Computing high frequency solutions of symmetric hyperbolic systems with polarized waves

Transcription

1 Computing high frequency solutions of symmetric hyperbolic systems with polarized waves Lel Jefferis Shi Jin December 5, Abstract We develop computational methods for high frequency solutions of general symmetric hyperbolic systems with eigenvalue degeneracies (multiple eigenvalues with constant multiplicities) in the dispersion matrices that correspond to polarized waves. Physical examples of such systems include the three dimensional elastic waves Maxwell equations. The computational methods are based on solving a coupled system of inhomogeneous Liouville equations which is the high frequency limit of the underlying hyperbolic systems by using the Wigner transform []. We first extend the level set methods developed in [5] for the homogeneous Liouville equation to the coupled inhomogeneous system, find an efficient simplification in one space dimension for the Eulerian formulation which reduces the computational cost of two-dimesnional phase space Liouville equations into that of two one-dimensional equations. For the Lagrangian formulation, we introduce a geometric method which allows a significant simplification in the numerical evaluation of the energy density flux. Numerical examples are presented in both one two space dimensions to demonstrate the validity of the methods in the high frequency regime. Introduction We will study the general symmetric hyperbolic system of the form A(x) u ε t + D j uε x j = u ε (x, ) = u (x)e is(x)/ε, (.) where u C n, x R d, A(x) is a n n symmetric positive definite matrix, the D j are n n symmetric constant matrices. Many physical systems such as Maxwell s equations, the elastic wave equations the acoustic equations all may be put into the symmetric hyperbolic form with the correct choices of A(x) D j. Here we are interested in high frequency solutions, where the high frequency is introduced by the wave length ε in the initial data in (.). In many physical applications ε is very small compared to the scale of the computational domain, the numerical meshes time steps need to resolve this small scale, thus computing the high frequency solutions, in particular in high dimensions, is prohibitively expensive. One efficient way to deal with high frequency wave problems is to solve the limiting equation by finding the asymptotic equation when ε. The Wigner transform, introduced in [6], is a powerful mathematical tool to study this limit [3], since it is valid global in time, even beyond caustic formation. The limiting equation is the Liouville equation which does not depend on ε, permitting large time steps mesh sizes. However, it encounters two major difficulties. First, the Liouville equation is defined in the phase space, This work was partially supported by NSF grants DMS-4546 DMS-79: NSF Research Network in Mathematical Sciences KI-Net: Kinetic description of emerging challenges in multiscale problems of natural sciences. Department of Mathematics, University of Wisconsin-Madison, 48 Lincoln Drive, Madison, WI 5376, USA (jefferis@math.wisc.edu) Department of Mathematics, Institute of Natural Sciences Ministry of Education Key Lab of Scientific Engineering Computing, Shanghai Jiao Tong University, Shanghai 4, China, Department of Mathematics, University of Wisconsin- Madison, 48 Lincoln Drive, Madison, WI 5376, USA (jin@math.wisc.edu) Assume the following summation convention: repeated Latin indices are summed whereas repeated Greek indices are not summed.

2 thus doubling the computational dimension. Second, for the WKB kind of initial data given in (.), the solution is a superposition of Dirac Delta functions [4, 3] which are difficult to resolve numerically with a high order accuracy. The first difficulty can be dealt with by local level set methods [, ] in the Eulerian formulation, or using Lagrangian formulation which is just the particle method. The second difficulty, the numerical resolution of Delta function, was elegantly hled by a singularity decomposition method, first introduced in [6] for the semiclassical limit of the Schrödinger equation, then extended to symmetric hyperbolic systems in [5]. The method is based on the observation that the solution to the linear Liouville equation of the form of the Delta functions can be decomposed into two solutions of the Liouville equations which are both bounded, one, roughly speaking, corresponding to strength (or amplitude of the wave) while the other one the kernel of the Delta function. The final solution is the combination of these two quantities involving the Delta function, but the Delta function needs to be computed only at the final output time, not during the time evolution! This significantly enhances the numerical resolution of the singular solutions to the Liouville equation. See [, 7] for reviews of computational high frequency waves or semiclassical limit of quantum waves. In this paper, we extend the singular decomposition method of [6, 5] to (.). Different from the problem studied in [5], here the dispersion matrix will have multiple eigenvalues while in [5] only simple eigenvalues are allowed. By assuming constant multiplicities of the multiple eigenvalues, the high frequency limit of (.), using the Wigner transform, becomes coupled systems of inhomogeneous Liouville equations (while in [5] they are decoupled homogeneous Liouville equations for different eigenvalues) []. The inhomogeneities arise due to the degeneracy of the eigenvalues, which describe the cross-polarization effects. We first show that the singular decomposition method developed in [6, 5] still applies here except that one needs to solve inhomogeneous Liouville system. We also found an efficient simplification in one space dimension for the Eulerian formulation which reduces the computational cost of two-dimensional phase space Liouville equations into that of two one-dimensional equations. For the Lagrangian formulation, we introduce a geometric method which allows a significant simplification in the numerical evaluation of the energy density flux. While the polarization effects usually appear in three dimensional space for elastic waves electromagnetic waves, our method, developed for one two dimensional hyperbolic systems, is the first of its kind to deal with polarized waves thus provides a foundation for three dimensional simulation which will be developed in the near future. Furthermore, similar effects also appear in molecular dynamics with quantum transitions [5] mixing of Bloch bs in solid state physics [4] where the multiplicity is a variable thus more elaborate numerical methods along the line of this paper are desirable. This paper is organized as follows. In Section, we will introduce then use the Wigner transform to obtain, in the limit that ε, a coupled system of Liouville equations which govern the evolution of this high frequency limit, as shown in []. In Section 3 we will prove the singularity decomposition result. In Section 4 we will discuss simplifications which occur in the one space dimension case. In Section 5 we will discuss d-dimensional implementation of our method in the Eulerian Lagrangian frames as well as introduce a geometric reduction of computational complexity for the evaluation of energy energy flux in the Lagrangian framework. In Section 6 we will show some illustrative numerical examples in one two dimensions. Finally, Section 7 will contain our conclusions. The Wigner Transform of Hyperbolic Systems This section is based primarily on the derivation which appears in []. The energy density d components of flux for a solution of (.) are given by E(t, x) = A(x)u ε(t, x), u ε (t, x) (.) F i (t, x) = Di u ε (t, x), u ε (t, x) (.) respectively where, is the stard Euclidian inner product on C n.bytakinganinnerproductof(.)

3 with u ε (t, x) one gets the energy conservation law E t + F =. (.3) Integration of (.3) shows conservation of total energy: d E(t, x)dx =. (.4) dt Next define the scaled Wigner transform W ε (t, x, k)dk = d π e ik y u ε (t, x εy/)u ε(t, x + εy/)dy (.5) where u = u T is the conjugate transpose of u. The matrix W (t, x, k) is Hermitian but only becomes positive definite in the limit as ε []. It has the useful property that W ε (t, x, k)dk = u ε (t, x)u ε(t, x). (.6) The energy density flux may be recovered from W (t, x, k) via E(t, x) = Tr(A(x)W ε (t, x, k))dk, (.7) F j (t, x) = Introduce a natural inner product for the system (.) as Tr(D j W ε (t, x, k))dk. (.8) u, v A = Au, v (.9) note that with this definition the energy (.) may be written as E = u, u A.Definethedispersion matrix as the sum L(x, k) =A (x)k i D i (.) note that L(x, k) is self-adjoint with respect to (.9): Lu, v A = u,lv A. (.) Therefore, all eigenvalues ω τ of L(x, k) are real the corresponding eigenvectors b τ can be chosen to be orthogonal with respect to, A : L(x, k)b τ (x, k) =ω τ (x, k)b τ (x, k), b τ, b β A = δ τβ. (.) We assume that the eigenvalues have constant multiplicity independent of x, k. This assumption is valid for many physical examples including all those studied in this paper. In general the eigenvalues of L(x, k) are multiple eigenvalues so let ω τ (x, k) be an eigenvalue of multiplicity r let the corresponding eigenvectors b τ,i, i =,...,r be orthonormal with respect to, A. Define the n n matrices B τ,ij = b τ,i b τ,j, (.3) with i, j =,...,r. In the limit that ε, thewignermatrixw ε (t, x, k) is approximated by W () (t, x, k) which may be written as W () (t, x, k) = τ,i,j w τ ij(t, x, k)b τ,ij (x, k). (.4) We define the r r coherence matrices as W τ (t, x, k) = w τ ij(t, x, k) r r. (.5) 3

4 Note that the multiplicity r of the eigenvalues ω τ depends on τ but this is not indicated explicitly. The entries of the coherence matrix Wij τ can be recovered via W τ ij(t, x, k) =W () (t, x, k),b τ,ij (x, k) (.6) where X, Y = Tr(AX AY ) is a matrix inner product. From here we can deduce that each of the coherence matrices (.6) satisfies the coupled system of Liouville equations t W τ + k ω τ x W τ x ω τ k W τ = N τ W τ W τ N τ (.7) where the skew-symmetric coupling matrix N τ (x, k) is given by Nmn(x, τ k) = b τ,n,d i bτ,m ω τ b τ,n, bτ,m ω τ δ nm. (.8) x i x i k i A xi ki In the sense of distributions, the weak limit of W ε (, x,k) as ε gives the initial condition Using (.6) this implies that W () (, x, k) =u (x)u (x)δ(k S (x)). (.9) W τ ij(, x, k) =Tr(Au u AB τ,ij )δ(k S (x)). (.) In summary, W τ is computed via (.7) with initial condition given by (.). Next W () follows from W τ via (.4) from which one may compute the approximate energy density flux using E () (t, x) = Tr(A(x)W () (t, x, k))dk, (.) F () j (t, x) = Tr(D j W () (t, x, k))dk (.) respectively. In the next section we show how to solve (.7) with singular initial conditions of the form (.). 3 A Singularity Decomposition Method From the previous section we wish to solve the Liouville equation (.7) together with the initial condition (.). Since accuracy is reduced when an initial condition containing a delta function is evolved numerically, it is desirable to decompose the original Liouville equation into two separate equations which have bounded initial conditions [5, 6]. To this end we have the following theorem. Theorem 3.. Define the Liouville operator L = t + k ω τ x x ω τ k (3.) the coupling matrix N τ as in (.8). Then the solution W τ (t, x, k) to the system LW τ = N τ W τ W τ N τ W τ (, x, k) =U τ (x, k)δ(g τ (x, k)) (3.) for a smooth function g(x, τ k) may be written as U τ (t, x, k)δ(g τ (t, x, k)) where the matrix function U τ vector function g τ are governed by LU τ = N τ U τ U τ N τ U τ (, x, k) =U τ (3.3) (x, k) respectively. Lg τ = g τ (, x, k) =g τ (x, k) (3.4) 4

5 Proof. (We drop the τ superscript notation for the duration of the proof) We begin by defining the weak equivalent to LW = NW WN (3.5) W (, x, k) =W (x, k) as requiring that for any d d matrix φ(t, x, k) where φ i,j C (R + R n R n ), the following holds R n Tr(W φ t= )dxdk R n Tr(W Lφ)dxdkdt = R n Tr((NW WN)φ)dxdkdt. (3.6) Note that we use the trace of the matrix product here to represent i,j φ j,iw i,j for ease of notation. Simple integration by parts shows that for smooth W,(3.5)(3.6)areequivalent. DefinetheHamiltonianflow H t (x, k ) which evolves any point (x, k ) in phase space forward to time t according the Hamiltonian system d dt x(t) = kω τ d dt k(t) =. (3.7) xω τ We change coordinates to (t, x, k ) defined by (t, x, k) =(t, H t (x, k )). Since the determinant of the Jacobian of this transform is we rewrite (3.6) as Tr(W φ t= )dx dk Tr(W d R n R dt φ)dx dk dt = Tr((NW WN)φ)dx dk dt (3.8) n R n where we have taken advantage of the fact that Lφ(t, x, k) = d dt φ(t, H t(x, k )). Notethatinthispaper d dt always represents the full derivative in time that one also has LU(t, x, k) = d dt U(t, H t(x, k )) = N(H t (x, k ))U(t, H t (x, k )) U(t, H t (x, k ))N(H t (x, k )) (3.9) Lg(t, x, k) = d dt g(t, H t(x, k )) =. (3.) Substituting the anzats U(t, x, k)δ(g(t, x, k)) into the left h side of (3.8) we obtain Tr (U (x, k )δ(g (x, k ))φ t= ) dx dk Tr U(t, H t (x, k ))δ(g(t, H t (x, k ))) ddt φ dx dk dt. R n R n (3.) Noting first that (3.) implies g(t, H t (x, k )) = g(, x, k ),observethattheabovespaceintegralsare weighted surface integrals in d-dimensional phase space over a d-dimensional sub-manifold defined by the zero set of g(, x, k ) that the weight given in terms of the inverse volume of g(, x, k ).Denoting this zero-set manifold as S, (3.) may be written as Where S Tr(Uφ t= ) vol( g) dσ S Tr(U d dt φ) dσdt (3.) vol( g) vol( g) = det(g) (3.3) G is the Gramian matrix defined as G i,j = g i g j with the gradients taken over the n space coordinates. (3.) is an application of the smooth Coarea formula vol(g(, x, k )) is non-zero because the initial conditions given to g. Furthernotethatvol(g(, x, k )) is independent of time as is the surface d S over which the integrals are performed. Also since S is fixed in time, dtφ is still the full derivative in time. Thus integrate by parts in time so that (3.) becomes S Tr d dt U φ dσdt. (3.4) vol( g) 5

6 Then using (3.9), Finally, S S d Tr dt U φ vol( g) dσdt = Tr((NU UN)φ) vol( g) dσdt = this completes the proof. = S Tr((NU UN)φ) dσdt. (3.5) vol( g) Tr((NUδ(g) Uδ(g)N)φ)dx dk dt R n Tr((NW WN)φ)dx dk dt R n (3.6) Remark. In our application of Theorem 3. to (.7) (.9), the initial amplitude U τ (x, k) for U is independent of k though the proof above does not require this to be the case. Also, when the eigenvalues of the dispersion matrix (.) are simple, the coupling matrix (.8) is zero the Theorem 3. reduces to ones proven in [5, 6] for homogeneous Liouville equations. For now, S (x) is assumed to be differentiable. A discussion of some cases where S (x) is discontinuous may be found in [5, 6]. Using Theorem 3., the computation of delta functions during time evolution is avoided delta functions only appear in post processing steps when one wishes to evaluate energy density (.) flux (.). In particular, our method uses the decomposition in Theorem 3. to replace solving t W τ + k ω τ x W τ x ω τ k W τ = N τ W τ W τ N τ Wij τ (, x, k) =Tr(Au u AB τ,ij (3.7) )δ(k S (x)) with instead solving both Next one obtains W τ (t, x, k) using LU τ = N τ U τ U τ N τ U τ (, x, k) ij = Tr(Au u AB τ,ij ) Lg τ = g τ (, x, k) =k S (x). (3.8) (3.9) W τ (t, x, k) =U τ (t, x, k)δ(g τ (t, x, k)), (3.) obtains W () (t, x, k) using (.4) then finally obtains energy density flux using (.) (.) respectively. A method which solves the Liouville equations (3.8) (3.9) directly is called Eulerian whereas a method which solves them along their respective characteristics is Lagrangian. Which approach is preferable, Eulerian or Lagrangian, depends greatly on the individual problem in the following sections we discuss both. 4 The Eulerian Formulation 4. d Dimensions The Eulerian formulation of our method starts by solving (3.8) (3.9) in d dimensions. For a given ω τ we denote the solutions to (3.8) (3.9) by U τ (t, x, k) g τ (t, x, k) respectively. Once these solutions are obtained, we recover W τ (t, x, k) via (3.) W () (t, x, k) via (.4). After this, energy density flux follow from (.) (.) respectively which requires detecting the zero set of g τ. At the initial time, the zero set of g τ is given by the initial condition in (3.9) so for each x there is exactly one k where g τ =. We call g τ is single valued when this property holds. At some later time, it may be the case that for some x there are multiple k where g τ = we call g τ multiple valued in this case [4, 3] (see Figure 4. for an example of a multiple valued solution). A point (x, k) where g τ (t, x, k) = 6

7 Figure 4.: In reference to evaluating (5.) (5.), if x = a then s a =3 if x = b then s b =5(note that τ notation has been dropped for simplicity). In reference to evaluating (5.) (5.), the simplices here are simply the x intervals delineated by the dots along the curve g(t,x,k)=. If x = a then x falls into 3 distinct intervals if x = b then x falls into 5 distinct intervals. x = c is an example of a point where g(t,x,k) has a caustic. vol ( k g τ (t, x, k)) = is called a caustic at these points, the energy density (.) flux (.) are not well defined (the point x = c in Figure 4. is an example of such a point). Away from caustics, however, if we define K τ (t, x) ={k : g τ (t, x, k) =} then the energy density flux may be evaluated at final time t = T as respectively where E () (T,x) = τ F () j (T,x) = τ k K τ (T,x) k K τ (T,x) Ũ τ (t, x, k) ij Tr Ũ τ (T,x, k) A(x) vol [ k g τ (4.) (T,x, k)] Tr D j Ũ τ (T,x, k) vol [ k g τ (4.) (T,x, k)] U τ ij(t, x, k)b τ,ij (x, k) (4.3) is a partially summed version of (.4). K τ could be approximated numerically in many ways one of which is the following. Define the fixed square grid in the phase variables by k z = hz where z i Z h> is the step size. Then for a given x g τ (t, x, k), definethesetss i with i =,...,d by S i = {k z : g τ i (t, x, k z )g τ i (t, x, k z+ei ) < } (4.4) where e i is i th unit vector in R d. Then for sufficiently small h, wehave K τ (t, x) d i=s i. (4.5) Alternately, one could evaluate (.) (.) by introducing numerical delta function in (3.) as is in [5, 6]. This alternate approach avoids the need to find K τ (t, x) ={k : g τ (t, x, k) =} explicitly but may introduce more error depending on the numerical delta function used. 7

8 To reduce the computational cost of solving the Liouville equations on d dimensions, one may take advantage of optimized techniques for solving Liouville equations found in, for example, [, 5, 6, 8,, ]. In particular, the local level set approach allows for computation only around the zero set of g τ (t, x, k) []. Beyond this, in one dimension the computations on two-dimensional phase space can be reduced to just one dimension as seen in Section One Dimension In one dimension, x, k g τ are all scalars so will be denoted x, k g τ respectively. Since in one dimension k appears as a scalar multiple in the dispersion matrix L given by (.), the eigenvalues of L will always be of the form ω τ (x, k) =kλ τ (x) the corresponding eigenvectors b τ will always be k independent. Consequently, the coupling matrix N τ given in (.8) is also k independent. Finally, from (.) it follows that the initial conditions of (3.8) are k independent so that (3.8) reduces to [ t + λ(x) x ]U τ = N τ U τ U τ N τ U τ (,x)=a τ (4.6) (x). For g τ of (3.4) we have the following simple Theorem: Theorem 4.. The solution to [ t + λ(x) x kλ (x) k ]g τ = g τ (x, k, ) = k x S (x) may be written as g τ (t, x, k) =kγ (t, x)+γ (t, x) with Γ Γ governed by [ t + λ(x) x ]Γ = Γ (x, ) = x S (x) respectively. [ t + λ(x) x λ (x)]γ = Γ (x, ) = Proof. The proof is a simple substitution of the assumed form for g τ into (4.7). Already, the computation has been reduced from one on two-dimensional phase space to one dimension but further simplification comes in the post processing steps for finding energy density or flux by noting that δ(g τ (x, k, t))dk = δ(kγ (x, t)+γ (x, t))dk = (4.) Γ (x, t) R R so that Γ needs not be computed. Also note that the function S (x) is no longer used in the computation. This suggests that in D at least, the high frequency limit (ε ) of the energy density flux is independent of S (x). ButhighermomentsofW τ clearly depend on S (x). In summary, in one dimension, the equations (3.8) (3.9) reduce to (4.6) (4.7) respectively. Then (4.7) may be solved by instead solving both (4.8) (4.9) then using g τ (t, x, k) =kγ (t, x)+γ (t, x). Thus all computations to acquire U τ (t, x, k) g τ (t, x, k) are performed in one dimension. Remark. The existence of an extension of the decomposition in Theorem 4. to higher dimensional space remains an open question. 5 The Lagrangian Formulation For the Lagrangian formulation, we solve (3.8) (3.9) by the method of characteristics. Since Theorem 3. implies that the only values of g τ (t, x, k) ki g τ (t, x, k) needed for computing energy density flux are those along the zero level set of g τ (t, x, k), we present two possible methods both of which reduce a computation on d-dimensional phase space to d dimensions. Then we discuss the numerical treatment of caustics using the Lagrangian approach. (4.7) (4.8) (4.9) 8

9 5. The Method of Characteristics For a given eigenvalue ω τ the Hamiltonian system defining the bicharacteristics of the Liouville equations (.7) is (3.7) with initial conditions given by (x(), k()) = (x, x S (x )). (5.) Note that even though the trajectory of x(t) k(t) depends on the eigenvalue ω τ,wedon tindicatethis explicitly. From (3.8), U τ along the characteristics is governed by with the initial conditions d dt U τ (t, x(t), k(t)) = N τ (t)u τ U τ N τ (t) (5.) U τ (, x(), k()) ij = Tr[A(x())u (x())u (x())a(x())b τ,ij (x(), k())] (5.3) where N τ (t) = N τ (x(t), k(t)) is the coupling matrix given by (.8). From (3.9), g τ along the bicharacteristics is governed by d dt gτ (t, x(t), k(t)) = g τ (5.4) (, x(), k()) =. For evaluation of the energy density flux, as shown in (4.) (4.), we also need approximations to ki g τ at points along the zero set of g τ at the final time. To obtain these, start from (3.9) take gradients with respect to x separately k to derive, for each component g τ i, L( x g τ i )= ( xkω τ )( x g τ i )+( xxω τ )( k g τ i ) L( k g τ i )=( kxω τ )( k g τ i ) ( kkω τ )( x g τ i ) (5.5) where L is given by (3.). Then along the characteristics, the above becomes d dt ( xgi τ )= ( xkω τ )( x gi τ )+( xxω τ )( k gi τ ) d dt ( kgi τ )=( kxω τ )( k gi τ ) ( kkω τ )( x gi τ ), (5.6) with initial conditions given by x g τ i (, x(), k()) = x i x S (x) k g τ i (, x(), k()) = e i (5.7) where e i is the i th unit vector in R d. Note that even though we only desire k gi τ,thefactthat(5.6)are coupled requires one to solve for x gi τ as well. Now, by solving (3.7), (5.), (5.6) for points along the zero set of g τ,onecefinethephase space energy density phase space flux corresponding to eigenvalue ω τ at x(t) as E τ (t, x(t), k(t)) = A(x(t)) Tr Ũ τ (t, x(t), k(t)) vol [ k g τ (5.8) (t, x(t), k(t))] F τ j (t, x(t), k(t)) = Tr D j Ũ τ (t, x(t), k(t)) vol [ k g τ (t, x(t), k(t))] (5.9) respectively with Ũ τ defined by (4.3). Note that with the initial condition (5.), the final point (x(t), k(t)) is different for each ω τ so that summing (5.8) (5.9) over τ is not straight forward as it was in the Eulerian case may be done analytically via the following construction. For a given point x, define(x τ s(t), k τ s(t)) with s =,...,s τ x for some positive integer s τ x as all points with the following properties: 9

10 . The evolution of (x τ s(t), k τ s(t)) is governed by (3.7) with eigenvalue ω τ. k τ s() = S (x τ s()) 3. At final time T, x τ s(t )=x Note that s τ x denotes the number of branches in the context of multiple valued solutions so that s τ x =before caustic formation s τ x after caustic formation (see Figure 4. for an illustration in one dimension). Now the energy density flux at time T may be expressed in terms of phase space energy density phase space flux ((5.7) (5.8) respectively) as E () (T,x) = τ s τ x E τ (T,x, k τ s(t )) (5.) s= F () j (T,x) = τ s τ x Fj τ (T,x, k τ s(t )) (5.) s= respectively. Finding all the points (x τ s(t), k τ s(t)) which satisfy the above three conditions is not necessarily easy. Because of this, in Section 5.3 we will introduce a more practical method to obtain energy density flux from these so called phase space energy density phase space flux before after g τ forms caustics. Even though we only need to solve (3.7), (5.) (5.6) for points on the set of g τ,theadditionof the d coupled ordinary differential equations (ODE) in (5.6) is a significant increase in computation. For example, in three dimensions, (5.6) represents 8 coupled ODEs! To avoid solving this considerable number of ODE, we present a geometric method next which avoids these ODEs completely. 5. A Geometric Method Here we present a method that entirely avoids solving any ODE beyond (3.7) (5.). In particular, we will show that one may use the solution to (3.7) (5.) to solve for the vol [ k g τ (t, x(t), k(t))] term needed for evaluating both (5.8) (5.9). Just as in Section 3., define the Hamiltonian flow H t (x, k ) which evolves any point in (x, k ) phase space forward to time t according the Hamiltonian system (3.7). Also define the pair of coordinates related by (t, x, k) =(t, H t (x, k )) the determinant of the Jacobian of the transform (t, x, k ) (t, x, k) is. If one parameterizes the zero set of g τ so that g τ (, x (s), k (s)) =, theng τ (t, H t (x (s), k (s))) = parameterizes the zero set for all time. At a fixed time t, denotethejacobianforthetransformh t as DH t. Then x x s =(DH k t ) s (5.) k [ x g τ (, x, k )] T [ k g τ (, x, k )] T =(DH t ) T [ x g τ (t, x, k)] T [ k g τ (t, x, k)] T. (5.3) x [ Note that each column vector in s is perpendicular to each column vector in x g τ (t, x, k)] T k [ k g τ (t, x, k)] T x [ that each column vector in s is perpendicular to each column vector in x g τ (, x, k )] T k [ k g τ (, x, k )] T We seek to establish useful relationships between the vectors appearing in (5.) (5.3) to this end we prove a few linear algebra results which appear in the Appendix. In light of these results, we note that these sets of vectors seen in (5.) (5.3) match the statement of Theorem 8. with α =d β = d (see the Appendix). Since det(dh t )=,weconcludethat x vol s k vol [ x g τ (, x, k )] T [ k g τ (, x, k )] T x = vol s k vol [ x g τ (t, x, k)] T [ k g τ (t, x, k)] T. (5.4).

11 Then Corollary 8.3 (see the Appendix) establishes that vol ( s x) vol ( k g τ (, x, k )) = vol ( s x ) vol ( k g τ (t, x, k)). (5.5) Now every term in (5.5) except the term vol ( k g τ (t, x, k)) may be computed exactly or approximately by simply solving (3.7) on a grid of points so that we obtain an approximation to vol ( k g τ (t, x, k)) without evolving k g τ (t, x, k) by the ODE system (5.6)! We start our computation with the initial grid given by x z () = {hz : z,z,...,z d [ N,...,N]} so that the corresponding points for k are given by k z () = S (x z ). For each eigenvalue ω τ define the evolution of the grid points according to (3.7) to a time t as x τ z(t) k τ z(t). NextsolveforU τ (t, x τ z(t), k τ z(t)) using using the ODE (5.). Then by using the initial conditions for g τ given in (3.9), (5.5) may be written as vol ( s x) = vol ( k g τ (t, x, k)) (5.6) so that for each eigenvalue ω τ,thephasespaceenergydensityphasespacefluxonthegridx τ z(t) may be written as E τ (t, x τ z(t), k τ z(t)) = A(x Tr τ Ũ τ (t, x τ z(t), k τ z(t)) z(t)) vol [ k g τ (t, x τ z(t), k τ z(t))] = (5.7) Tr A(x τ z(t))ũτ (t, x τ z(t), k τ z(t)) vol [ s x τ z(t)] Fj τ (t, xτ z(t), k τ z(t)) = Tr D j Ũ τ (t, x τ z(t), k τ z(t)) vol [ k g τ (t, x τ z(t), k τ z(t))] = Tr D j Ũ (5.8) τ (t, x τ z(t), k τ z(t)) vol [ s x τ z(t)] where Ũ τ is defined by (4.3) s x τ z(t) here may be approximated as the d d matrix given by h d x τ (z +,z,...,z d ) (t) xτ (z,z,...,z d ) (t) xτ (z,z,...,z d +) (t) xτ (z,z,...,z d ) (t). (5.9) In summary, for a given eigenvalue ω τ by use of Theorem 8., ki g τ (t, x(t), k(t)) may be approximated by simply solving the characteristic equations (3.7) on an initial grid x z. But this must be done anyway to solve U τ (t, x(t), k(t)), thus the approximation to ki g τ (t, x(t), k(t)) is obtained for free without additional computation. Again, in Section 5.3 we will show how to obtain energy density flux from these so called phase space energy density phase space flux before after g τ forms caustics. Remark. We observe that some elements of this geometric method relate to those presented in [9] which rely on the property of conservation of charge, but the equations studied therein are homogeneous scalar Liouville equations whereas here we have introduced a method for coupled systems of inhomogeneous Liouville equations of the form (3.) in which charge is not conserved. Thus this method is more general than that in [9]. 5.3 Summing Before or After Caustic Formation We now show how to obtain energy density (.) flux (.) from the phase space energy density phase space flux defined by (5.7) (5.8) respectively before or after the formation of caustics. As in Section 5., define an initial mesh x z () = {hz : z,z,...,z d [ N,...,N]} so that the corresponding points for k are given by k z () = S (x z ).Foreacheigenvalueω τ define the evolution of the grid points according to (3.7) to a time t as x τ z(t) k τ z(t). Then solving for phase space energy density flux for the eigenvalue ω τ is given by (5.7) (5.8). The grid at the final time T given by x τ z(t ) k τ z(t ) is different in general for each τ. Alsoforeachτ, thephasespaceenergydensityfluxatfinaltimet give by (5.7) (5.8) are possibly multiple valued. Thus to evaluate the energy density flux at any x,

12 .8 V5.8 V3.6 V3.6.4 T3.4 x..8 V T T V4 x..8 V V4 T T T3 V V.4.. V x x Figure 5.: The left plot shows a triangulation on an initial grid of 5 points at t =. The right plot shows the data set at a later time T>. At the initial time, the three triangles do not overlap whereas at the later time, the solution has become multiple valued the triangles do overlap (in particular, T lies completely inside of T ). To evaluate the energy density, for example, at the point at time T,observethatthe lies in triangle T only. Thus use the energy density at the vertices of T to interpolate the value at. To evaluate the energy density, at the point at time T,however,observethatthe lies in triangles T, T T 3. Thus one must use the energy density at the vertices of each triangle to interpolate the value at three times (once for each triangle) then sum the results together as indicated by (5.). first one must fix τ sum (5.7) (5.8) over all of branches second one must sum all these results over τ. Asystematicwaytocomputethissummationisdescribednext. Define a triangulation of the initial grid x τ z() as T τ () where each member of T τ () is a simplex of dimension d denoted by its d +vertices which are members of x τ z(). Then denote by T τ (t) the collection of simplices who have the corresponding vertices in x τ z(t). Note that for t>, T τ (t) may not be a proper triangulation since its simplices may overlap one another (see Figure 4. for a one dimensional illustration Figure 5. for a two dimensional illustration). Further note that such an overlap occurs exactly when g τ becomes multiple valued. If a point x falls inside of a simplex S T τ (t) then since the solution to (5.7) (5.8) has been computed at the vertices of S, denotethelinearinterpolationofphasespaceenergy density phase space flux onto the point x relative to the vertices of S as LI[E τ (t, x τ z(t), k τ z(t)), S, x] LI[Fj τ (t, xτ z(t), k τ z(t)), S, x] respectively. Then finally one obtains the energy density flux at the point x by E () (T,x) = LI[E τ (T,x τ z(t ), k τ z(t )), S, x] (5.) τ respectively. F () j (T,x) = τ {S T τ (t):x S} {S T τ (t):x S} LI[Fj τ (T,x τ z(t ), k τ z(t )), S, x] (5.) Remark. This triangulation method has been implemented in our numerical examples some additional details about this approach can be found in [9].

13 6 Numerical Results The following few examples show the effectiveness of the Eulerian Lagrangian formulations in one two dimensions are based on two model problems shown below. 6. One-Dimensional Model Problem The following model system is one of the most simple one-dimensional systems which has repeated eigenvalues of the dispersion matrix as well as a nontrivial coupling matrix. In reference to (.) define D = I A(x) where with R = A = RMR T (6.) M = a b b b a b b b a cos(θ) sin(θ) sin(θ) cos(θ) (6.) (6.3) where we assume that <b<a that a, b, θ are arbitrary function of x. The eigenvalues eigenvectors of the dispersion matrix L orthonormal with respect to, A are ω = k(a b) b, = a brv, b, = a brv, ω = k(a +b) b = (6.4) a +brv where The coupling matrix for ω is N = v, = (,, ) v, = 6 (,, ) v = 3 (,, ). (a b)θ 3 (a b)θ 3 (6.5). (6.6) Example In this example we demonstrate the convergence of the full solution of (.) to the solution obtained via our new method in the high frequency limit (ε ). We take the following parameters θ(x) = sin(πx) a =,b= domain: x [.5,.5] boundary conditions: periodic in x not needed in k Initial conditions u (x) =(e 4 tan(πx),, ), S (x) =x, ε =/(4π) ε =/(π) Final time: t f =.5 Remark. Boundary conditions are not needed in k because of the reduction of dimension we derived in Section 4. 3

14 .5.45 Energy Density Our Method Lax Wendroff ε = /(4π) Lax Wendroff ε = /(π).5.45 Energy Flux Our Method Lax Wendroff ε = /(4π) Lax Wendroff ε = /(π) Energy Density.3.5. Energy Flux x x Figure 6.: (Example ) Energy density energy flux of the one-dimensional model problem comparing the full solution to our new method s solution in the high frequency limit. The full solution on the original system was computed using the Lax-Wendroff scheme the solution from our new method was computed in the Eulerian frame using the simplifications outlined in Sections 3 4. The results are shown in Figure 6. where the Lax-Wendroff solution is seen to be converging to our new method s solution as ε decreases. Example Next we show an example where a, b are no longer constants take the following parameters: θ(x) =sin(πx) a = (.5+.4sin(4πx)), b=(.5+.4sin(4πx)) domain: x [.5,.5] boundary conditions: periodic in x not needed in k Initial conditions u (x) =(e 4 tan(πx),, ), S (x) =x, =/(4π) Final time: t f =.5 The full solution on the original system was computed using the Lax-Wendroff scheme while our new method s solution was computed in the Eulerian frame using the simplifications outlined in Section 3. The results (which show as good agreement) are shown in Figure 6.. Remark. We also computed Example with the Lagrangian formulation of our method using the interpolation described in Section 5.3 we obtained similar results. 6. Two-Dimensional Model Problem The following model system is a two-dimensional analogue of Maxwell s equations which are shown in []. In reference to (.) define /a A = /a /b, D =, D = (6.7) /b 4

15 Energy Density Energy Flux.5 Lax Wendroff Our Method.5 Lax Wendroff Our Method.4.4 Energy Density.3 Energy Flux x x Figure 6.: (Example ) Energy density of the one-dimensional model problem with non-constant a b functions. where a, b > are arbitrary functions of x. The dispersion matrix is ak ak L(x, k) = ak ak bk bk (6.8) bk bk The eigenvalues eigenvectors of the dispersion matrix orthonormal with respect to, A are ω = ab k + k b, = a k, a k b k +k,, k +k b, = a k, a k b k +k, k +k, ω = ab k + k b, = a k, a k b k +k,, k +k b, = a, a b,, k k +k k k +k (6.9) The coupling matrix for ω is N = N. N = b a b a (a k +k x k a x k ) (a k +k x k a x k ) (6.) Remark. When k = in the one two-dimensional model problems, the multiplicity of the eigenvalues changes; This violates the assumption we made when we developed our new method. This issue is resolved by the fact that in solving via our new method, g(t, x, k) given by (3.9) is initialized with zero set away from the origin so conservation of energy of Hamiltonian flows guarantees that this zero set stays away from the origin for all time. Thus even though the origin may be part of the computational domains of (3.8) (3.9), it will not be used for evaluating the physical observables via (.7) (.8). [5] also discusses this issue in further detail. 5

16 Example 3 In this example, we show a solution to the two-dimensional model problem computed with the Lagrangian formulation of our new method outlined in Section 5. Since our new method is tailored to compute multiple valued observables, we choose an example where the solution to (3.4) becomes multiple valued; something that never happens in one dimension. We take the following parameters: a(x,x )=.5+.4 cos(4πx )sin(4π(x )), b(x,x )= domain: (x,x ) [.5,.5] [.5,.5] boundary conditions: periodic in x x. Initial conditions u (x,x )=(e 5(tan(πx) +tan(π(x +.5)) ),,, ), S (x,x )=x + x Final time: t f =.8,.6,.4,.3 The result is shown in Figure 6.3 where we have chosen to plot the energy density. The two components of the flux for the final time t =.3 are shown in Figure 6.4. A three-dimensional projection of the zero set of g corresponding to ω at the final time is shown in Figure 6.5, as well as a contour plot of the same surface. The contour plot shows the multiple valued regions in the x space which correspond to the caustics appearing in Figure 6.3. A reference solution with ε =/(8π) was computed found to have a good agreement with our method until time neared the caustic formation time t =.4 which is what we expect. Computing a reference solution with a stard finite difference method for ε =/(8π) after caustic formation was not computationally feasible due to heavy diffusion of these methods for this example. Thus we computed, for sake of comparison, the exact solution for t =.3 ε =/(8π) using a Gaussian beam type method which we will introduce in a forthcoming paper. The result may be seen in Figure 6.6 where one sees interference fringes appearing in the multiple valued region of the solution. As ε shrinks, the interference fringes in this region is expected to limit weakly to the solution shown in Figure Conclusion We have extended the singularity decomposition idea in [5] to the case of high frequency solutions of symmetric hyperbolic systems with repeated eigenvalues of the dispersion matrix in both Eulerian Lagrangian frameworks. Such problems arise in many physically important problems such as Maxwell s equations the elastic wave equations. Furthermore we introduce a highly efficient Lagrangian method with a geometric reduction of computational complexity to the numerical evaluation of the energy energy flux. Numerical examples in both one two space dimensions are given to show the validity of the new computational methods. 8 Appendix: A Few Linear Algebra Identities Lemma 8.. Suppose M is an α α matrix with det(m) =.WriteM M in block form as A B Ã B M = M C D = C D (8.) where A Ã are β β matrices for some β<α. Then det(a) =det(m)det( D). (8.) Proof. Examine the two cases: Case: Assume that det(a) =. Then there exists v a = where Av a = so take A B va = C D Cv a (8.3) 6

17 Energy Density t=.8 Energy Density t= x. x x x Energy Density t=.4 Energy Density t= x. x x x Figure 6.3: (Example 3) Energy density plotted at different times. Caustics form around t =.4. 7

18 Flux t=.3 (x Component) Flux t=.3 (x Component) x x x x Figure 6.4: (Example 3) The two components of energy flux at time t =.3. Figure 6.5: (Example 3) Zero set projection of the g function corresponding to ω. On the left, the multiple valued surface is projected into three dimensions by plotting x, x versus k. On the right, the same surface is represented as a contour plot with axis limits set equal to those in Figure 6.3 Figure 6.4 for comparison. 8

19 Energy Density t= x x Figure 6.6: (Example 3) Energy density at t =.3 with ε = /(8π) computed with Gaussian beams. where Cv a = since det(m) =. Then by definition à B C D = Cv a BCva DCv a = va (8.4) which implies that det( D) =since Cv a =. Thus (8.) holds in this case. Case: Assume that det(a) =. Then note first that A B A I A det =det B C D C I D CA =det(a)det(d CA B)=det(M) (8.5) B where we point out for the sake of the following steps that (8.5) implies det(d CA B) =. Then, from the block matrix inversion formula: A B A B(D CA B) C D (D CA B) =. (8.6) I In particular D =(D CA B) so that Thus (8.) again holds. det(m)det( D) =det(m)det((d CA B) )= det(m) det(d CA =det(a). (8.7) B) Theorem 8.. Given β,α N with β<α,let{v,...,v β, v β+,...,v α } {w,...,w β, w β+,...,w α } be sets of linearly independent vectors in R α where w i w j = for all i {,...,β},j {β +,...,α}. (8.8) Let M be an α α matrix with det(m) = where v i = Mw i for all i {,...,β} (8.9) Then w j = M T v j for all j {β +,...,α}. (8.) vol(v,...,v β )vol(w β+,...,w α )= det(m) vol(v β+,...,v α )vol(w,...,w β ) (8.) 9

20 Proof. Because of (8.8), without loss of generality we may choose an orthonormal basis wherein W (w,...,w β )= (w β+,...,w α )=. (8.) W4 α β α (α β) where W is a β β matrix W 4 is an (α β) (α β) matrix. The matrices M M may be written in bock form as A B Ã B M = M C D = C D (8.3) where A Ã are β β matrices. Then from the definitions, AW CT W (v,...,v β )= (v CW β+,...,v α )= 4 D T W 4. (8.4) Next note that M T A M = T A + C T C B T A + D T C A T B + C T D B T B + D T D M M T ÃÃ = T + B B T Ã C T T + B D CÃT + D B T C CT + D D T (8.5) (8.6) so that since det(m T M)=det(M) Lemma 8. gives that det(a T A + C T C)=det(M) det( C C T + D D T ). (8.7) Then AW vol(v,...,v β )vol(w β+,...,w α ) = vol CW vol W4 = det(w T AT AW + W T CT CW ) det(w 4 ) = det(a T A + C T C) det(w ) det(w 4 ) = det(m) det( C C T + D D T ) det(w ) det(w 4 ) = det(m) det(w4 T C C T W 4 + W4 T D D T W 4 ) det(w ) CT W = det(m) vol 4 W D T vol W 4 = det(m) vol(v β+,...,v α )vol(w,...,w β ) (8.8) which proves the result. Corollary 8.3. Let w i v i be the vectors given in Theorem 8.. Written in block form they become W W3 (w,...,w β )=, (w β+,...,w α )= (8.9) (v,...,v β )= W V V α β α β, (v β+,...,v α )= V3 V 4 W 4 α (α β), (8.) α (α β) where W V are both β β matrices while W 4 V 4 are both (α β) (α β). Then vol(v )vol(w 4 )= det(m) vol(v 4 )vol(w ). (8.) Proof. By orthogonality W T W 3 + W T W 4 =. (8.)

21 If vol(w )=vol(w 4 )=, (8.) holds trivially. Thus assume vol(w ) = so that W vol(w 4 )vol = det(w W 4 ) det(w T W + W T W ) = det(w 4 ) det(w ) det(i +(W ) T W T W W ) = det(w 4 ) det(w ) det(i + W W (W ) T W T ) = det(w ) det(w4 T W 4 + W4 T W W (W ) T W T W 4) = det(w ) det(w4 T W 4 + W3 T W W (W ) T W T W 3) = det(w ) det(w4 T W 4 + W3 T W 3) W3 = vol(w )vol W 4 (8.3) where the third line of (8.3) follows from Sylvester s determinant theorem the fifth line follows from W W3 (8.). Note that linear independence implies that vol = vol = so that (8.3) gives vol(w 4 ) =. Since similar steps give the same result as (8.3) in the case where vol(w 4 ) =,onegets that vol(w ) = iff vol(w 4 ) =.Byfirstnotingthat W v i v j = for all i {,...,β},j {β +,...,α} (8.4) is also guaranteed by the statement of Theorem 8., an equivalent result to (8.3) then holds for the v i vectors so that in summary W W3 V V3 vol(w 4 )vol = vol(w W )vol vol(v W 4 )vol = vol(v 4 V )vol. (8.5) V 4 W 4 Finally, from Theorem 8. vol V V W3 vol W 4 V3 = det(m) vol V 4 W vol W. (8.6) Multiplying both sides of (8.6) by vol(v )vol(w 4 ) using (8.5) gives V W3 vol(v )vol(w 4 )vol vol = det(m) vol(v 4 )vol(w )vol V which after cancelation gives (8.). References W 4 V V W3 vol W 4 (8.7) [] L.-T. Cheng, H. Liu, S. Osher. Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of schrödinger equations. Communication in Mathematical Science, (3):593 6, 3. [] B. Engquist, O. Runborg. Computational high frequency wave propagation. Acta Numerica, :8 66, 3. [3] P. Gérard, P. A. Markowich, N. J. Mauser, F. Poupaud. Homogenization limits wigner transforms. Communications on Pure Applied Mathematics, 5(4):33 379,997. [4] S. Jin, X. Li. Multi-phase computations of the semiclassical limit of the schrödinger equation related problems: Whitham vs wigner. Physica D, 8:46 85, 3. [5] S. Jin, H. Liu, S. Osher, R. Tsai. Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. Journal of Computational Physics, :497 58,5. [6] S. Jin, H. Liu, S. Osher, R. Tsai. Computing multivalued physical observables for the semiclassical limit of the schrödinger equation. J. Comput. Phys., 5: 4,5. [7] S. Jin, P. Markowich, C. Sparber. Mathematical computational methods for semiclassical schrödinger equations. Acta Numerica, : 89,.

22 [8] S. Jin, S. Osher. A level set method for the computation of multivalued solutions to quasi-linear hyperbolic pdes hamilton-jacobi equations. Communication in Mathematical Science, (3):575Ð59, 3. [9] S. Jin, D. Wei. A particle method for the semiclassical limit of the schrödinger equation the vlasov-poisson equation. To appear in SIAM Journal of Numerical Analysis. [] C. Min. Local level set method in high dimension codimension. Journal of Computational Physics, ():368 38, 4. [] S. Osher, L.-T. Cheng, M. Kang, H. Shim, Y.-H. Tsai. Geometric optics in a phase-space-based level set eulerian framework. Journal of Computational Physics, 79():6 648,. [] L. Ryzhik, G. Papanicolaou, J. Keller. Transport equations for elastic other waves in rom media. Wave Motion, 4(4):37 37,996. [3] C. Sparber, P. A. Markowich, N. J. Mauser. Wigner functions versus wkb-methods in multivalued geometrical optics. Asymptotic Analysis, 33():53 87, 3. [4] G. Sundaram, Q. Niu. Wave-packet dynamics in slowly perpurbed crystals: Gradient corrections berry-phase effects. Phys Rev. B, 59: ,999. [5] J. Tully. Molecular dynamics with electronic transitions. J. Chem. Phys., 93:6 7,99. [6] E. Wigner. On the quantum correction for the thermodynamic equilibrium. Physical Review, 4: , 93.