splitting methods. For the sake of notation, we shall introduce the method only in the case of one space dimension, d = 1.

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Generalizations to d > 1 are straightforward for tensor product grids and the results remain valid without modiﬁcations. In the following, we shall therefore study the one-dimensional version of equation (2.1), i.e., iε∂t uε = −

ε2 ∂xx uε + V (x)uε , 2

uε (0, x) = uεin (x),

(5.1)

for x ∈ [a, b], 0 < a < b < +∞, equipped with periodic boundary conditions uε (t, a) = uε (t, b),

∂x uε (t, a) = ∂x uε (t, b),

∀ t ∈ R.

We choose the spatial mesh size ∆x = h > 0 with h = (b − a)/M for some M ∈ 2N, and an ε-independent time step ∆t ≡ k > 0. The spatio-temporal grid points are then given by xj := a + jh, j = 1, . . . , M,

tn := nk, n ∈ N.

In the following, let uε,n be the numerical approximation of uε (xj , tn ), for j j = 1, . . . , M .

First-order time-splitting spectral method (SP1 ) The Schr¨ odinger equation (5.1) is solved by a splitting method, based on the following two steps. Step 1. From time t = tn to time t = tn+1 , ﬁrst solve the free Schr¨odinger equation iε∂t uε +

ε2 ∂xx uε = 0. 2

(5.2)

Step 2. On the same time interval, i.e., t ∈ [tn , tn+1 ], solve the ordinary diﬀerential equation (ODE) iε∂t uε − V (x)uε = 0,

(5.3)

with the solution obtained from Step 1 as initial data for Step 2. Equation (5.3) can be solved exactly since |u(t, x)| is left invariant under (5.3), u(t, x) = |u(0, x)| e iV (x)t . In Step 1, the linear equation (5.2) will be discretized in space by a (pseudo-) spectral method (see, e.g., Fornberg (1996) for a general introduction) and consequently integrated in time exactly. More precisely, one obtains at time t = tn+1 u(tn+1 , x) ≈ uε,n+1 = e iV (xj )k/ε uε,∗ j j ,

j = 0, 1, 2, . . . , M,

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ε with initial value uε,0 j = uin (xj ), and

1 M

uε,∗ j = where γ =

2πl b−a

and

u ε,n =

u ε,n M

M/2−1 2

iγ (xj −a) e iεkγ /2 u ε,n , e

=−M/2

denote the Fourier coeﬃcients of uε,n , i.e., −iγ (xj −a) uε,n , j e

=−

j=1

M M ,..., − 1. 2 2

Note that the only time discretization error of this method is the splitting error, which is ﬁrst-order in k = ∆t, for any ﬁxed ε > 0. Strang splitting (SP2 ) In order to obtain a scheme which is second-order in time (for ﬁxed ε > 0), one can use the Strang splitting method , i.e., on the time interval [tn , tn+1 ] we compute = e iV (xj )k/2ε uε,∗∗ uε,n+1 j j ,

j = 0, 1, 2, . . . , M − 1,

where uε,∗∗ = j with

u ε,∗

1 M

M/2−1 2

iγ (xj −a) e iεkγ /2 u ε,∗ , e

=−M/2

denoting the Fourier coeﬃcients of uε,∗ given by iV (xj )k/2ε ε,n uj . uε,∗ j =e

Again, the overall time discretization error comes solely from the splitting, which is now (formally) second-order in ∆t = k for ﬁxed ε > 0. Remark 5.1. Extensions to higher-order (in time) splitting schemes can be found in the literature: see, e.g., Bao and Shen (2005). For rigorous investigations of the long time error estimates of such splitting schemes we refer to Dujardin and Faou (2007a, 2007b) and the references given therein. In comparison to ﬁnite diﬀerence methods, the main advantage of such splitting schemes is that they are gauge-invariant: see the discussion in Section 4 above. Concerning the stability of the time-splitting spectral approximations with variable potential V = V (x), one can prove the following lemma (see Bao, Jin and Markowich (2002)), in which we denote U = (u1 , . . . , uM ) and · l2 the usual discrete l2 -norm on the interval [a, b], i.e., 1/2 M b−a 2 U l2 = |uj | . M j=1

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Lemma 5.2. The time-splitting spectral schemes SP1 and SP2 are unconditionally stable, i.e., for any mesh size h and any time step k, we have ε U ε,n l2 = U ε,0 l2 ≡ Uin l2 ,

n ∈ N,

and consequently ε,0 uε,n int L2 (a,b) = uint L2 (a,b) ,

n ∈ N,

where uε,n int denotes the trigonometric polynomial interpolating ε,n ε,n {(x1 , uε,n 1 ), (x1 , u1 ), . . . , (xM , uM )}.

In other words, time-splitting spectral methods satisfy mass-conservation on a fully discrete level. 5.2. Error estimate of SP1 in the semiclassical limit To get a better understanding of the stability of spectral methods in the classical limit ε → 0, we shall establish the error estimates for SP1. Assume that the potential V (x) is (b − a)-periodic, smooth, and satisﬁes m d ≤ Cm , (A) dxm V ∞ L [a,b] for some constant Cm > 0. Under these assumptions it can be shown that the solution uε = uε (t, x) of (5.1) is (b−a)-periodic and smooth. In addition, we assume m +m ∂ 1 2 ε Cm +m ≤ m11+m22 , (B) ∂tm1 ∂xm2 u ε 2 C([0,T ];L [a,b])

for all m, m1 , m2 ∈ N ∪ {0}. Thus, we assume that the solution oscillates in space and time with wavelength ε, but no smaller. Remark 5.3. The latter is known to be satisﬁed if the initial data uεin only invoke oscillations of wavelength ε (but no smaller). Theorem 5.4. Let V (x) satisfy assumption (A) and let uε (t, x) be a solution of (5.1) satisfying (B). Denote by uε,n int the interpolation of the discrete approximation obtained via SP1. Then, if ∆t ∆x = O(1), = O(1), ε ε as ε → 0, we have that, for all m ∈ N and tn ∈ [0, T ], m ε T ∆x CT ∆t u (tn ) − uε,n 2 , + int L (a,b) ≤ Gm ∆t ε(b − a) ε

(5.4)

where C > 0 is independent of ε and m and Gm > 0 is independent of ε, ∆x, ∆t.

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1.2

1

1 0.5

0.8 0.6

0

0.4 −0.5

0.2 0

−1

0

0.2

0.4

0.6

0.8

1

0

0.2

(a)

0.4

0.6

0.8

1

(b)

Figure 5.1. Numerical solution of (a) ρε and (b) j ε at t = 0.54, as given in Example 5.5. In the ﬁgure, the solution computed by using SP2 for 1 , is superimposed, with the limiting ρ and j obtained by ε = 0.0008, h = 512 taking moments of the Wigner measure solution of (3.8).

The proof of this theorem is given in Bao et al. (2002), where a similar result is also shown for SP2. Now let δ > 0 be a desired error bound such that uε (tn ) − uε,n int L2 [a,b] ≤ δ holds uniformly in ε. Then Theorem 5.4 suggests the following meshing strategy on O(1) time and space intervals: ∆t ∆x = O(δ), = O δ 1/m (∆t)1/m , (5.5) ε ε where m ≥ 1 is an arbitrary integer, assuming that Gm does not increase too fast as m → ∞. This meshing is already more eﬃcient than what is needed for ﬁnite diﬀerences. In addition, as will be seen below, the conditions (5.5) can be strongly relaxed if, instead of resolving the solution uε (t, x), one is only interested in the accurate numerical computation of quadratic observable densities (and thus asymptotically correct expectation values). Example 5.5. This is an example from Bao et al. (2002). The Schr¨ odinger equation (2.1) is solved with V (x) = 10 and the initial data ρin (x) = exp(−50(x − 0.5)2 ), 1 Sin (x) = − ln(exp(5(x − 0.5)) + exp(−5(x − 0.5))), 5

x ∈ R.

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The computational domain is restricted to [0, 1] equipped with periodic boundary conditions. Figure 5.1 shows the solution of the limiting position density ρ and current density j obtained by taking moments of w, satisfying the Liouville equation (3.8). This has to be compared with the oscillatory ρε and j ε , obtained by solving the Schr¨odinger equation (2.1) using SP2. As one can see, these oscillations are averaged out in the weak limits ρ, j. 5.3. Accurate computation of quadratic observable densities using time-splitting We shall again invoke the theory of Wigner functions and Wigner measures. To this end, let uε (t, x) be the solution of (5.1) and let wε (t, x, ξ) be the corresponding Wigner transform. Keeping in mind the results of Section 3, we see that the ﬁrst-order splitting scheme SP1 corresponds to the following time-splitting scheme for the Wigner equation (3.2). Step 1. For t ∈ [tn , tn+1 ], ﬁrst solve the linear transport equation ∂t wε + ξ ∂x wε = 0.

(5.6)

Step 2. On the same time interval, solve the non-local (in space) ordinary diﬀerential equation ∂t wε − Θε [V ]wε = 0,

(5.7)

with initial data obtained from Step 1 above. In (5.6), the only possible ε-dependence stems from the initial data. In addition, in (5.7) the limit ε → 0 can be easily carried out (assuming sufﬁcient regularity of the potential V (x)) with k = ∆t ﬁxed. In doing so, one consequently obtains a time-splitting scheme of the limiting Liouville equation (3.8) as follows. Step 1. For t ∈ [tn , tn+1 ], solve ∂t w + ξ ∂x w0 = 0. Step 2. Using the outcome of Step 1 as initial data, solve, on the same time interval, ∂t w − ∂x V ∂ξ w0 = 0. Note that in this scheme no error is introduced other than the splitting error, since the time integrations are performed exactly. These considerations, which can easily be made rigorous (for smooth potentials), show that a uniform time-stepping (i.e., an ε-independent k = ∆t) of the form ∆t = O(δ)

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combined with the spectral mesh size control given in (5.5) yields the following error: ε,n L2 (a,b) ≤ δ, wε (tn ) − wint

uniformly in ∆t as ε → 0. Essentially this implies that a ﬁxed number of grid points in every spatial oscillation of wavelength ε combined with ε-independent time-stepping is suﬃcient to guarantee the accurate computation of (expectation values of) physical observables in the classical limit. This strategy is therefore clearly superior to ﬁnite diﬀerence schemes, which require k/ε → 0 and h/ε → 0, even if one only is interested in computing physical observables. Remark 5.6. Time-splitting methods have been proved particularly successful in nonlinear situations: see the references given in Section 15.4 below.

6. Moment closure methods We have seen before that a direct numerical calculation of uε is numerically very expensive, particularly in higher dimensions, due to the mesh and time step constraint (5.5). In order to circumvent this problem, the asymptotic analysis presented in Sections 2 and 3 can be invoked in order to design asymptotic numerical methods which allow for an eﬃcient numerical simulation in the limit ε → 0. The initial value problem (3.8)–(3.9) is the starting point of the numerical methods to be described below. Most recent computational methods are derived from, or closely related to, this equation. The main advantage is that (3.8)–(3.9) correctly describes the limit of quadratic densities of uε (which in itself exhibits oscillations of wavelength O(ε)), and thus allows a numerical mesh size independent of ε. However, we face the following major diﬃculties in the numerical approximation. (1) High-dimensionality. The Liouville equation (3.8) is deﬁned in phase space, and thus the memory requirement exceeds the current computational capability in d ≥ 3 space dimensions. (2) Measure-valued initial data. The initial data (3.9) form a delta measure and the solution at later time remains one (for a single-valued solution), or a summation of several delta functions (for a multivalued solution (3.10)). In the past few years, several new numerical methods have been introduced to overcome these diﬃculties. In the following, we shall brieﬂy describe the basic ideas of these methods.

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6.1. The concept of multivalued solutions In order to overcome the problem of high-dimensionality one aims to approximate w(t, x, p) by using averaged quantities depending only on t, x. This is a well-known technique in classical kinetic theory, usually referred to as moment closure. A basic example is provided by the result of Theorem 3.4, which tells us that, as long as the WKB analysis of Section 2.2 is valid (i.e., before the appearance of the ﬁrst caustic), the Wigner measure is given by a mono-kinetic distribution in phase space, i.e., w(t, x, ξ) = ρ(t, x)δ(ξ − v(t, x)), where one identiﬁes ρ = |a|2 and v = ∇S. The latter solve the pressure-less Euler system ∂t ρ + div(ρv) = 0, ∂t v + (v · ∇)v + ∇V = 0,

ρ(0, x) = |ain |2 (x), v(0, x) = ∇Sin (x),

(6.1)

which, for smooth solutions, is equivalent to the system of transport equation (2.14) coupled with the Hamilton–Jacobi equation (2.12), obtained through the WKB approximation. Thus, instead of solving the Liouville equation in phase space, one can as well solve the system (6.1), which is posed on physical space Rt × Rdx . Of course, this can only be done until the appearance of the ﬁrst caustic, or, equivalently, the emergence of shocks in (6.1). In order to go beyond that, one might be tempted to use numerical methods based on the unique viscosity solution (see Crandall and Lions (1983)) for (6.1). However, the latter does not provide the correct asymptotic description – the multivalued solution – of the wave function uε (t, x) beyond caustics. Instead, one has to pass to so-called multivalued solutions, based on higher-order moment closure methods. This fact is illustrated in Figure 6.1, which shows the diﬀerence between viscosity solutions and multivalued solutions. Figures 6.1(a) and 6.1(b) are the two diﬀerent solutions for the following eikonal equation (in fact, the zero-level set of S): ∂t S + |∇x S| = 0,

x ∈ R2 .

(6.2)

This equation, corresponding to H(ξ) = |ξ|, arises in the geometric optics limit of the wave equation and models two circular fronts moving outward in the normal direction with speed 1; see Osher and Sethian (1988). As one can see, the main diﬀerence occurs when the two fronts merge. Similarly, Figures 6.1(c) and 6.1(d) show the diﬀerence between the viscosity and the multivalued solutions to the Burgers equation ∂t v +

1 ∂x v 2 = 0, 2

x ∈ R.

(6.3)

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Eikonal equation (a)

(b)

Burgers equation (c)

(d)

Figure 6.1. Multivalued solution (left) versus viscosity solution (right). (a, b) Zero-level set curves (at diﬀerent times) of solutions to the eikonal equation (6.2). (c, d) Two solutions to the Burgers equation (6.3) before and after the formation of a shock.

This is simply the second equation in the system (6.1) for V (x) = 0, written in divergence form. The solution begins as a sinusoidal function and then forms a shock. Clearly, the solutions are diﬀerent after the shock formation. 6.2. Moment closure The moment closure idea was ﬁrst introduced by Brenier and Corrias (1998) in order to deﬁne multivalued solutions to the Burgers equation, and seems to be the natural choice in view of the multiphase WKB expansion given in (2.21). The method was then used numerically in Engquist and Runborg (1996) (see also Engquist and Runborg (2003) for a broad review) and Gosse (2002) to study multivalued solutions in the geometrical optics regime of hyperbolic wave equations. A closely related method is given in Benamou (1999), where a direct computation of multivalued solutions to Hamilton– Jacobi equations is performed. For the semiclassical limit of the Schr¨odinger equation, this was done in Jin and Li (2003) and then Gosse et al. (2003).

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In order to describe the basic idea, let d = 1 and deﬁne m (t, x) = ξ w(t, x, ξ) dξ, = 1, 2, . . . , L ∈ N,

(6.4)

R

i.e., the th moment (in velocity) of the Wigner measure. By multiplying the Liouville equation (3.8) by ξ and integrating over Rξ , one obtains the following moment system: ∂t m0 + ∂x m1 = 0, ∂t m1 + ∂x m2 = −m0 ∂x V, .. . ∂t mL−1 + ∂x mL = −(L − 1)mL−2 ∂x V. Note that this system is not closed , since the equation determining the th moment involves the ( + 1)th moment. The δ-closure As already mentioned in (3.10), locally away from caustics the Wigner measure of uε as ε → 0 can be written as w(t, x, ξ) =

J

ρj (t, x)δ(ξ − vj (t, x)),

(6.5)

j=1

where the number of velocity branches J can in principle be determined a priori from ∇Sin (x). For example, in d = 1, it is the total number of inﬂection points of v(0, x): see Gosse et al. (2003). Using this particular form (6.5) of w with L = 2J provides a closure condition for the moment system above. More precisely, one can express the last moment m2J as a function of all of the lower-order moments (Jin and Li 2003), i.e., m2J = g(m0 , m1 , . . . , m2J−1 ).

(6.6)

This consequently yields a system of 2J × 2J equations (posed in physical space), which eﬀectively provides a solution of the Liouville equation, before the generation of a new phase, yielding a new velocity vj , j > J. It was shown in Jin and Li (2003) that this system is only weakly hyperbolic, in the sense that the Jacobian matrix of the ﬂux is a Jordan block, with only J distinct eigenvalues v1 , v2 , . . . , vJ . This system is equivalent to J pressureless gas equations (6.1) for (ρj , vj ) respectively. In Jin and Li (2003) the explicit ﬂux function g in (6.6) was given for J ≤ 5. For larger J a numerical procedure was proposed for evaluating g. Since the moment system is only weakly hyperbolic, with phase jumps which are under-compressive shocks (Gosse et al. 2003), standard shock-capturing schemes such as the Lax–Friedrichs scheme and the Godunov scheme

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face severe numerical diﬃculties as in the computation of the pressure-less gas dynamics: see Bouchut, Jin and Li (2003), Engquist and Runborg (1996) or Jiang and Tadmor (1998). Following the ideas of Bouchut et al. (2003) for the pressure-less gas system, a kinetic scheme derived from the Liouville equation (3.8) with the closure condition (6.6) was used in Jin and Li (2003) for this moment system. The Heaviside closure Another type of closure was introduced by Brenier and Corrias (1998) using the following ansatz, called H-closure, to obtain the J-branch velocities vj , with j = 1, . . . , J: w(t, x, ξ) =

J

(−1)j−1 H(vj (t, x) − ξ).

(6.7)

j=1

This type of closure condition for (3.8) arises from an entropy-maximization principle: see Levermore (1996). Using (6.7), one arrives at (6.6) with L = J. The explicit form of the corresponding function g(m0 , . . . , m2J−1 ) for J < 5 is available analytically in Runborg (2000). Note that this method decouples the computation of velocities vj from the densities ρj . In fact, to obtain the latter, Gosse (2002) has proposed solving the following linear conservation law (see also Gosse and James (2002) and Gosse et al. (2003)): ∂t ρj + ∂x (ρj vj ) = 0,

for j = 1, . . . , N.

The numerical approximation to this linear transport with variable or even discontinuous ﬂux is not straightforward. Gosse et al. (2003) used a semiLagrangian method that uses the method of characteristics, requiring the time step to be suﬃciently small for the case of non-zero potentials. The corresponding method is usually referred to as H-closure. Note that in d = 1 the H-closure system is a non-strictly rich hyperbolic system, whereas the δ-closure system described before is only weakly hyperbolic. Thus one expects a better numerical resolution from the H-closure approach, which, however, is much harder to implement in the higher dimension. In d = 1, the mathematical equivalence of the two moment systems was proved in Gosse et al. (2003). Remark 6.1. Multivalued solutions also arise in the high-frequency approximation of nonlinear waves, for example, in the modelling of electron transport in vacuum electronic devices: see, e.g., Granastein, Parker and Armstrong (1999). There the underlying equations are the Euler–Poisson equations, a nonlinearly coupled hyperbolic–elliptic system. The multivalued solution of the Euler–Poisson system also arises for electron sheet initial data, and can be characterized by a weak solution of the Vlasov–Poisson equation: see Majda, Majda and Zheng (1994). Similarly, the work of Li,

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W¨ohlbier, Jin and Booske (2004) uses the moment closure ansatz (6.6) for the Vlasov–Poisson system; see also W¨ohlbier, Jin and Sengele (2005). For multivalued (or multiphase) solution of the semiclassical limit of nonlinear dispersive waves using the closely related method of Whitham’s modulation theory, we refer to Whitham (1974) and Flaschka, Forest and McLaughlin (1980). Finally, we mention that multivalued solutions also arise in supply chain modelling: see, e.g., Armbruster, Marthaler and Ringhofer (2003). In summary, the moment closure approach yields an Eulerian method deﬁned in the physical space which oﬀers a greater eﬃciency compared to the computation in phase space. However, when the number of phases J ∈ N becomes very large and/or in dimensions d > 1, the moment systems become very complex and thus diﬃcult to solve. In addition, in high space dimensions, it is very diﬃcult to estimate a priori the total number of phases needed to construct the moment system. Thus it remains an interesting and challenging open problem to develop more eﬃcient and general physicalspace-based numerical methods for the multivalued solutions.

7. Level set methods 7.1. Eulerian approach Level set methods have been recently introduced for computing multivalued solutions in the context of geometric optics and semiclassical analysis. These methods are rather general, and applicable to any (scalar) multi-dimensional quasilinear hyperbolic system or Hamilton–Jacobi equation (see below). We shall now review the basic ideas, following the lines of Jin and Osher (2003). The original mathematical formulation is classical; see for example Courant and Hilbert (1962). Computation of the multivalued phase Consider a general d-dimensional Hamilton–Jacobi equation of the form ∂t S + H(x, ∇S) = 0,

S(0, x) = Sin (x).

(7.1)

For example, in present context of semiclassical analysis for Schr¨ odinger equations, 1 H(x, ξ) = |ξ|2 + V (x), 2 while for applications in geometrical optics (i.e., the high-frequency limit of the wave equation), H(x, ξ) = c(x)|ξ|, with c(x) denoting the local sound (or wave) speed. Introducing, as before, a velocity v = ∇S and taking the gradient of (7.1), one gets an equivalent

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equation (at least for smooth solutions) in the form (Jin and Xin 1998): ∂t v + (∇ξ H(x, v) · ∇)v + ∇x H(x, v) = 0,

v(0, x) = ∇x Sin (x).

(7.2)

Then, in d ≥ 1 space dimensions, deﬁne level set functions φj , for j = 1, . . . , d, via ∀(t, x) ∈ R × Rd :

φj (t, x, ξ) = 0 at ξ = vj (t, x).

In other words, the (intersection of the) zero-level sets of all {φj }dj=1 yield the graph of the multivalued solution vj (t, x) of (7.2). Using (7.2) it is easy to see that φj solves the following initial value problem: ∂t φj + {H(x, ξ), φj }φj = 0,

φj (x, ξ, 0) = ξj − vj (0, x),

(7.3)

which is simply the phase space Liouville equation. Note that in contrast to (7.2), this equation is linear and thus can be solved globally in time. In doing so, one obtains, for all t ∈ R, the multivalued solution to (7.2) needed in the asymptotic description of physical observables. See also Cheng, Liu and Osher (2003). Computation of the particle density It remains to compute the classical limit of the particle density ρ(t, x). To do so, a simple idea was introduced in Jin, Liu, Osher and Tsai (2005a). This method is equivalent to a decomposition of the measure-valued initial data (3.9) for the Liouville equation. More precisely, a simple argument based on the method of characteristics (see Jin, Liu, Osher and Tsai (2005b)) shows that the solution to (3.8)–(3.9) can be written as w(t, x, ξ) = ψ(t, x, ξ)

d

δ(φj (t, x, ξ)),

j=1

where φj (t, x, ξ) ∈ Rn , j = 1, . . . , d, solves (7.3) and the auxiliary function ψ(t, x, ξ) again satisﬁes the Liouville equation (3.8), subject to initial data: ψ(0, x, ξ) = ρin (x). The ﬁrst two moments of w with respect to ξ (corresponding to the particle ρ and current density J = ρu) can then be recovered through d ρ(t, x) = ψ(t, x, ξ) δ(φj (t, x, ξ)) dξ, Rd

u(t, x) =

1 ρ(t, x)

j=1

ξψ(t, x, ξ) Rd

d

δ(φj (t, x, ξ)) dξ.

j=1

Thus the only time one has to deal with the delta measure is at the numerical output, while during the time evolution one simply solves for φj and ψ, both

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of which are smooth L∞ -functions. This avoids the singularity problem mentioned earlier, and gives numerical methods with much better resolution than solving (3.8)–(3.9) directly, e.g., by approximating the initial delta function numerically. An additional advantage of this level set approach is that one only needs to care about the zero-level sets of φj . Thus the technique of local level set methods developed in Adalsteinsson and Sethian (1995) and Peng, Merriman, Osher, Zhao and Kang (1999) can be used. One thereby restricts the computational domain to a narrow band around the zero-level set, in order to reduce the computational cost to O(N ln N ), for N computational points in the physical space. This is a nice alternative for dimension reduction of the Liouville equation. When solutions for many initial data need to be computed, fast algorithms can be used: see Fomel and Sethian (2002) or Ying and Cand`es (2006). Example 7.1. Consider (2.1) in d = 1 with periodic potential V (x) = cos(2x + 0.4)), and WKB initial data corresponding to Sin (x) = sin(x + 0.15), 1 π 2 π 2 + exp − x − . ρin (x) = √ exp − x + 2 2 2 π Figure 7.1 shows the time evolution of the velocity and the corresponding density computed by the level set method described above. The velocity eventually develops some small oscillations with higher frequency, which require a ﬁner grid to resolve. Remark 7.2. The outlined ideas have been extended to general linear symmetric hyperbolic systems in Jin et al. (2005a). So far, however, level set methods have not been formulated for nonlinear equations, except for the one-dimensional Euler–Poisson equations (Liu and Wang 2007), where a three-dimensional Liouville equation has to be used in order to calculate the corresponding one-dimensional multivalued solutions. 7.2. The Lagrangian phase ﬂow method While the Eulerian level set method is based on solving the Liouville equation (3.8) on a ﬁxed mesh, the Lagrangian (or particle) method , is based on solving the Hamiltonian system (3.6), which is simply the characteristic ﬂow of the Liouville equation (3.7). In geometric optics this idea is referred to as ray tracing (Cerven´ y 2001), and the curves x(t, y, ζ), ξ(t, y, ζ) ∈ Rd , obtained by solving (3.6), are usually called bi-characteristics. Remark 7.3. Note that ﬁnding an eﬃcient way to numerically solve Hamiltonian ODEs, such as (3.6), is a problem of great (numerical) interest in its own right: see, e.g., Leimkuhler and Reich (2004).

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(a)

(b)

Figure 7.1. Example 7.1: (a) multivalued velocity v at time T = 0.0, 6.0, and 12.0, (b) corresponding density ρ.

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Here we shall brieﬂy describe a fast algorithm, called the phase ﬂow method in Ying and Cand`es (2006), which is very eﬃcient if multiple initial data, as is often the case in practical applications, are to be propagated by the Hamiltonian ﬂow (3.6). Let Ft : R2d → R2d be the phase ﬂow deﬁned by Ft (y, ζ) = (x(t, y, ζ), ξ(t, y, ζ)), t ∈ R. A manifold M ⊂ Rdx × Rdξ is said to be invariant if Ft (M) ⊂ M. For the autonomous ODEs, such as (3.6), a key property of the phase map is the one-parameter group structure, Ft ◦ Fs = Ft+s . Instead of integrating (3.6) for each individual initial condition (y, ζ), up to, say, time T , the phase ﬂow method constructs the complete phase map FT . To this end, one ﬁrst constructs the Ft for small times using standard ODE integrators and then builds up the phase map for larger times via a local interpolation scheme together with the group property of the phase ﬂow. Speciﬁcally, ﬁx a small time τ > 0 and suppose that T = 2n τ . Step 1. Begin with a uniform or quasi-uniform grid on M. Step 2. Compute an approximation of the phase map Fτ at time τ . The value of Fτ at each grid point is computed by applying a standard ODE or Hamiltonian integrator with a single time step of length τ . The value of Fτ at any other point is deﬁned via a local interpolation. Step 3. For k = 1, . . . , n, construct F2k τ using the group relation F2k τ = F2k−1 τ ◦ F2k−1 τ . Thus, for each grid point (y, ζ), F2k τ (y, ζ) = F2k−1 τ (F2k−1 τ (y, ζ)), while F2k τ is deﬁned via a local interpolation at any other point. When the algorithm terminates, one obtains an approximation of the whole phase map at time T = 2n τ . This method is clearly much faster than solving each for initial condition independently.

8. Gaussian beam methods: Lagrangian approach A common numerical problem with all numerical approaches based on the Liouville equation with mono-kinetic initial data (3.8)–(3.9) is that the particle density ρ(t, x) blows up at caustics. Another problem is the loss of phase information when passing through a caustic point, i.e., the loss of the Keller–Maslov index (Maslov 1981). To this end, we recall that the Wigner measure only sees the gradient of the phase: see (3.10). The latter can be ﬁxed by incorporating this index into a level set method as was done in Jin and Yang (2008)). Nevertheless, one still faces the problem that any numerical method based on the Liouville equation is unable to handle wave

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interference eﬀects. The Gaussian beam method (or Gaussian wave packet approach, as it is called in quantum chemistry: see Heller (2006)), is an eﬃcient approximate method that allows an accurate computation of the wave amplitude around caustics, and in addition captures the desired phase information. This now classical method was developed in Popov (1982), Ralston (1982) and Hill (1990), and has seen increasing activity in recent years. In the following, we shall describe the basic ideas, starting with its classical Lagrangian formulation.

8.1. Lagrangian dynamics of Gaussian beams Similar to the WKB method, the approximate Gaussian beam solution is given in the form ϕε (t, x, y) = A(t, y) e iT (t,x,y)/ε ,

(8.1)

where the variable y = y(t, y0 ) will be determined below and the phase T (t, x, y) is given by 1 T (t, x, y) = S(t, y) + p(t, y) · (x − y) + (x − y) M (t, y)(x − y) + O(x − y|3 ). 2 This is reminiscent of the Taylor expansion of the phase S around the point y, upon identifying p = ∇S ∈ Rd , M = ∇2 S, the Hessian matrix. The idea is now to allow the phase T to be complex-valued (in contrast to WKB analysis), and choose the imaginary part of M ∈ Cn×n positive deﬁnite so that (8.1) indeed has a Gaussian proﬁle. Plugging the ansatz (8.1) into the Schr¨ odinger equation (2.1), and ignoring the higher-order terms in both ε and (y − x), one obtains the following system of ODEs: dy = p, dt dM = −M 2 − ∇2y V, dt dS 1 = |p|2 − V, dt 2

dp = −∇y V, dt

(8.2) (8.3)

dA 1 = − Tr(M ) A, dt 2

(8.4)

where p, V, M, S and A have to be understood as functions of (t, y(t, y0 )). The latter deﬁnes the centre of a Gaussian beam. Equations (8.2)–(8.4) can be considered as the Lagrangian formulation of the Gaussian beam method, with (8.2) furnishing a classical ray-tracing algorithm. We further note that (8.3) is a Riccati equation for M . We state the main properties of (8.3), (8.4) in the following theorem, the proof of which can be found in Ralston (1982) (see also Jin, Wu and Yang (2008b)).

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Theorem 8.1. Let P (t, y(t, y0 )) and R(t, y(t, y0 )) be the (global) solutions of the equations dP dR (8.5) = R, = −(∇2y V )P, dt dt with initial conditions P (0, y0 ) = Id,

R(0, y0 ) = M (0, y0 ),

(8.6)

where Id is the identity matrix and Im(M (0, y0 )) is positive deﬁnite. Assume that M (0, y0 ) is symmetric. Then, for each initial position y0 , we have the following properties. (1) P (t, y(t, y0 )) is invertible for all t > 0. (2) The solution to equation (8.3) is given by M (t, y(t, y0 )) = R(t, y(t, y0 ))P −1 (t, y(t, y0 )).

(8.7)

(3) M (t, y(t, y0 )) is symmetric and Im(M (t, y(t, y0 ))) is positive deﬁnite for all t > 0. (4) The Hamiltonian H = 12 |p|2 + V is conserved along the y-trajectory, as is (A2 det P ), i.e., A(t, y(t, y0 )) can be computed via 1/2 A(t, y(t, y0 )) = (det P (t, y(t, y0 )))−1 A2 (0, y0 ) , (8.8) where the square root is taken as the principal value. In particular, since (A2 det P ) is a conserved quantity, we infer that A does not blow up along the time evolution (provided it is initially bounded). 8.2. Lagrangian Gaussian beam summation It should be noted that a single Gaussian beam given by (8.1) is not an asymptotic solution of (2.1), since its L2 (R2d )-norm goes to zero, in the classical limit ε → 0. Rather, one needs to sum over several Gaussian beams, the number of which is O(ε−1/2 ). This is referred to as the Gaussian beam summation; see for example Hill (1990). In other words, one ﬁrst needs to approximate given initial data through Gaussian beam proﬁles. For WKB initial data (2.18), a possible way to do so is given by the next theorem, proved by Tanushev (2008). Theorem 8.2.

Let the initial data be given by uεin (x) = ain (x) e iSin (x)/ε ,

with ain ∈ C 1 (Rd ) ∩ L2 (Rd ) and Sin ∈ C 3 (Rd ), and deﬁne ϕε (x, y0 ) = ain (y0 ) e iT (x,y0 )/ε ,

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where 1 T (x, y0 ) = Tα (y0 ) + Tβ · (x − y0 ) + (x − y0 ) Tγ (x − y0 ), 2 Tα (y0 ) = Sin (y0 ), Tβ (y0 ) = ∇x Sin (y0 ), Tγ (y0 ) = ∇2x Sin (y0 ) + i Id. Then

ε uin − (2πε)−d/2

rθ (· − y0 )ϕ (·, y0 ) dy0 d

1

ε

R

L2

≤ Cε 2 ,

where rθ ∈ C0∞ (Rd ), rθ ≥ 0 is a truncation function with rθ ≡ 1 in a ball of radius θ > 0 around the origin, and C is a constant related to θ. In view of Theorem 8.2, one can specify the initial data for (8.2)–(8.4) as y(0, y0 ) = y0 ,

p(0, y0 ) = ∇x Sin (y0 ),

∇2x Sin (y0 )

M (0, y0 ) = S(0, y0 ) = Sin (y0 ),

+ i Id, A(0, y0 ) = ain (y0 ).

(8.9) (8.10) (8.11)

Then, the Gaussian beam solution approximating the exact solution of (2.1) is given by rθ (x − y(t, y0 ))ϕε (t, x, y(t, y0 )) dy0 . uεG (t, x) = (2πε)−d/2 Rd

In discretized form this reads uεG (t, x)

−d/2

≈ (2πε)

Ny0

rθ (x − y(t, y0j ))ϕε (t, x, y0j )∆y0 ,

j=1

where the y0j are equidistant mesh points, and Ny0 is the number of the beams initially centred at y0j . Remark 8.3. Note that the cut-oﬀ error introduced via rθ becomes large when the truncation parameter θ is taken too small. On the other hand, a big θ for wide beams makes the error in the Taylor expansion of T large. As far as we know, it is still an open mathematical problem to determine an optimal size of θ when beams spread. However, for narrow beams one can take a fairly large θ, which makes the cut-oﬀ error almost zero. For example, a one-dimensional constant solution can be approximated by ∆y0 −(x − y0 )2 −(x − y0j )2 1 √ √ 1= exp exp dy0 ≈ , 2ε 2ε 2πε 2πε R j in which rθ ≡ 1.

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8.3. Higher-order Gaussian beams The above Gaussian beam method can be extended to higher order in ε: see Tanushev (2008), Jin et al. (2008b) and Liu and Ralston (2010). For notational convenience we shall only consider the case d = 1. Consider the Schr¨odinger equation (2.1) with initial data uεin (x) = e iSin (x)/ε

N

εj aj (x),

x ∈ R.

j=0

Let a ray y(t, y0 ) start at a point y0 ∈ R. Expand Sin (x) in a Taylor series around y0 : Sin (x) =

k+1

Sβ (y0 )(x − y0 )β + O(|x − y0 |β+1 ).

β=0

Then, a single kth-order Gaussian beam takes the form

[k/2]−1

ϕεk (t, x, y)

=

εj Aj (t, y) e iT (t,x,y)/ε ,

j=1

where the phase is given by 1 1 Tβ (t, y)(x−y)β , T (t, x, y) = T0 (t, y)+p(t, y)(x−y)+ M (t, y)(x−y)2 + 2 β! k+1

β=3

and the amplitude reads Aj =

k−2j−1 β=0

1 Aj,β (t, y)(x − y)β . β!

Here the bi-characteristic curves (y(t, y0 ), p(t, y(t, y0 ))) satisfy the Hamiltonian system (8.2) with initial data y(0, y0 ) = y0 ,

p(0, y0 ) = ∂y S0 (y0 ).

In addition, the equations for the phase coeﬃcients along the bi-characteristic curves are given by dT0 p2 = − V, dt 2 dM = −M 2 − ∂y2 V, dt β dTβ (β − 1)! =− Tγ Tβ−γ+2 − ∂yβ V, dt (γ − 1)!(β − γ)! γ=2

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for β = 3, . . . , k + 1. These equations are equipped with initial data T0 (0, y0 ) = S0 (y0 ),

M (0, y0 ) = ∂y2 S0 (y0 ) + i Id,

Tβ (0, y0 ) = Sβ (y0 ).

Finally, the amplitude coeﬃcients are obtained recursively by solving the transport equations for Aj,β with β ≤ k − 2j − 1, starting from dA0,0 1 = − Tr(M (t, z))A0,0 , dt 2

(8.12)

with initial data Aj,β (0, y0 ) = aβ (y0 ). In d = 1, the kth-order Gaussian beam superposition is thus formed by ε −1/2 rθ (x − y(t, y0 ))ϕεk (t, x, y(t, y0 )) dy0 , (8.13) uG,k (t, x) = (2πε) R ∞ C0 (R; R)

is some cut-oﬀ function. For this type where, as before, rθ ∈ of approximation, the following theorem was proved in Liu, Runborg and Tanushev (2011). Theorem 8.4. If uε (t, x) denotes the exact solution to the Schr¨odinger equation (2.1) and uεG,k is the kth-order Gaussian beam superposition, then sup uε (t, ·) − uεG,k (t, ·) ≤ C(T )εk/2 ,

(8.14)

|t|
for any T > 0.

9. Gaussian beam methods: Eulerian approach 9.1. Eulerian dynamics of Gaussian beams The Gaussian beam method can be reformulated in an Eulerian framework. To this end, let us ﬁrst deﬁne the linear Liouville operator as L = ∂ t + ξ · ∇ y − ∇ y V · ∇ξ . In addition, we shall denote Φ := (φ1 , . . . , φd ), where φj is the level set function deﬁned in (7.1) and (7.3). Using this, Jin et al. (2005b) and Jin and Osher (2003) showed that one can obtain from the original Lagrangian formulation (8.2)–(8.4) the following (inhomogeneous) Liouville equations for velocity, phase and amplitude, respectively: LΦ = 0, 1 LS = |ξ|2 − V, 2 1 LA = Tr (∇ξ Φ)−1 ∇y Φ A. 2

(9.1) (9.2) (9.3)

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In addition, if one introduces the quantity (Jin et al. 2005b) f (t, y, ξ) = A2 (t, y, ξ) det(∇ξ Φ), then f (t, y, ξ) again satisﬁes the Liouville equation, i.e., Lf = 0.

(9.4)

Two more inhomogeneous Liouville equations, which are the Eulerian version of (8.5) for P and R, were introduced in Leung, Qian and Burridge (2007) to construct the Hessian matrix. More precisely, one ﬁnds LR = −(∇2y V )P,

(9.5)

LP = R.

(9.6)

Note that the equations (9.1)–(9.4) are real , while (9.5) and (9.6) are complex and consist of 2n2 equations. Gaussian beam dynamics using complex level set functions In Jin et al. (2008b) the following observation was made. Taking the gradient of the equation (9.1) with respect to y and ξ separately, we have L(∇y Φ) = ∇2y V ∇ξ Φ,

(9.7)

L(∇ξ Φ) = −∇y Φ.

(9.8)

Comparing (9.5)–(9.6) with (9.7)–(9.8), one observes that −∇y Φ and ∇ξ Φ satisfy the same equations as R and P , respectively. Since the Liouville operator is linear, one can allow Φ to be complex-valued and impose for −∇y Φ, ∇ξ Φ the same initial conditions as for R and P , respectively. By doing so, R = −∇x Φ, P = ∇ξ Φ hold true for any time t ∈ R. In view of (8.6) and (8.10), this suggests the following initial condition for Φ: Φ0 (y, ξ) = −iy + (ξ − ∇y S0 ).

(9.9)

With this observation, one can now solve (9.1) for complex Φ, subject to initial data (9.9). Then the matrix M can be constructed by M = −∇y Φ(∇ξ Φ)−1 ,

(9.10)

where the velocity v = ∇y S is given by the intersection of the zero-level contours of the real part of Φ, i.e., for each component φj , Re(φj (t, y, ξ)) = 0,

at ξ = v(t, y) = ∇y S.

(9.11)

Note that in order to compute v, S and M , one only needs to solve d complex-valued homogeneous Liouville equations (9.1). The Eulerian level

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set method of Jin et al. (2008b) (in complex phase space) can then be summarized as follows. Step 1. Solve (9.1) for Φ complex , with initial condition (9.9), and obtain the velocity v by the intersection of the zero-level sets of Re φj , j = 1, . . . , n. Step 2. Use −∇y Φ and ∇ξ Φ to construct M by (9.10) (note that these quantities are already available from the ﬁrst step after discretizing the Liouville equation for Φ). Step 3. Integrate the velocity v along the zero-level sets (Gosse 2002, Jin and Yang 2008) to get the phase S. To do so, one performs a numerical integration following each branch of the velocity. The integration constants are obtained by the boundary condition and the fact that the multivalued phase is continuous when passing from one branch to the other. For example, if one considers a bounded domain [a, b] in space dimension d = 1, the phase function is given by x 1 t 2 S(t, x) = −V (a)t − v (τ, a) dτ + v(t, y) dy + S(0, a). (9.12) 2 0 a For more details on this and its extension to higher dimensions, see Jin and Yang (2008). Step 4. Solve (9.4) with the initial condition f0 (y, ξ) = A20 (y, ξ). Then amplitude A is given by A = (det(∇ξ Φ)−1 f )1/2 ,

(9.13)

where the square root has to be understood as the principal value. (We also refer to Jin, Wu and Yang (2011) for a more elaborate computation of A.) Note that all functions appearing in Steps 2–4 only need to be solved locally around the zero-level sets of Re φj , j = 1, . . . , n. Thus, the entire algorithm can be implemented using the local level set methods of Osher, Cheng, Kang, Shim and Tsai (2002) and Peng et al. (1999). For a given mesh size ∆y, the computational cost is therefore O((∆y)−d ln(∆y)−1 ), about the same as the local level set methods for geometrical optics computation: see Jin et al. (2005b). Remark 9.1. If one is only interested in computing the classical limit of (the expectation values of) physical observables, one observes that the only in (9.12) which aﬀects a quadratic observable density for ﬁxed time t is term x a v(t, y) dy. Thus, as long as one is only interested in physical observables, one can simply take x S(t, x) = v(t, y) dy (9.14) a

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in the numerical simulations. That M and A are indeed well-deﬁned via (9.10) and (9.13) is justiﬁed by the following theorem (which can be seen as the Eulerian version of Theorem 8.1), proved in Jin et al. (2008b). Theorem 9.2. Let Φ = Φ(t, y, ξ) ∈ C be the solution of (9.1) with initial data (9.9). Then, the following properties hold: (1) ∇ξ Φ is non-degenerate for all t ∈ R, (2) Im −∇y Φ(∇ξ Φ)−1 is positive deﬁnite for all t ∈ R, y, ξ ∈ Rd . Although det Re(∇ξ Φ) = 0 at caustics, the complexiﬁed Φ makes ∇ξ Φ non-degenerate, and the amplitude A, deﬁned in (9.13), does not blow up at caustics. 9.2. Eulerian Gaussian beam summation As before, we face the problem of Gaussian beam summation, i.e., in order to reconstruct the full solution uε a superposition of single Gaussian beams has to be considered. To this end, we deﬁne a single Gaussian beam, obtained through the Eulerian approach, by ϕ˜ε (t, x, y, ξ) = A(t, y, ξ) eiT (t,x,y,ξ)/ε ,

(9.15)

where A is solved via (9.13), and 1 T (t, x, y, ξ) = S(t, y, ξ) + ξ · (x − y) + (x − y) M (t, y, ξ)(x − y). 2 Then, the wave function is constructed via the following Eulerian Gaussian beam summation formula (Leung et al. 2007): u ˜εG (t, x)

−d/2

= (2πε)

Rd ×Rd

rθ (x − y)ϕ˜ε (t, x, y, ξ)

d

δ(Re(φj )) dξ dy,

j=1

which is consistent with the Lagrangian summation formula (8.2). Indeed, the above given double integral for uεG can be evaluated as a single integral in y as follows. We again denote by vj , j = 1, . . . , J the jth branch of the multivalued velocity and write J ϕ˜ε (t, x, y, vk ) ε −d/2 dy. (9.16) u ˜G (t, x) = (2πε) rθ (x − y) | det(Re(∇ξ Φ)ξ=vj )| Rd j=1

However, since det Re(∇ξ Φ) = 0 at caustics, a direct numerical integration of (9.16) loses accuracy around caustics. To get better accuracy, one can

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split (9.16) into two parts, J (2πε)−d/2 rθ (x − y) I1 = j=1

I2 =

J j=1

where

Ω1

Ω2

(2πε)−d/2 rθ (x − y)

Ω1 := y

Ω2 := y

ϕ˜ε (t, x, y, vk ) dy, det(Re(∇ξ Φ)ξ=vj )

(9.17)

ϕ˜ε (t, x, y, vk ) dy, det(Re(∇ξ Φ)ξ=vj )

(9.18)

: det(Re(∇p φ)(t, y, pj )) ≥ τ , : det(Re(∇p φ)(t, y, pj )) < τ ,

with τ being a small parameter. The latter is chosen suﬃciently small to minimize the cost of computing (9.18), yet large enough to make I1 a regular integral. The regular integral I1 can then be approximated by a standard quadrature rule, such as the trapezoid quadrature rule, while the singular integral I2 is evaluated by the semi-Lagrangian method introduced in Leung et al. (2007). Remark 9.3. When the velocity contours are complicated due to large numbers of caustics, implementation of the local semi-Lagrangian method is hard. In such situations one can use a discretized δ-function method for numerically computing (9.18), as was done in Wen (2010). In this method one needs to numerically solve (9.2) in order to obtain the phase function, since all values of φj near the support of δ(Re(φj )) are needed to evaluate (9.18). Example 9.4. This is an example from Jin et al. (2008b). It considers the free motion of particles in d = 1 with V (x) = 0. The initial conditions for the Schr¨ odinger equation (2.1) are induced by ρin (x) = exp(−50x2 ),

vin (x) = ∂x S0 (x) = − tanh(5x).

Figure 9.1 shows the l∞ -errors between the square modulus of uε , the exact solution of the Schr¨odinger equation (2.1), and the approximate solution constructed by: (i) the level set method described in Section 7, (ii) the level set method with Keller–Maslov index built in (Jin and Yang 2008), and (iii) the Eulerian Gaussian beam method described above. As one can see, method (ii) improves the geometric optics solution (i) away from caustics, while the Gaussian beam method oﬀers a uniformly small error even near the caustics. Compared to the Lagrangian formulation based on solving the ODE system (8.2)–(8.4), the Eulerian Gaussian beam method has the advantage of maintaining good numerical accuracy since it is based on solving PDEs on

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(a)

(b)

(c)

Figure 9.1. Example 9.4: numerical errors between the solution of the Schr¨ odinger equation and (a) the geometrical optics solution, (b) the geometrical optics with phase shift built in, and (c) the Gaussian beam method. Caustics are around x = ±0.18.

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ﬁxed grids. Moreover, higher-order (in ε) Eulerian Gaussian beam methods have been constructed: see Liu and Ralston (2010) and Jin et al. (2011). Remark 9.5. Research on Gaussian beam methods is of great recent interest in the applied mathematics community: see, e.g., Leung et al. (2007), Jin et al. (2008b) and Leung and Qian (2009) for Eulerian formulations, Tanushev (2008), Motamed and Runborg (2010), Liu and Ralston (2010), Bougacha, Akian and Alexandre (2009) and Liu et al. (2011) for error estimates, and Tanushev, Engquist and Tsai (2009), Ariel, Engquist, Tanushev and Tsai (2011), Qian and Ying (2010) and Yin and Zheng (2011) for initial data decompositions. 9.3. Frozen Gaussian approximations The construction of the Gaussian beam approximation is based on the truncation of the Taylor expansion of the phase T (t, x, y) (8.1) around the beam centre y. Hence it loses accuracy when the width of the beam becomes large, i.e., when the imaginary part of M (t, y) in (8.1) becomes small so that the Gaussian function is no longer localized . This happens, for example, when the solution of the Schr¨odinger equation spreads (namely, the ray determined by (8.2) diverges), which can be seen as the time-reversed situation of caustic formation. The corresponding loss in the numerical computation can be overcome by re-initialization every once in a while: see Tanushev et al. (2009), Ariel et al. (2011), Qian and Ying (2010) and Yin and Zheng (2011). However, this approach increases the computational complexity in particular, when beams spread quickly. The frozen Gaussian approximation (as it is referred to in quantum chemistry), ﬁrst proposed in Heller (1981), uses Gaussian functions with ﬁxed widths to approximate the exact solution uε . More precisely, instead of using Gaussian beams only in the physical space, the frozen Gaussian approximation uses a superposition of Gaussian functions in phase space. That is why the method is also known by the name coherent state approximation. To this end, we ﬁrst decompose the initial data into several Gaussian functions in phase space, 1 2 ε ψ (y0 , p0 ) = uεin (y) e−( ip0 ·(y−y0 )− 2 |y−y0 | )/ε dy, Rd

and then propagate the centre of each function (y(t), p(t)) along the Hamiltonian ﬂow (8.2), subject to initial data at (y0 , p0 ). The frozen Gaussian beam solution takes the form uεFG (t, x) = −3d/2

1

(2πε)

Rd ×Rd

a(t, y0 , p0 )ψ(y0 , p0 ) e( ip(t)·(x−y(t))− 2 |x−y(t)|

2 )/ε

dp0 dy0 ,

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where the complex-valued amplitude a(t, y0 , p0 ), is the so-called Herman– Kluk pre-factor : see Herman and Kluk (1984). Since the width of the Gaussians is ﬁxed, one does not encounter the problem of beam spreading here. However, since this method is based in phase space, the computational cost is considerably higher than the standard Gaussian beam methods. For subsequent developments in this direction, see Herman and Kluk (1984), Kay (1994, 2006), Robert (2010), Swart and Rousse (2009) and Lu and Yang (2011).

10. Asymptotic methods for discontinuous potentials Whenever a medium is heterogeneous, the potential V can be discontinuous, creating a sharp potential barrier or interface where waves can be partially reﬂected and transmitted (as in the Snell–Descartes Law of Refraction). This gives rise to new mathematical and numerical challenges not present in the smooth potential case. Clearly, the semiclassical limit (3.8)–(3.9) does not hold at the barrier. Whenever V is discontinuous, the Liouville equation (3.8) contains characteristics which are discontinuous or even measure-valued. To this end, we recall that the characteristic curves (x(t), ξ(t)) are determined by the Hamiltonian system (3.6). The latter is a nonlinear system of ODEs whose right-hand side is not Lipschitz due to the singularity in ∇x H(x, ξ), and thus the classical well-posedness theory for the Cauchy problem of the ODEs fails. Even worse, the coeﬃcients in the Liouville equation are in general not even BV (i.e., of bounded variation), for which almost everywhere solutions were introduced by DiPerna and Lions (1989) and Ambrosio (2004). Analytical studies of semiclassical limits in situations with interface were carried out by, e.g., Bal, Keller, Papanicolaou and Ryzhik (1999b), Miller (2000), Nier (1995, 1996) and Benedetto, Esposito and Pulvirenti (2004) using more elaborate Wigner transformation techniques, such as two-scale Wigner measures. 10.1. The interface condition In order to allow for discontinuous potentials, one ﬁrst needs to introduce a notion of solutions of the underlying singular Hamiltonian system (3.6). This can be done by providing interface conditions for reﬂection and transmission, based on Snell’s law. The solution so constructed will give the correct transmission and reﬂection of waves through the barrier, obeying the laws of geometrical optics. Figure 10.1 (taken from Jin, Liao and Yang (2008a)) shows typical cases of wave transmissions and reﬂections through an interface. When the interface is rough, or in higher dimensions, the scattering can be diﬀusive, in which the transmitted or reﬂected waves can move in all directions: see

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Reflection

A

B

Reflection

A

B

Transmission

Transmission

Incident

Incident

(a)

(b)

Figure 10.1. Wave transmissions and reﬂections through an interface.

Figure 10.1(b). However, in this section we will not discuss the diﬀusive interface, which was treated analytically in Bal et al. (1999b) and numerically in Jin et al. (2008a). An Eulerian interface condition Jin and Wen (2006a) provide an interface condition connecting the Liouville equations at both sides of a given (sharp) interface. Let us focus here only on the case of space dimension d = 1 and consider a particle moving with velocity ξ > 0 towards the barrier. The interface condition at a given ﬁxed time t is given by w(t, x+ , ξ + ) = αT w(t, x− , ξ − ) + αR w(t, x+ , −ξ + ).

(10.1)

Here the superscripts ‘±’ represent the right and left limits of the quantities, αT ∈ [0, 1] and αR ∈ [0, 1] are the transmission and reﬂection coeﬃcients respectively, satisfying αR + αT = 1. For a sharp interface x+ = x− , however, ξ + and ξ − are connected by the Hamiltonian preservation condition H(x+ , ξ + ) = H(x− , ξ − ).

(10.2)

The latter is motivated as follows. In classical mechanics, the Hamiltonian H = 12 ξ 2 + V (x) is conserved along the particle trajectory, even across the barrier. In this case, αT , αR = 0 or 1, i.e., a particle can be either completely transmitted or completely reﬂected. In geometric optics (corresponding to H(x, ξ) = c(x)|ξ|), condition (10.2) is equivalent to Snell’s Law of Refraction for a ﬂat interface (Jin and Wen 2006b), i.e., waves can be partially transmitted or reﬂected.

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Remark 10.1. In practical terms, the coeﬃcients αT and αR are determined from the original Schr¨odinger equation (2.1) before the semiclassical limit is taken. Usually one invokes stationary scattering theory to do so. Thus (10.1) represents a multiscale coupling between the (macroscopic) Liouville equation and the (microscopic) Schr¨odinger (or wave) equation. Furthermore, by incorporating the diﬀraction coeﬃcients, determined from the geometrical theory of diﬀraction developed in Keller and Lewis (1995), into the interface condition, one could even simulate diﬀraction phenomena near boundaries, interfaces or singular geometries (Jin and Yin 2008a, 2008b, 2011). The well-posedness of the initial value problem of the singular Liouville equation with the interface condition (10.1) was established in Jin and Wen (2006a), using the method of characteristics. To determine a solution at (x, p, t) one traces back along the characteristics determined by the Hamiltonian system (3.6) until hitting the interface. At the interface, the solution bifurcates according to the interface condition (10.1). One branch of the solution thereby corresponds to the transmission of waves and the other to the reﬂection. This process continues until one arrives at t = 0. The interface condition (10.1) thus provides a generalization of the method of characteristics. A Lagrangian Monte Carlo particle method for the interface A notion of the solution of the (discontinuous) Hamiltonian system (3.6) was introduced in Jin (2009) (see also Jin and Novak (2006)) using a probability interpretation. One thereby solves the system (3.6) using a standard ODE or Hamiltonian solver, but at the interface, the following Monte Carlo solution can be constructed (we shall only give a solution in the case of ξ − > 0; the other case is similar). (1) With probability αR , the particle (wave) is reﬂected with x → x,

ξ − → −ξ − .

(10.3)

(2) With probability αT , the particle (wave) is transmitted, with x → x,

ξ + obtained from ξ − using (10.2).

(10.4)

Although the original problem is deterministic, this probabilistic solution allows one to go beyond the interface with the new value of (x, ξ) deﬁned in (10.3)–(10.4). This is the Lagrangian formulation of the solution determined by using the interface condition (10.1). This is the basis of a (Monte-Carlobased) particle method for thin quantum barriers introduced in Jin and Novak (2007).

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10.2. Modiﬁcation of the numerical ﬂux at the interface A typical one-dimensional semi-discrete ﬁnite diﬀerence method for the Liouville equation (3.7) is ∂t wij + ξj

− + wi+ − wi− 1 1 ,j ,j 2

2

∆x

− DVi

wi,j+ 1 − wi,j− 1 2

2

∆ξ

= 0.

Here wij is the cell average or pointwise value of w(t, xi , ξj ) at ﬁxed t. The numerical ﬂuxes wi+ 1 ,j , wi,j+ 1 are typically deﬁned by a (ﬁrst-, or higher2 2 order) upwind scheme, and DVi is some numerical approximation of ∂x V at x = xi . When V is discontinuous, such schemes face diﬃculties when the Hamiltonian is discontinuous, since ignoring the discontinuity of V in the actual numerical computation will result in solutions which are inconsistent with the notion of the physically relevant solution, deﬁned in the preceding subsection. Even with a smoothed Hamiltonian, it is usually impossible (at least in the case of partial transmission and reﬂection) to obtain transmission and reﬂection with the correct transmission and reﬂection coeﬃcients. A smoothed V will also give a severe time step constraint like ∆t ∼ O(∆x∆ξ): see, e.g., Cheng, Kang, Osher, Shim and Tsai (2004). This is a parabolic-type CFL condition, despite the fact that we are solving a hyperbolic problem. A simple method to solve this problem was introduced by Jin and Wen (2005, 2006a). The basic idea is to build the interface condition (10.1) into the numerical ﬂux , as follows. Assume V is discontinuous at xi+1/2 . First one should avoid discretizing V across the interface at xi+1/2 . One possible discretization is DVk =

− + − Vk−1/2 Vk+1/2

∆x

,

for k = i, i + 1,

where, for example, ± Vk+1/2 =

lim

x→xk+1/1 ±0

V (x).

The numerical ﬂux in the ξ direction, Wi,j±1/2 , can be the usual numerical ﬂux (for example, the upwind scheme or its higher-order extension). To ± deﬁne the numerical ﬂux wi+1/2,j , without loss of generality, consider the − case ξj > 0. Using the upwind scheme, wi+ = wij . However, 1 ,j 2

+ wi+1/2,j

=

+ w(x+ i+1/2 , ξj )

= =

− + + αT w(x− i+1/2 , ξj ) + αR w(xi+1/2 , −ξj ) αT wi (ξj− ) + αR wi+1,−j ,

while ξ − is obtained from (10.2) with ξj+ = ξj . Since ξ − may not be a

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grid point, one has to deﬁne it approximately. A simple approach is to locate the two cell centres that bound it, then use a linear interpolation to approximate the needed numerical ﬂux at ξj− . The case of ξj < 0 is treated similarly. The detailed algorithm to generate the numerical ﬂux is given in Jin and Wen (2005, 2006a). This numerical scheme overcomes the aforementioned analytic and numerical diﬃculties. In particular, it possesses the following properties. (1) It produces the correct physical solution crossing the interface (as deﬁned in the previous subsection). In particular, in the case of geometric optics, this solution is consistent with the Snell–Descartes Law of Refraction at the interface. (2) It allows a hyperbolic CFL condition ∆t = O(∆x, ∆ξ). The idea outlined above has its origin in so-called well-balanced kinetic schemes for shallow water equations with bottom topography: see Perthame and Simeoni (2001). It has been applied to computation of the semiclassical limit of the linear Schr¨ odinger equation with potential barriers in Jin and Wen (2005), and the geometrical optics limit with complete transmission/reﬂection in Jin and Wen (2006b), for thick interfaces, and Jin and Wen (2006a), for sharp interfaces. Positivity and both l1 and l∞ stabilities were also established, under a hyperbolic CFL condition. For piecewise constant Hamiltonians, an l1 -error estimate of the ﬁrst-order ﬁnite diﬀerence of this type was established in Wen (2009), following Wen and Jin (2009). Remark 10.2. Let us remark that this approach has also been extended to high-frequency elastic waves (Jin and Liao 2006), and high-frequency waves in random media with diﬀusive interfaces (Jin et al. 2008a). When the initial data are measure-valued, such as (3.9), the level set method introduced in Jin et al. (2005b) becomes diﬃcult for interfaces where waves undergo partial transmissions and reﬂections, since one needs to increase the number of level set functions each time a wave hits the interface. A novel method to get around this problem has been introduced in Wei, Jin, Tsai and Yang (2010). It involves two main ingredients. (1) The solutions involving partial transmissions and partial reﬂections are decomposed into a ﬁnite sum of solutions, obtained by solving problems involving only complete transmissions or complete reﬂections. For the latter class of problems, the method of Jin et al. (2005b) applies. (2) A re-initialization technique is introduced such that waves coming from multiple transmissions and reﬂections can be combined seamlessly as new initial value problems. This is implemented by rewriting the sum of several delta functions as one delta measure with a suitable weight, which can easily be implemented numerically.

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10.3. Semiclassical computation of quantum barriers Correct modelling of electron transport in nano-structures, such as resonant tunnelling diodes, superlattices or quantum dots, requires the treatment of quantum phenomena in highly localized regions within the devices (so-called quantum wells), while the rest of the device can be dealt with by classical mechanics. However, solving the Schr¨ odinger equation in the entire physical domain is usually too expensive, and thus it is attractive to use a multiscale approach as given in Ben Abdallah, Degond and Gamba (2002): that is, solve the quantum mechanics only in the quantum well, and couple the solution to classical mechanics outside the well. To this end, the following semiclassical approach for thin quantum barriers was proposed in Jin and Novak (2006). Step 1. Solve the time-independent Schr¨ odinger equation (either analytically or numerically) for the local barrier well to determine the scattering data, i.e., the transmission and reﬂection coeﬃcients αT , αR . Step 2. Solve the classical Liouville equation elsewhere, using the scattering data generated in Step 1 and the interface condition (10.1) given above. The results for d = 1 and d = 2 given in Jin and Novak (2006, 2007) demonstrate the validity of this approach whenever the well is either very thin (i.e., of the order of only a few ε) or well separated. In higher dimensions, the interface condition (10.1) needs to be implemented in the direction normal to interface, and the interface condition may be non-local for diﬀusive transmissions or reﬂections (Jin and Novak 2007). This method correctly captures both the transmitted and the reﬂected quantum waves, and the results agree (in the sense of weak convergence) with the solution obtained by directly solving the Schr¨odinger equation with small ε. Since one obtains the quantum scattering information only in a preprocessing step (i.e., Step 1), the rest of the computation (Step 2) is classical, and thus the overall computational cost is the same as for computing classical mechanics. Nevertheless, purely quantum mechanical eﬀects, such as tunnelling, can be captured. If the interference needs to be accounted for, then such Liouville-based approaches are not appropriate. One attempt was made in Jin and Novak (2010) for one-dimensional problems, where a complex Liouville equation is used together with an interface condition using (complex-valued ) quantum scattering data obtained by solving the stationary Schr¨ odinger equation. Its extension to multi-dimensional problems remains to be done. A more general approach could use Gaussian beam methods, which do capture the phase information. This is an interesting line of research yet to be pursued.

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11. Schr¨ odinger equations with matrix-valued potentials and surface hopping Problems closely related to those mentioned in Section 10 arise in the study of semiclassical Schr¨odinger equations with matrix-valued potentials. This type of potential can be seen as a highly simpliﬁed model of the full manybody quantum dynamics of molecular dynamics. Using the celebrated Born–Oppenheimer approximation to decouple the dynamics of the electrons from the one for the much heavier nuclei (see, e.g., Spohn and Teufel (2001)), one ﬁnds that the electrons are subject to external forces which can be modelled by a system of Schr¨odinger equations for the nuclei along the electronic energy surfaces. The nucleonic Schr¨ odinger system has matrixvalued potentials, which will be treated in this section. To this end, we consider the following typical situation, namely, a timedependent Schr¨odinger equation with R2×2 -matrix-valued potential (see, e.g., Spohn and Teufel (2001), or Teufel (2003)): 2 ε ε iε∂t u = − ∆x + V (x) uε , uε (0, x) = uεin ∈ L2 (R2 , C2 ), (11.1) 2 for (t, x) ∈ R+ × R2 and ∆x = diag(∆x1 + ∆x2 , ∆x1 + ∆x2 ). The unknown uε (t, x) ∈ C2 and V is a symmetric matrix of the form 1 v (x) v2 (x) , (11.2) V (x) = tr V (x) + 1 v2 (x) −v1 (x) 2 with v1 (x), v2 (x) ∈ R. The matrix V then has two eigenvalues, λ(±) (x) = tr V (x) ± v1 (x)2 + v2 (x)2 . Remark 11.1. In the Born–Oppenheimer approximation, m the dimensionless semiclassical parameter ε > 0 is given by ε = M , where m and M are the masses of an electron and a nucleus respectively (Spohn and Teufel 2001). Then, all oscillations are roughly characterized by the frequency 1/ε, which typically ranges between one hundred and one thousand. 11.1. Wigner matrices and the classical limit for matrix-valued potentials In this section, we shall discuss the inﬂuence of matrix-valued potentials on the semiclassical limit of the Schr¨odinger equations (11.1). Introduce the Wigner matrix as deﬁned in G´erard et al. (1997): ε ε ε ε −2 ε ε ¯ W [u ](x, ξ) = (2π) ψ x− η ⊗ψ x+ η e iη·ξ dy, (x, ξ) ∈ R2x ×R2ξ . 2 2 2 R We also let W denote the corresponding (weak) limit W ε [uε ] −→ W ∈ L∞ (R; M+ (R2x × R2ξ ; R2 )). ε→0

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(a)

(b)

Figure 11.1. The conical crossings for potentials (a) Viso and (b) VJT deﬁned in (11.3). The crossing point is (0, 0) for both potentials.

In order to describe the the dynamics of the limiting matrix-valued measure W (t, x, ξ), ﬁrst note that the complex 2 × 2 matrix-valued symbol of (11.1) is given by i P (x, ξ) = |ξ|2 + iV (x). 2 The two eigenvalues of −iP (x, ξ) are |ξ|2 |ξ|2 + tr V (x) ± v1 (x)2 + v2 (x)2 = + λ± (x). 2 2 These eigenvalues λn , n = 1, 2 govern the time evolution of the limiting measure W (t), as proved in G´erard et al. (1997). They act as the correct classical Hamiltonian function in phase space, corresponding to the two energy levels, respectively. In the following, we shall say that two energy levels cross at a point x∗ ∈ R2 if λ+ (x∗ ) = λ− (x∗ ). Such a crossing is called conical if the vectors ∇x v1 (x∗ ) and ∇x v2 (x∗ ) are linearly independent. In Figure 11.1 we give two examples of conical crossings for potentials of the form x1 x2 x1 x2 Viso = , VJT = . (11.3) x2 −x1 x2 −x1 λ1,2 (x, ξ) =

If all the crossings are conical, the crossing set S = {x ∈ R2 : λ+ (x) = λ− (x)} is a sub-manifold of co-dimension two in R2 : see Hagedorn (1994). Assume that the Hamiltonian ﬂows with Hamiltonians λn leave invariant the set Ω = (R2x × R2ξ )\S.

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For (x, ξ) ∈ Ω, denote by χn (x, ξ) the column eigenvector corresponding to the eigenvalue λn (x, ξ), and the matrix Πn (x, ξ) = χn (x, ξ)(χτ (x, ξ)) is the orthogonal projection onto the eigenspace associated to λn (x, ξ). By Theorem 6.1 of G´erard et al. (1997), the matrix-valued Wigner measure W (t) commutes with the projectors Πn , outside the crossing set S, and can thus be decomposed as W (t, ·) = Π1 W (t, ·)Π1 + Π2 W (t, ·)Π2 . Since the eigenspaces are both one-dimensional, the decomposition is simpliﬁed to be W (t, ·) = W1 (t, ·)Π1 + W2 (t, ·)Π2 . The scalar functions Wn (t, x, ξ) given by Wn (t, x, ξ) = tr(Πn W (t, x, ξ)) are then found to satisfy the Liouville equations ∂t Wn + ∇ξ λn · ∇x Wn − ∇x λn · ∇ξ Wn = 0,

(t, x, ξ) ∈ R+ × Ω,

(11.4)

subject to initial data Wn (0) = tr(Πn W ),

(x, ξ) ∈ Ω.

(11.5)

The scalar functions Wn , n = 1, 2, are the phase space probability densities corresponding to the upper and lower energy levels, respectively. One can recover from them the particle densities ρn via Wn (t, x, ξ) dξ, n = 1, 2. (11.6) ρn (t, x) = R2ξ

In other words, the Liouville equations (11.4) yield the propagation of the Wigner measures W1 (t, ·) and W2 (t, ·) on any given time interval, provided that their supports do not intersect the eigenvalue crossing set S. However, analytical and computational challenges arise when their supports intersect the set S. In S the dynamics of W1 , W2 are coupled due to the non-adiabatic transitions between the two energy levels, and an additional hopping condition is needed (analogous to the interface condition considered in Section 10 above). The Landau–Zener formula Lasser, Swart and Teufel (2007) give a heuristic derivation of the nonadiabatic transition probability. The derivation is based on the Hamiltonian

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system corresponding to the Liouville equations (11.4), i.e., x˙ n (t) = ∇ξ λn (t) = ξn (t), ξ˙n (t) = −∇x λn (t), n = 1, 2.

(11.7)

The basic idea is to insert the trajectories (x(t), ξ(t)) of the Hamiltonian systems (11.7) into the trace-free part of the potential matrix (11.2) to obtain a system of ordinary diﬀerential equations, given by d ε v1 (x(t)) v2 (x(t)) uε (t). iε u (t) = v2 (x(t)) −v1 (x(t)) dt The non-adiabatic transitions occur in the region where the spectral gap between the eigenvalues becomes minimal. The function h(x(t)) = |λ+ (x(t)) − λ− (x(t))| = 2|ϑ(x(t))| measures the gap between the eigenvalues in phase space along the classical trajectory (x(t), ξ(t)), where ϑ(x) = (v1 (x), v2 (x)) and | · | denotes the Euclidean norm. The necessary condition for a trajectory to attain the minimal gap is given by d |ϑ(x(t))|2 = ϑ(x(t)) · ∇x ϑ(x(t))ξ(t) = 0, dt ˙ where ∇x ϑ(x(t) is the Jacobian matrix of the vector ϑ(x(t)), and ξ(t) = x(t). Hence, a crossing manifold in phase space containing these points is given by

S ∗ = (x, ξ) ∈ R2x × R2ξ : ϑ(x(t)) · ∇x ϑ(x(t))ξ(t) = 0 . The transition probability when one particle hits S ∗ is assumed to be given by π (ϑ(x0 ) ∧ ∇x ϑ(x0 )ξ0 )2 ε T (x0 , ξ0 ) = exp − , (11.8) ε |∇x ϑ(x0 )ξ0 |3 which is the famous Landau–Zener formula (Landau 1932, Zener 1932). Note that T decays exponentially in x and ξ and 1 (x, ξ) ∈ S ∗ }, lim T ε = T0 = ε→0 0 (x, ξ) ∈ / S∗. In other words, as ε → 0, the transition between the energy bands only occurs on the set S ∗ , which is consistent with the result in the previous subsections. 11.2. Numerical approaches Lagrangian surface hopping One widely used numerical approach to simulation of the non-adiabatic dynamics at energy crossings is the surface hopping method, ﬁrst proposed by

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Tully and Preston (1971), and further developed in Tully (1990) and Sholla and Tully (1998). The basic idea is to combine the classical transports of the system on the individual potential energy surfaces λ± (x) that follow (11.7) with an instantaneous transitions at S ∗ from one energy surface to another. The rate of transition is determined by the Landau–Zener formula (11.8) whenever available, or computed by some quantum mechanical simulation locally around S ∗ . The hoppings were performed in a Monte Carlo procedure based on the transition rates. For a review of surface hopping methods see Drukker (1999). More recently, surface hopping methods have generated increasing interest among the mathematical community. For molecular dynamical simulations, Horenko, Salzmann, Schmidt and Sch¨ utte (2002) adopted the partial Wigner transform to reduce a full quantum dynamics into the quantum–classical Liouville equation, and then the surface hopping is realized by approximating the quantum Liouville equation using phase space Gaussian wave packets. From the analytical point of view, Lasser and Teufel (2005) and Fermanian Kammerer and Lasser (2003) analysed the propagation through conical surface crossings using matrix-valued Wigner measures and proposed a corresponding rigorous surface hopping method based on the semiclassical limit of the time-dependent Born–Oppenheimer approximation. In Lasser et al. (2007) and Kube, Lasser and Weber (2009) they used a particle method to solve the Liouville equation, in which each classical trajectory was subject to a deterministic branching (rather than Monte Carlo) process. Branching occurs whenever a trajectory attains one of its local minimal gaps between the eigenvalue surfaces. The new branches are consequently re-weighted according to the Landau–Zener formula for conical crossings These Lagrangian surface hopping methods are very simple to implement, and in particular, very eﬃcient in high space dimensions. However, they require either many statistical samples in a Monte Carlo framework or the increase of particle numbers whenever hopping occurs. In addition, as is typical for Lagrangian methods, a complicated numerical re-interpolation procedure is needed whenever the particle trajectories diverge, in order to maintain uniform accuracy in time. Eulerian surface hopping The Eulerian framework introduced in Jin, Qi and Zhang (2011) consists of solving the two Liouville equations (11.4), with a hopping condition that numerically incorporates the Landau–Zener formula (11.8). Note that the Schr¨odinger equation (11.1) implies conservation of the total mass, which in the semiclassical limit ε → 0, locally away from S ∗ , yields d (W1 + W2 )(t, x, ξ) dξ dx = 0. (11.9) dt

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For this condition to hold for all x, ξ, the total ﬂux in the direction normal to S ∗ needs to be continuous across S ∗ . To ensure this, the Landau–Zener transition at S ∗ should be formulated as a continuity condition for the total ﬂux in the normal direction en . Deﬁne the ﬂux function jn (x, ξ) ∈ R2 for each eigenvalue surface via jn (x, ξ) = (∇ξ λn , −∇x λn ) Wn (x, ξ),

n = 1, 2.

Assume, before hopping, that the particle remains on one of the eigenvalue surfaces, i.e., − − − (1) j1 (x− 0 , ξ0 ) = 0 and j2 (x0 , ξ0 ) = 0, or − − − (2) j1 (x− 0 , ξ0 ) = 0 and j2 (x0 , ξ0 ) = 0.

Then the interface condition is given by + − j1 (x− j1 (x+ T (x0 , ξ0 ) 0 , ξ0 ) · en = 1 − T (x0 , ξ0 ) 0 , ξ0 ) · e n , + − T (x0 , ξ0 ) 1 − T (x0 , ξ0 ) j2 (x+ j2 (x− 0 , ξ0 ) · e n 0 , ξ0 ) · e n where (x± , ξ ± ) are the (pre- and post-hopping) limits to (x0 , ξ0 ) ∈ S ∗ along the direction en . Remark 11.2. There is a restriction of this approach based on the Liouville equation and the Landau–Zener transition probability. The interference eﬀects generated when two particles from diﬀerent energy levels arrive at S ∗ at the same time are thereby not accounted for, and thus important quantum phenomena, such as the Berry phase, are missing. One would expect that Gaussian beam methods could handle this problem, but this remains to be explored.

12. Schr¨ odinger equations with periodic potentials So far we have only considered ε-independent potential V (x). The situation changes drastically if one allows for potentials varying on the fast scale y = x/ε. An important example concerns the case of highly oscillatory potentials VΓ (x/ε), which are periodic with respect to a periodic lattice Γ ⊂ Rd . In the following we shall therefore consider Schr¨odinger equations of the form x ε ε2 ε ε u + V (x)uε , uε (0, x) = uεin (x). iε∂t u = − ∆u + VΓ (12.1) 2 ε Here V ∈ C ∞ denotes some smooth, slowly varying potential and VΓ is a rapidly oscillating potential (not necessarily smooth). For deﬁniteness we shall assume that for some orthonormal basis {ej }dj=1 , VΓ satisﬁes VΓ (y + 2πej ) = VΓ (y) i.e., Γ =

(2πZ)d .

∀ y ∈ Rd ,

(12.2)

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Remark 12.1. Equations of this form arise in solid state physics, where they are used to describe the motion of electrons under the action of an external ﬁeld and a periodic potential generated by the ionic cores. This problem has been extensively studied from both physical and mathematical points of view: see, e.g., Ashcroft and Mermin (1976), Bensoussan, Lions and Papanicolaou (1978), Teufel (2003) and the references given therein. One of the most striking dynamical eﬀects due to the inclusion of a periodic potential VΓ is the occurrence of so-called Bloch oscillations. These are oscillations exhibited by electrons moving in a crystal lattice under the inﬂuence of a constant electric ﬁeld V (x) = F · x, F ∈ Rd (see Section 12.2 below). 12.1. Emergence of Bloch bands In order to better understand the inﬂuence of VΓ , we recall here the basic spectral theory for periodic Schr¨odinger operators of the form (Reed and Simon 1976): 1 Hper = − ∆y + VΓ (y). 2 With VΓ obeying (12.2), we have the following. (1) The fundamental domain of the lattice Γ = (2πZ)d is the interval Y = [0, 2π]d . (2) The dual lattice Γ∗ can then be deﬁned as the set of all wave numbers k ∈ R for which plane waves of the form e ik·x have the same periodicity as the potential VΓ . This yields Γ∗ = Zd in our case. (3) The fundamental domain of the dual lattice, Y ∗ , i.e., the (ﬁrst) Brillouin zone, is the set of all k ∈ R closer to zero than to any other d dual lattice point. In our case Y ∗ = − 12 , 12 , equipped with periodic boundary conditions, i.e., Y ∗ Td . By periodicity, it is suﬃcient to consider the operator Hper on the fundamental domain Y only, where we impose the following quasi-periodic boundary conditions: f (y + 2πej ) = e2 ikj π f (y) ∀ y ∈ R, k ∈ Y ∗ . (12.3) It is well known (Wilcox 1978) that under mild conditions on VΓ , the operator H admits a complete set of eigenfunction {ψm (y, k)}m∈N , parametrized by k ∈ Y ∗ . These eigenfunctions provide, for each ﬁxed k ∈ Y ∗ , an orthonormal basis in L2 (Y ). Correspondingly there exists a countable family of real eigenvalues {E(k)}m∈N , which can be ordered as E1 (k) ≤ E2 (k) ≤ · · · ≤ Em (k) ≤ · · · , where the respective multiplicities are accounted for in the ordering. The set {Em (k) : k ∈ Y } ⊂ R is called the mth energy band of the operator Hper .

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4 3

5

2

4

1

3 2

0

1

−1 −0.5

−0.4

−0.3

−0.2

−0.1

0

k

0.1

0.2

0.3

0.4

0.5

Figure 12.1. The eigenvalues Em (k), m = 1, . . . , 8 for Mathieu’s model: VΓ = cos y.

Concerning the dependence on k ∈ Y ∗ , it has been shown (Wilcox 1978) that for any m ∈ N there exists a closed subset X ⊂ Y ∗ such that Em (k) is analytic and ψm (·, k) can be chosen to be a real analytic function for all k ∈ Y ∗ \X. Moreover, Em−1 < Em (k) < Em+1 (k)

∀ k ∈ Y ∗ \X.

If this condition indeed holds for all k ∈ Y ∗ , then Em (k) is called an isolated Bloch band. Moreover, it is known that meas X = meas {k ∈ Y ∗ : En (k) = Em (k), n = m} = 0.

(12.4)

This set of Lebesgue measure zero consists of the so-called band crossings. See Figure 12.1 for an example of bands for Mathieu’s model with potential VΓ = cos y. Note that due to (12.3) one can rewrite ψm (y, k) as ψm (y, k) = e ik·y χm (y, k)

∀ m ∈ N,

(12.5)

for some 2π-periodic function χm (·, k), usually called Bloch functions. In terms of χm (y, k) the spectral problem for Hper becomes (Bloch 1928) H(k)χm (y, k) = Em (k)χm (y, k), χm (y + 2πej , k) = χm (y, k) ∀ k ∈ Y ∗ ,

(12.6)

where H(k) denotes the shifted Hamiltonian 1 H(k) := e− ik·y Hper e ik·y = (−i∇y + k)2 + VΓ (y). 2

(12.7)

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Most importantly, the spectral data obtained from (12.6) allow us to decompose the original space L2 (Rd ) into a direct sum of so-called band Hilbert ∞ 2 d spaces: L (R ) = m=1 Hm . This is the well-known Bloch decomposition method , which implies that ∀ f ∈ L2 (Rd ) : f = fm , fm ∈ H m . (12.8) m∈N

The corresponding projection of f ∈ L2 (Rd ) onto the mth band space Hm is thereby given via (Reed and Simon 1976) f (ζ)ψ m (ζ, k) dζ ψm (y, k) dk. (12.9) fm (y) = Y∗

Rd

In the following, we shall also denote Cm (k) := f (ζ)ψ m (ζ, k) dζ,

(12.10)

Rd

the coeﬃcient of the Bloch decomposition. 12.2. Two-scale WKB approximation Equipped with the basic theory of Bloch bands, we recall here an extension of the WKB method presented in Section 2.2 to the case of highly oscillatory periodic potentials. Indeed, it has been shown in Bensoussan et al. (1978) and Guillot, Ralston and Trubowitz (1988) that solutions to (12.1) can be approximated (at least locally in time) by x ε→0 ε (12.11) u (t, x) ∼ a(t, x)χm , ∇S e iS(t,x)/ε + O(ε), ε where χm is parametrized via k = ∇S(t, x). The phase function S thereby solves the semiclassical Hamilton–Jacobi equation ∂t S + Em (∇S) + V (x) = 0,

S(0, x) = Sin (x).

(12.12)

The corresponding semiclassical ﬂow Xtsc : y → x(t, y) is given by x(t, ˙ y) = ∇k Em (k(t, y)), ˙ y) = −∇x V (x(t, y)), k(t,

x(0, x) = y,

(12.13a)

k(0, y) = ∇Sin (y).

(12.13b)

The wave vector k ∈ Y ∗ is usually called crystal momentum. In the case of a constant electric ﬁeld V = F · x, equation (12.13b) yields k(t, y) = k − tF,

k = ∇Sin (y).

Note that since k ∈ Y ∗ Td , this yields a periodic motion in time of x(t, y), the so-called Bloch oscillations.

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In addition, the leading-order amplitude in (12.11) is found to be the solution of the semiclassical transport equation (Carles, Markowich and Sparber 2004), ∂t a + ∇k Em (∇S) · ∇a +

1 divx (∇k Em (∇x φm ))am = (βm · ∇x V (x)) am , 2 (12.14)

where βm (t, x) := χm (y, ∇S), ∇k χm (y, ∇S) L2 (Y )

(12.15)

denotes the so-called Berry phase term (Carles et al. 2004, Panati, Spohn and Teufel 2006), which is found to be purely imaginary βm (t, x) ∈ (iR)d . It is, importantly, related to the quantum Hall eﬀect (Sundaram and Niu 1999). The amplitude a is therefore necessarily complex-valued and exhibits a non-trivial phase modulation induced by the geometry of VΓ ; see also Shapere and Wilczek (1989) for more details. Note that (12.14) yields the following conservation law for ρ = |a|2 : ∂t ρ + div(ρ∇k Em (∇S)) = 0. The outlined two-scale WKB approximation again faces the problem of caustics. Furthermore, there is an additional problem of possible bandcrossings at which ∇k Em is no longer deﬁned. The right-hand side of (12.11) can therefore only be regarded as a valid approximation for (possibly very) short times only. Nevertheless, it shows the inﬂuence of the periodic potential, which can be seen to introduce additional high-frequency oscillations (Γ-periodic) within uε . Remark 12.2. These techniques have also been successfully applied in weakly nonlinear situations (Carles et al. 2004). 12.3. Wigner measures in the periodic case The theory of Wigner measures discussed in Section 3 can be extended to the case of highly oscillatory potentials. The theory of the so-called Wigner band series was developed in Markowich, Mauser and Poupaud (1994) and G´erard et al. (1997). The basic idea is to use Bloch’s decomposition and replace the continuous momentum variable ξ ∈ Rd by the crystal momentum k ∈ Y ∗. A more general approach, based on space adiabatic perturbation theory (Teufel 2003), yields in the limit ε → 0 a semiclassical Liouville equation of the form sc ∂t w + {Hm , w} = 0,

w(0, x, k) = win (x, k),

(12.16)

where w(t, x, k) is the mth band Wigner measure in the Γ-periodic phase

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space Rdx × Y ∗ , {·, ·} denotes the corresponding Poisson bracket, and sc = Em (k) + V (x) Hm

(12.17)

denotes the mth band semiclassical Hamiltonian.

13. Numerical methods for Schr¨ odinger equations with periodic potentials 13.1. Bloch-decomposition-based time-splitting method

The introduction of a highly oscillatory potential VΓ xε poses numerical challenges for the numerical computation of semiclassical Schr¨ odinger equations. It has been observed by Gosse (2006) and Gosse and Markowich (2004) that conventional split-step algorithms do not perform well. More precisely, to guarantee convergence of the scheme, time steps of order O(ε) are required. In order to overcome this problem, a new time-splitting algorithm based on Bloch’s decomposition method has been proposed in Huang, Jin, Markowich and Sparber (2007) and further developed in Huang, Jin, Markowich and Sparber (2008) and Huang, Jin, Markowich and Sparber (2009). The basic idea is as follows. Step 1. For t ∈ [tn , tn+1 ] one ﬁrst solves iε∂t uε = −

x ε ε2 ∆uε + VΓ u . 2 ε

(13.1)

The main point is that by using the Bloch decomposition method, Step 1 can be solved exactly, i.e., only up to numerical errors. In fact, in each band space Hm , equation (13.1) is equivalent to iε∂t uεm = Em (−i∇)uεm ,

uεm ∈ Hm ,

(13.2)

where uεm ≡ Pεm uε is the (appropriately ε-scaled) projection of uε ∈ L2 (Rd ) onto Hm deﬁned in (12.9) and Em (−i∇) is the Fourier multiplier corresponding to the symbol Em (k). Using standard Fourier transformation, equation (13.2) is easily solved by uεm (t, x) = F −1 e iEm (k)t/ε (Fum )(0, k) , (13.3) where F −1 is the inverse Fourier transform. In other words, one can solve (13.1) by decomposing uε into a sum of band-space functions uεm , each of which is propagated in time via (13.3). After resummation this yields uε (tn+1 , x). Once this is done, we proceed as usual to take into account V (x). Step 2. On the same time interval as before, we solve the ODE iε∂t uε = V (x)uε ,

(13.4)

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where the solution obtained in Step 1 serves as initial condition for Step 2. In this algorithm, the dominant eﬀects from the dispersion and the periodic lattice potential are computed in one step. It thereby maintains their strong interaction and treats the non-periodic potential as a perturbation. Because the split-step error between the periodic and non-periodic parts is relatively small, the time steps can be chosen considerably larger than a conventional time-splitting algorithm (see below). As in a conventional splitting method (see Section 5), the numerical scheme conserves the particle density ρε = |uε |2 on the fully discrete level. More importantly, if V (x) = 0, i.e., no external potential, the algorithm preserves the particle density (and hence the mass) in each individual band space Hm . Remark 13.1. Clearly, the algorithm given above is only ﬁrst-order in time, but this can easily be improved by using the Strang splitting method (see Section 5). In this case, the method is unconditionally stable and comprises spectral convergence for the space discretization as well as secondorder convergence in time. Numerical calculation of Bloch bands In the numerical implementation of this algorithm a necessary prerequisite is the computation of Bloch bands Em (k) and Bloch eigenfunction χm (y, k). This requires us to numerically solve the eigenvalue problem (12.6). In one space dimension d = 1 we proceed as in Gosse and Markowich (2004), by expanding VΓ ∈ C 1 (R) in its Fourier series 2π 1 iλy VΓ (y) = V (λ) e , V (λ) = VΓ (y) e− iλy dy. 2π 0 λ∈Z

Clearly, if VΓ ∈ C ∞ (R) the corresponding Fourier coeﬃcients V (λ) decay faster than any power, as λ → ±∞, in which case we only need to take into account a few coeﬃcients to achieve suﬃcient accuracy. Likewise, expand the Bloch eigenfunction χm (·, k) in its respective Fourier series χ m (λ, k) e iλy . χm (y, k) = λ∈Z

For λ ∈ {−Λ, . . . , Λ − 1} ⊂ Z, one consequently approximates the spectral problem (12.6) by the algebraic eigenvalue problem     χ m (−Λ) χ m (−Λ) χ χ    m (1 − Λ)  m (1 − Λ) H(k)  (13.5)  = Em (k)  , .. ..     . . χ m (Λ − 1)

χ m (Λ − 1)

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where the 2Λ × 2Λ matrix H(k) is given by H(k) =  V (0) + 12 (k − Λ)2 V (−1) ··· 1 2  V (1) V (0) + 2 (k − Λ + 1) · · ·   . .. .. ..  . . V (2Λ − 1) V (2Λ − 2) ···

V (1 − 2Λ) V (2 − 2Λ) .. .

   . 

V (0) + 12 (k + Λ − 1)2

The matrix H(k) has 2Λ eigenvalues. Clearly, this number has to be large enough to have suﬃciently many eigenvalues Em (k) for the simulation, i.e., we require m ≤ 2Λ. Note, however, that the number Λ is independent of the spatial grid (in particular, independent of ε), thus the numerical costs of this eigenvalue problem are often negligible compared to those of the evolutionary algorithms (see below for more details). In higher dimensions d > 1, computing the eigenvalue problem (12.6) along these lines becomes numerically too expensive to be feasible. In many physical applications, however, the periodic potential splits into a sum of one-dimensional potentials, i.e., VΓ (y) =

d

VΓ (yj ),

Vj (yj + 2π) = Vj (yj ),

j=1

where y = (y1 , y2 , . . . , yd ) ∈ Rd . In this case, Bloch’s spectral problem can be treated separately (using a fractional step-splitting approach) for each coordinate yj ∈ R, as outlined before. Remark 13.2. In practical applications, the accurate numerical computation of Bloch bands is a highly non-trivial task. Today, though, there exists a huge amount of numerical data detailing the energy band structure of the most important materials used in the design of semiconductor devices, for example. In the context of photonic crystals the situation is similar. Thus, relying on such data one can in principle completely avoid the above eigenvalue computations and their generalizations to more dimensions. To this end, one should also note that, given the energy bands Em (k), we do not need any knowledge about VΓ to solve (12.1) numerically. Also, we remark that it was shown in Huang et al. (2009) that the Bloch-decomposition-based time-splitting method is remarkably stable with respect to perturbations of the spectral data. Implementation of the Bloch-decomposition-based time-splitting method In the numerical implementation we shall assume that VΓ admits the decomposition (13.1). In this case we can solve (12.1) by using a fractional step method, treating each spatial direction separately, i.e., one only needs

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to study the one-dimensional equation

x ε ε2 ε u iε∂t u = − ∂xx u + Vj 2 ε ε

(13.6)

on the time interval [tn , tn+1 ]. This equation will be considered on a one-dimensional computational domain (a, b) ⊂ R, equipped with periodic boundary conditions (necessary to invoke fast Fourier transforms). We suppose that there are L ∈ N lattice cells within (a, b) and numerically compute uε at L × R grid points, for some R ∈ N. In other words we assume that there are R grid points in each lattice cell, which yields the discretization 1 −1 k = − + , where = {1, . . . , L} ⊂ N, 2 L 2π(r − 1) , where r = {1, . . . , R} ⊂ N, yr = R and thus un ≡ u(tn ) are evaluated at the grid points x,r = ε(2π( − 1) + yr ).

(13.7)

(13.8)

Note that in numerical computations one can use R L whenever ε 1, i.e., only a few grid points are needed within each cell. Keeping in mind the basic idea of using Bloch’s decomposition, one has the problem that the solution uε of (13.6) does not in general have the same periodicity properties as ϕm . A direct decomposition of uε in terms of this new basis of eigenfunctions is therefore not possible. This problem can be overcome by invoking the following unitary transformation for f ∈ L2 (R): f (ε(y + 2πγ)) e− i2πkγ , y ∈ Y, k ∈ Y ∗ , f (y) → f(y, k) := γ∈Z

with the properties f(y + 2π, k) = e2 iπk f(y, k),

f(y, k + 1) = f(y, k).

In other words f(y, k) admits the same periodicity properties with respect to k and y as the eigenfunction ψm (y, k). In addition, the following inversion formula holds: f (ε(y + 2πγ)) = (13.9) f(y, k) e i2πkγ dk. Y∗

Moreover, one easily sees that the Bloch coeﬃcient, deﬁned in (12.10), can be equivalently written as f(y, k)ψ m (y, k) dy, Cm (k) = (13.10) Y

which, in view of (12.5), resembles a Fourier integral. In fact, all of these formulae can be easily implemented by using the fast Fourier transform.

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The numerical algorithm needed to perform Step 1 outlined above is then as follows. Step 1.1. First compute u ε at time tn by u ε (tn , x,r , k ) =

L

uε (tn , xj,r ) e− i2πk ·(j−1) ,

j=1

where x,r is as in (13.8). ε (t , k ) via (13.10): Step 1.2. Compute the coeﬃcient Cm n ε Cm (tn , k ) ≈

R 2π ε u (tn , x,r , k )χm (yr , k ) e− ik yr . R r=1

ε (t , k) up to time tn+1 according to (13.2), Step 1.3. Evolve Cm n ε ε (tn+1 , k ) = Cm (tn , k ) e− iEm (k )∆t/ε . Cm

Step 1.4. u ε can be obtained at time tn+1 by summing over all band contributions, u (tn+1 , x,r , k ) = ε

M

ε Cm (tn+1 , k )χm (yr , k ) e ik yr .

m=1

Step 1.5. Perform the inverse transformation (13.9): 1 ε u (tn+1 , xj,r , kj ) e i2πkj (−1) . u (tn+1 , x,r , k ) ≈ L L

ε

j=1

This concludes the numerical procedure performed in Step 1. The Bloch-decomposition-based time-splitting method was found to be converging for ∆x = O(ε) and ∆t = O(1); see Huang et al. (2007) for more details. In other words, the time steps can be chosen independently of ε, a huge advantage in comparison to the standard time-splitting method used in Gosse (2006), for instance. Moreover, the numerical experiments done in Huang et al. (2007) show that, of only a few Bloch bands, Em (k) are suﬃcient to achieve very high accuracy, even in cases where V (x) is no longer smooth (typically m = 1, . . . , M , with M ≈ 8 is suﬃcient). Applications of this method are found in the simulation of lattice Bose–Einstein condensates (Huang et al. 2008) and of wave propagation in (disordered) crystals (Huang et al. 2009). Remark 13.3. For completeness, we recall the numerical complexities for the algorithm outlined above: see Huang et al. (2007). The complexities of Steps 1.1 and 1.5 are O(RL ln L), the complexities of Steps 1.2 and 1.4

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are O(M LR ln R), and for Step 1.3 it is O(M L). The complexity of the eigenvalue problem (13.5) is O(Λ3 ). However, since Λ (or R) is independent of ε and since (13.5) needs to be solved only once (as a preparatory step), the computational costs for this step are negligible. In addition, since M and R are independent of ε, one can choose R L and M L, whenever ε 1. Finally, one should notice that the complexities in each time step are comparable to the usual time-splitting method. 13.2. Moment closure in Bloch bands It is straightforward to adapt the moment closure method presented in Section 6 to the case of periodic potentials. To this end, one considers the semiclassical Liouville equation (12.16), i.e., ∂t w + ∇k Em (k) · ∇x w − ∇x V (x) · ∇k w = 0, and close it with the following ansatz for the Wigner measure: w(t, x, k) =

J

|aj (t, x)|2 δ (k − vj (t, x)),

j=1

where we let δ denote the Γ∗ -periodic delta distribution, i.e., δ(· − γ ∗ ). δ = γ ∗ ∈Γ∗

By following this idea, Gosse and Markowich (2004) showed the applicability of the moment closure method in the case of periodic potentials in d = 1 (see also Gosse (2006)). In addition, self-consistent Schr¨odinger–Poisson systems were treated in Gosse and Mauser (2006). As mentioned earlier, extending this method to higher space dimensions d > 1 is numerically challenging. 13.3. Gaussian beams in Bloch bands The Gaussian beam approximation, discussed in Sections 8 and 9, can also be extended to the Schr¨odinger equation with periodic potentials. To this end, one adopts the Gaussian beam within each Bloch band of the Schr¨odinger equation (12.1). In the following, we shall restrict ourselves to the case d = 1, for simplicity. Lagrangian formulation As for the two-scale WKB ansatz (12.11), we deﬁne x i Tm (t,x,ym )/ε ε ∗ ϕm (t, x, ym ) = Am (t, ym )χm ∂x Tm , e , ε

(13.11)

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where ym = ym (t, y0 ), and 1 Tm (t, x, ym ) = Sm (t, ym ) + pm (t, ym )(x − ym ) + Mm (t, ym )(x − ym )2 . 2 Here Sm ∈ R, pm ∈ R, am ∈ C, Mm ∈ C. In addition, we denote by χ∗m the function obtained by evaluating the usual Bloch function χm (y, k) (with real-valued k ∈ Y ∗ ) at the point y = x/ε and k = ∂x Tm ∈ C. To this end, we impose the following condition: χ∗m (y, z) = χm (y, z)

for z ∈ R.

One can derive a similar derivation of the Lagrangian formulation as in Section 8.1 that corresponds to the semiclassical Hamiltonian (12.17). For more details we refer to Jin, Wu, Yang and Huang (2010b). Here we only mention that, in order to deﬁne the initial values for the Gaussian beams, one ﬁrst decomposes the initial condition, which is assumed to be given in two-scale WKB form, i.e., x iSin (x)/ε ε e uin (x) = bin x, , ε in terms of Bloch waves with the help of the stationary phase method (Bensoussan et al. 1978): ∞ x ε→0 ε in , ∂x Sin e iSin (x)/ε + O(ε), uin (x) ∼ am (x)χm ε m=1

where the coeﬃcient ain m (x)

= Y

bin (x, y)χm (y, ∂x Sin ) dy.

When one computes the Lagrangian beam summation integral, the complexvalued ∂x Tm = pm + (x − ym )Mm can be approximated by the real-valued pm√with a Taylor truncation error of order O(|x − ym |). Since |x − ym | is of O( ε) (see Tanushev (2008) and Jin et al. (2008b)), this approximation does not destroy the total accuracy of the Gaussian beam method, yet it provides the beneﬁt that the eigenfunction χ∗m (k, z) is only evaluated for real-valued k (and thus consistent with the Bloch decomposition method). However, because of this, the extension of this method to higher order becomes a challenging task. Eulerian formulation Based on the ideas presented in Section 9, an Eulerian Gaussian beam method for Schr¨ odinger equations with periodic potentials has been introduced in Jin et al. (2010b). It involves solving the (multivalued) velocity

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um = ∂x Sm from the zero-level set of the function Φm , which satisﬁes the homogeneous semiclassical Liouville equation (12.16) in the form Lm Φm = 0, where the mth band Liouville operator Lm is deﬁned as L m = ∂ t + ∂ k Em ∂ x − ∂ x V ∂ ξ ,

(13.12)

and Φm is the complex-valued d-dimensional level set function for the velocity corresponding to the mth Bloch band. They also solve the following inhomogeneous Liouville equations for the phase Sm and amplitude am : Lm Sm = k∂k Em − V, 1 Lm am = − ∂k2 Em Mm am + βm am ∂y V, 2 where Lm is deﬁned by (13.12) and βm denotes the Berry phase term as given by (12.15). Here, the Hessian Mm ∈ C is obtained from Mm = −

∂y Φm . ∂ξ Φm

14. Schr¨ odinger equations with random potentials Finally, we shall consider (small) random perturbations of the potential V (x). It is well known that in one space dimension, linear waves in a random medium get localized even when the random perturbations are small: see, e.g., Fouque, Garnier, Papanicolaou and Sølna (2007). Thus the analysis here is restricted to three dimensions. (The two-dimensional case is diﬃcult because of criticality, i.e., the mean-ﬁeld approximation outlined below is most likely incorrect.) 14.1. Scaling and asymptotic limit Consider the Schr¨ odinger equation with a random potential VR : √ x ε ε2 ε ε ε iε∂t u = − ∆u + V (x)u + ε VR u , x ∈ R3 . 2 ε

(14.1)

Here VR (y) is a mean zero, stationary random function with correlation length of order one. Its correlation length is assumed to be of the same √ order as the wavelength. The ε-scaling given above is critical in the sense that the inﬂuence of the random potential is of the same order as the one given by V (x) (see also the remark below). We shall also assume that the ﬂuctuations are statistically homogeneous and isotropic so that VR (x)VR (y) = R(|x − y|),

(14.2)

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where · · denotes statistical average and R(|x|) is the covariance of random ﬂuctuations. The power spectrum of the ﬂuctuations is deﬁned by −3 e iξ·x R(x) dx. (14.3) R(ξ) = (2π) R3

is a function of |ξ| When (14.2) holds, the ﬂuctuations are isotropic and R only. Remark 14.1. Because of the statistical homogeneity, the Fourier transform of the random potential VR is a generalized random process with orthogonal increments + p). VR (ξ)VR (p) = R(ξ)δ(ξ

(14.4)

If the amplitude of these ﬂuctuations is large, then purely random scattering will dominate and waves will be localized: see Fr¨ ohlich and Spencer (1983). On the other hand if the random ﬂuctuations are too weak, they will not aﬀect the transport of waves at all. Thus, in order to have scattering induced by the random potential and the inﬂuence of the slowly varying background V (x) aﬀect the (energy transport of the) waves in comparable ways, the √ ﬂuctuations in the random potential must be of order ε. Using the ε-scaled Wigner transformation, we can derive the analogue of (3.2) in the following form: ∂t wε + ξ · ∇x wε − Θε [V + VR ]wε = 0,

(14.5)

where the pseudo-diﬀerential operator Θε is given by (3.3). The behaviour of this operator as ε → 0 is very diﬀerent from the case without VR , as can already be seen on the level of formal multiscale analysis: see Ryzhik, Papanicolaou and Keller (1996). Let y = x/ε be a fast spatial variable, and introduce an expansion of wε (t) in the following form: wε (t, x, ξ) = w(t, x, ξ) + ε1/2 w(1) (t, x, y, ξ) + εw(2) (t, x, y, ξ) + · · · . (14.6) Note that we hereby assume that the leading term w does not depend on the fast scale. We shall also assume that the initial Wigner distribution ε (x, ξ) tends to a smooth, non-negative function w (x, ξ), which decays win in suﬃciently fast at inﬁnity. Then, as ε → 0, one formally ﬁnds that wε (t) , i.e., the averaged solution to (14.5), is close to a limiting measure w(t), which satisﬁes the following linear Boltzmann-type transport equation: ∂t w + ξ · ∇x w − ∇x V · ∇ξ w = Qw. Here, the linear scattering operator Q is given by − p)δ(|ξ|2 − |p|2 )(w(x, p) − w(x, ξ)) dp, R(ξ Qw(x, ξ) = 4π R3

(14.7)

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with diﬀerential scattering cross-section − p)δ(|ξ|2 − |p|2 ) σ(k, p) = 4π R(ξ and total scattering cross-section − p)δ(|ξ|2 − |p|2 ) dp. Σ(k) = 4π R(ξ R3

(14.8)

(14.9)

Note that the transport equation (14.7) has two important properties. First, the total mass (or energy, depending on the physical interpretation) is conserved, i.e., E(t) = w(t, x, ξ) dξ dx = E(0). (14.10) R3 ×R3

Second, the positivity of w(t, x, ξ) is preserved. Rigorous mathematical results concerning the passage from (14.1) to the transport equation (14.7) can be found in Spohn (1977), Dell’Antonio (1983) and Ho, Landau and Wilkins (1993) (which contains extensive references), and, more recently, in Erd¨os and Yau (2000). 14.2. Coupling with other media One can also study the problems when there are other media, including periodic media and interfaces (ﬂat or random). In the case of periodic media coupled with random media, one can use the above multiscale analysis combined with the Bloch decomposition method to derive a system of radiative transport equations: see Bal, Fannjiang, Papanicolaou and Ryzhik (1999a). In the limit system ε → 0, the transport part is governed by the semiclassical Liouville equation (12.16) in each Bloch band, while the right-hand side is a non-local scattering operator (similar to the one above) coupling all bands. One can also consider random high-frequency waves propagating through a random interface. Away from the interface, one obtains the radiative transport equation (14.7). At the interface, due to the randomness of the interface, one needs to consider a diﬀusive transmission or reﬂection process, in which waves can be scattered in all directions (see Figure 10.1). To this end, a non-local interface condition has to be derived: see Bal et al. (1999b). So far there have been few numerical works on random Schr¨odinger equations of the form (14.1). Bal and Pinaud (2006) studied the accuracy of the radiative transport equation (14.7) as an approximation to (14.1). Jin et al. (2008a) discretized a non-local interface condition for diﬀusive scattering, similar in spirit to the treatment described in Section 10.2. We also mention that the temporal resolution issue for time-splitting approximations of the Liouville equations with random potentials was rigorously studied in Bal

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and Ryzhik (2004). Finally, we refer to Fouque et al. (2007) and Bal, Komorowski and Ryzhik (2010) for a comprehensive reading on high-frequency waves in random media.

15. Nonlinear Schr¨ odinger equations in the semiclassical regime So far we have only considered linear Schr¨odinger equations. Nonlinear models, however, are almost as important, since they describe a large number of physical phenomena in nonlinear optics, quantum superﬂuids, plasma physics or water waves: see, e.g., Sulem and Sulem (1999) for a general overview. The inclusion of nonlinear eﬀects poses new challenges for mathematical and numerical study. 15.1. Basic existence theory In the following we consider nonlinear Schr¨ odinger equations (NLS) in the form ε2 ∆uε + V (x)uε + εα f (x, |uε |2 )uε , uε (0, x) = uεin (x), (15.1) 2 with α ≥ 0 some scaling parameter, measuring (asymptotically) the strength of the nonlinearity. More precisely, we shall focus on two important classes of nonlinearities: iε∂t uε = −

(1) local gauge-invariant nonlinearities, where f = ±|uε |2σ , with σ > 0, (2) Hartree-type nonlinearities, where f = V0 ∗ |uε |, with V0 = V0 (x) ∈ R some given convolution kernel. The ﬁrst case of nonlinearities has important applications in Bose–Einstein condensation and nonlinear optics, whereas the latter typically appears as a mean-ﬁeld description for the dynamics of quantum particles, say electrons 1 (in which case one usually has V0 (x) = ± |x| in d = 3). Concerning the existence of solutions to (15.1), we shall in the following only sketch the basic ideas. For more details we refer to Cazenave (2003) and Carles (2008). As a ﬁrst step we represent the solution to (15.1) using Duhamel’s formula: t ε ε α u (t) = U (t)uin − iε U ε (t − s) f ε (x, |uε |2 )(s)uε (s) ds, (15.2) 0

U ε (t)

where is the unitary semi-group generated by the linear Hamiltonian 1 H = − 2 ∆ + V (see the discussion in Section 2.1). The basic idea is to prove that the right-hand side of (15.2) deﬁnes a contraction mapping (in some suitable topology). To this end, one has to distinguish between the case of sub- and supercritical nonlinearities. The existence of a unique solution

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uε ∈ C([0, ∞); L2 (Rd )) can be guaranteed, provided (sub-critical case) σ < 2 ∞ d q d d and/or V0 suﬃciently smooth and decaying, say V0 ∈ L (R ) ∩ L (R ), with q > d4 . Remark 15.1. Unfortunately, this subcritical range of nonlinearities does not allow inclusion of several physically important examples, such as a cubic nonlinearity σ = 1 in three space dimensions (needed, for instance, in the description of Bose–Einstein condensates). In the case of supercritical nonlinearities σ > d2 , the space L2 (Rd ) is generally too large to guarantee existence of solutions. Typically, one restricts the initial data to uε ∈ H 1 (Rd ), i.e., initial data with ﬁnite kinetic energy, in order to control the nonlinear terms by means of Sobolev’s embedding. Assuming that uε ∈ H 1 , local-in-time existence can be guaranteed by using Strichartz estimates (see Cazenave (2003) for more details), and one obtains uε ∈ C([−T ε , T ε ]; H 1 (Rd )), for some (possibly small) existence time T ε > 0. 2 in order to use Remark 15.2. Strictly speaking, we require σ < d−2 Sobolev’s embedding to control the nonlinearity. Note, however, that the case of cubic NLS in d = 3 is allowed.

Even though local-in-time existence is relatively easy to achieve even in the supercritical case, the existence of a global-in-time solution to (15.1) generally does not hold. The reason is the possibility of ﬁnite-time blow-up, i.e., the existence of a time |T ∗ | < ∞ such that lim ∇uε = +∞.

t→T ∗

For T > T ∗ , a strong solution to (15.1) ceases to exist, and one can only hope for weak solutions, which, however, are usually not uniquely deﬁned. In order to understand better which situations blow-up can occur in, consider, for simplicity, the case of a cubic NLS with V (x) = 0: ε2 (15.3) ∆uε ± |uε |2 uε , uε (0, x) = uεin (x). 2 Given a local-in-time solution to this equation, one can invoke the conservation of the (nonlinear) energy, i.e., ε2 1 ε ε 2 |∇u (t, x)| dx ± |uε (t, x)|4 dx = E[uεin ]. E[u (t)] = 2 Rd 2 Rd iε∂t uε = −

Since E[uεin ] < const., by assumption, we can see that in the case where the nonlinearity comes with ‘+’, the a priori bound on the energy rules out the possibility of ﬁnite-time blow-up. These nonlinearities are usually referred to as defocusing. On the other hand, in the case of a focusing nonlinearity, i.e., f = −|uε |2 , we can no longer guarantee the existence of global-in-time

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solutions (in the supercritical regime). Rather, we have to expect ﬁnitetime blow-up to occur in general. Clearly, this phenomenon will also have a signiﬁcant impact on numerical simulations, in particular for ε 1. 15.2. WKB analysis of nonlinear Schr¨ odinger equations In order to understand the inﬂuence of nonlinear terms in the limit ε → 0, one can again invoke a WKB approximation of the form (2.10): ε→0

uε (t, x) ∼ aε (t, x) e iS(t,x)/ε . Let us assume for simplicity that α ∈ N. Then, plugging the ansatz given above into (15.1), and ordering equal powers in ε, we see that S solves 0 if α > 0, 1 2 (15.4) ∂t S + |∇S| + V (x) = 2 2 f (x, |a| ) if α = 0. We see that in the case α = 0 we can no longer solve the Hamilton–Jacobi equation for the phase S independently of the amplitude a. In other words, the amplitude inﬂuences the geometry of the rays or characteristics. This is usually referred to as supercritical geometric optics (Carles 2008), not to be confused with the supercritical regime concerning the existence of solutions, as outlined in Section 15.1 above. Weakly nonlinear geometric optics In contrast, the situation for α > 0 (subcritical geometric optics) is similar to the linear situation, in the sense that the rays of geometric optics are still given by (2.13) and thus independent of the nonlinearity. In this case, the method of characteristics guarantees the existence of a smooth S ∈ C ∞ ([−T, T ] × Rd ) up to caustics, and one can proceed with the asymptotic expansion to obtain the following transport equation for the leading-order amplitude: 0 if α > 1, a (15.5) ∂t a + ∇S · ∇a + ∆S = 2 2 −if (x, |a| ) if α = 1. One sees that if α > 1, nonlinear eﬀects are asymptotically negligible for ε 1. The solution is therefore expected to be roughly the same as in the linear situation (at least before caustics). For α = 1, however, nonlinear eﬀects show up in the leading-order amplitude. Note, however, that by multiplying (15.5) with a ¯ and taking the real part, one again ﬁnds ∂t ρ + div(ρ∇S) = 0, which is the same conservation law for ρ = |a|2 as in the linear case. The nonlinear eﬀects for α = 1 are therefore given solely by nonlinear phase

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modulations of the leading-order amplitude. In the case of a simple cubic nonlinearity, one explicitly ﬁnds (Carles 2000) a0 (y(t, x)) iG(t,x) e , a(t, x) = Jt (y(t, x))

|t| ≤ T,

(15.6)

where the slowly varying phase G is given by t |a0 (y(τ, x))|2 dτ. G(t, x) = − Jτ (y(τ, x)) 0 This regime is therefore often called weakly nonlinear geometric optics, and indeed it is possible to prove a rigorous approximation result analogous to Theorem 2.3 in this case too. Supercritical geometric optics On the other hand, the situation for α = 0 is much more involved. Indeed, it can be easily seen that a naive WKB approximation breaks down since the system of amplitude equation is never closed, i.e., the equation for an , obtained at O(εn ), involves an+1 , and so on (a problem which is reminiscent of the moment closure problem discussed in Section 6 above). This diﬃculty was overcome by Grenier (1998) and Carles (2007a, 2007b), who noticed that there exists an exact representation of uε in the form uε (t, x) = aε (t, x) e iS

ε (t,x)/ε

,

(15.7)

with real-valued phase S ε (possibly ε-dependent) and complex-valued amplitude aε . (Essentially, the right-hand side of (15.7) introduces an additional degree of freedom by invoking complex amplitudes.) Plugging (15.7) into (15.1) with α = 0, one has the freedom to split the resulting equations for aε and S ε as follows. On plugging the ansatz given above into (15.1), and ordering equal powers in ε, one sees that S solves 1 ∂t S ε + |∇S ε |2 + V (x) + f (x, |aε |2 ) = 0, 2 a ε ∂t aε + ∇S ε · ∇aε + ∆S ε − i ∆aε = 0. 2 2 Formally, we expect the limit of this system as ε → 0 to give a semiclassical approximation of uε , at least locally in time. Indeed this can be done by ﬁrst rewriting these equations into a new system for ρε = |aε |2 and v ε = ∇S ε . Under some assumptions on the nonlinearity f (satisﬁed, for instance, in the cubic case), the obtained equations form a strictly hyperbolic system in which one can rigorously pass to the limit as ε → 0 to obtain ∂t v + v · ∇v + V (x) + f (ρ) = 0, ∂t ρ + div(ρv) = 0.

(15.8)

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This is a system of (irrotational) Euler equations for a classical ﬂuid with pressure law p(ρ) = f (ρ)/ρ. Following this approach, one can prove that as long as (15.8) admits (local-in-time) smooth solutions ρ, v, the quantum mechanical densities (2.6)–(2.7) indeed converge strongly in C([0, T ]; L1 (Rd )), with ε→0 ε→0 ρε −→ ρ, J ε −→ ρ. Reconstructing from these limits an approximate solution of the nonlinear Schr¨odinger equation in WKB form, i.e., uεapp (t, x) = ρ(t, x) e iS(t,x)/ε , v(t, x) := ∇S(t, x), generally requires some care (see Carles (2007b) for more details). Essentially one needs to take into account a higher-order corrector to ensure that the formal approximation is indeed correct up to errors of order O(ε), independent of t ∈ [0, T ]. The semiclassical limit for α = 0 for |t| > T , i.e., beyond the formation of shocks in (15.8), is a very challenging mathematical problem. In the one-dimensional case the only available result is given by Jin, Levermore and McLaughlin (1999), using the inverse scattering technique. The semiclassical limit of focusing NLS is more delicate, but can also be obtained by inverse scattering: see Kamvissis, McLaughlin and Miller (2003). Remark 15.3. Let us close this subsection by noticing that the analysis for Hartree-type nonlinearities is found to require slightly less sophistication than for local ones (Alazard and Carles 2007) and that the intermediate case 0 < α < 1 can be understood as a perturbation of the situation for α = 0 (Carles 2007b). In addition, a threshold condition for global smooth solutions to (15.8) has been determined in Liu and Tadmor (2002). 15.3. Wigner measure techniques for nonlinear potentials One might hope to extend the results for Wigner measures given in Section 3 to nonlinear problems. This would have the considerable advantage of avoiding problems due to caustics. Unfortunately, this idea has not been very successful so far, the reason being that Wigner measures are obtained as weak limits only, which in general is not suﬃcient to pass to the limit in nonlinear terms. Indeed, one can even prove an ill-posedness result for Wigner measures in the nonlinear case (Carles 2001). A notable exception is the case of Hartree nonlinearities f = V0 ∗ |uε | with smooth interaction kernels V0 ∈ Cb1 (Rd ) (Lions and Paul 1993). In this case the Wigner measure associated to uε is found to be a solution of the self-consistent Vlasov equation: ∂t w + ξ · ∇ξ w − ∇x (V0 ∗ ρ) · ∇ξ w = 0, ρ(t, x) = w(t, x, dξ). Rd

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The physically more interesting case of non-smooth interaction kernels, such 1 as V0 ∼ |x| , which describes the coupling to a Poisson equation, is not covered by this result and can only be established in the particular case of the fully mixed quantum state; see Lions and Paul (1993) and Markowich and Mauser (1993) for more details, and also Zhang (2002) for a similar result valid for short times only.

15.4. Numerical challenges Due to the introduction of nonlinear eﬀects, the numerical diﬃculties discussed in Sections 4 and 5 are enhanced. The main numerical obstacles are the formation of singularities in focusing nonlinear Schr¨odinger equations and the creation of new scales at caustics for both focusing and defocusing nonlinearities. Basic numerical studies of semiclassical NLS were conducted in Bao et al. (2002) and Bao, Jin and Markowich (2003b), with the result that ﬁnite diﬀerence methods typically require prohibitively ﬁne meshes to even approximate observables well in semiclassical defocusing and focusing NLS computations. In the case when these very restrictive meshing constraints are bypassed, the usual ﬁnite diﬀerence schemes for NLS can deliver incorrect approximations in the classical limit ε → 0, without any particular sign of instability (Carles and Gosse 2007). Time-splitting spectral schemes are therefore the preferred method. To this end, we refer to Jin, Markowich and Zheng (2004) for the application of the time-splitting spectral method to the Zakharov system, to Huang, Jin, Markowich, Sparber and Zheng (2005) for the numerical solution of the Dirac–Maxwell system, and to Bao et al. (2003b) for numerical studies of nonlinear Schr¨odinger equations. In addition, let us mention Bao, Jaksch and Markowich (2003a, 2004), where numerical simulations of the cubically nonlinear Gross–Pitaevskii equation (appearing in the description of Bose–Einstein condensates) are given using time-splitting trigonometric spectral methods. A numerical study of ground state solutions of the Gross–Pitaevskii equation can be found in Bao, Wang and Markowich (2005). Remark 15.4. Note, however, that in the nonlinear case, even for ε > 0 ﬁxed, a rigorous convergence analysis of splitting methods is considerably more diﬃcult than for linear Schr¨ odinger equations: see, e.g., Lubich (2008), Gauckler and Lubich (2010) and Faou and Grebert (2010). Due to the nonlinear creation of new highly oscillatory scales in the limit ε → 0, time-splitting methods suﬀer from more severe meshing restrictions for NLS than for linear Schr¨odinger equations, particularly after the appearance of the ﬁrst caustics in the corresponding WKB approximation; see Bao et al. (2003b) and Carles and Gosse (2007) for more details. In the weakly

¨ dinger equations Methods for semiclassical Schro

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nonlinear regime the following meshing strategy is suﬃcient: ∆x = O(ε),

∆t = O(ε)

(to be compared with (5.5)). In the regime of supercritical geometric optics, however, one typically requires (even for quadratic observable densities) ∆x = O(ε),

∆t = o(ε),

i.e., a severe restriction on the time step. In addition, one may need to invoke the Krasny ﬁlter (Krasny 1986), i.e., high Fourier-mode cut-oﬀs, to avoid artifacts (such as symmetry breaking) in focusing NLS computations (Bao et al. 2003b). The latter, however, violates the conservation of mass: a clear drawback from the physics point of view. In order to overcome this problem, higher-order methods (in time) have to be deployed, such as exponential time-diﬀerencing or the use of integrating factors, and we refer to Klein (2008) for a comparison of diﬀerent fourth-order methods for cubic NLS in the semiclassical regime. Remark 15.5. In the closely related problem of the complex Ginzburg– Landau equation in the large space and time limit, the situation is known to be slightly better, due to the dissipative nature of the equation; see Degond, Jin and Tang (2008) for a numerical investigation