Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds

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1 Journal of Computational Physics (6) Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds Shi Jin a,b, *, in Wen b, a Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 8 Lincoln Drive, Madison, WI , USA b Department of Mathematical Sciences, Tsinghua University, Beijing 8, PR China Received 8 June 5; received in revised form October 5; accepted October 5 Available online 6 December 5 Abstract In this paper, we construct two classes of Hamiltonian-preserving numerical schemes for a Liouville equation with discontinuous local wave speed. This equation arises in the phase space description of geometrical optics, and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics. We etend our previous work in [S. Jin,. Wen, Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials, Commun. Math. Sci. 3 (5) 85 35] for the semiclassical limit of the Schrödinger equation into this system. The designing principle of the Hamiltonian preservation by building in the particle behavior at the interface into the numerical flu is used here, and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition. When a plane wave hits a flat interface, the Hamiltonian preservation is shown to be equivalent to SnellÕs law of refraction in the case when the ratio of wave length over the width of the interface goes to zero, when both length scales go to zero. Positivity, and stabilities in both l and l norms, are established for both schemes. The approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular (discontinuous and measure-valued) coefficients. Benchmark numerical eamples are given, with analytic solution constructed, to study the numerical accuracy of these schemes. Ó 5 Elsevier Inc. All rights reserved. Keywords: Geometrical optics; Liouville equation; Level set method; Hamiltonian-preserving schemes; SnellÕs law of refraction. Introduction In this paper, we construct and study numerical schemes for the Liouville equation in d-dimension: f t þr v H r f r H r v f ¼ ; t > ; ; v R d ; ð:þ * Corresponding author. Tel.: ; fa: addresses: jin@math.wisc.edu (S. Jin), wenin@amss.ac.cn (. Wen). Present address: Institute of Computational Mathematics and Scientific and Engineering Computing, Chinese Academy of Science, Beijing 8, China. -999/$ - see front matter Ó 5 Elsevier Inc. All rights reserved. doi:.6/j.jcp.5..

2 where the Hamiltonian H is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ð; v; tþ ¼cðÞjvj ¼cðÞ v þ v þþv d ð:þ with c() > being the local wave speed. f(t,, v) is the density distribution of particles depending on position, time t and the slowness vector v. In this paper, we are interested in the case when c() contains discontinuities corresponding to different indices of refraction at different media. This discontinuity will generate an interface at the point of discontinuity of c(), and as a consequence waves crossing this interface will undergo transmissions and reflections. The incident and transmitted waves obey SnellÕs law of refraction. The bicharacteristics of the Liouville equation (.) satisfy the Hamiltonian system: d dt ¼ cðþ v jvj ; S. Jin,. Wen / Journal of Computational Physics (6) dv dt ¼ jvjr c. In classical mechanics the Hamiltonian (.) of a particle remains a constant along particle trajectory, even across an interface. This Liouville equation arises in the phase space description of geometrical optics. It is the high frequency limit of the wave equation u tt cðþ Du ¼ ; t > ; R d. ð:þ Recently, several phase space based level set methods are based on this equation, see [3,6,,3]. High frequency limit of wave equations with transmissions and reflections at the interfaces was studied in [,3,39]. A Liouville equation based level set method for the wave front, but with only reflection, was introduced in [7]. It was also suggested to smooth out the local wave speed in [3]. The Liouville equation (.) is a linear wave equation, with the characteristic speed determined by bicharacteristic (.3). Ifc() is smooth, then the standard numerical methods (for eample, the upwind scheme and its higher order etensions) for linear wave equations give satisfactory results. However, if c() is discontinuous, then the conventional numerical schemes suffer from two problems. Firstly, the characteristic speed c of the Liouville equation is infinity at the discontinuous point of wave speed. When numerically approimating c crossing the interface (for eample by smoothing out c() [3]), the numerical derivative of c is of O(/), with the mesh size in the physical space. Thus an eplicit scheme needs time step Dt =O(Dv) with Dv the mesh size in particle slowness space. This is very epensive. Moreover, a conventional numerical scheme in general does not preserve a constant Hamiltonian across the interface, usually leading to poor or even incorrect numerical resolutions by ignoring the discontinuities of c(). Theoretically, there is a uniqueness issue for weak solutions to these linear hyperbolic equations with singular wave speeds [6,9,9,3,35]. It is not clear which weak solution a standard numerical discretization that ignores the discontinuity of c() will select. We also remark that Hamiltonian or sympletic schemes have been introduced for Hamiltonian ODEs and PDEs in order to preserve the Hamiltonian or sympletic structures, see for eample [5,8]. To our knowledge, no such schemes have been constructed for Hamiltonian systems with discontinuous Hamiltonians. In this paper, we construct a class of numerical schemes that are suitable for the Liouville equation (.) with a discontinuous local wave speed c(). An important feature of our schemes is that they are consistent with the constant Hamiltonian across the interface. This gives a selection criterion for a unique solution to the governing equation. As done in [] for the Liouville equation for the semiclassical limit of the linear Schrödinger equation, we call such schemes Hamiltonian-preserving schemes. A key idea of these schemes is to build the behavior of a particle at the interface either cross over with a changed velocity or be reflected with a negative velocity into the numerical flu. This idea was formerly used by Perthame and Semioni in their work [33] to construct a well-balanced kinetic scheme for the shallow water equations with a (discontinuous) bottom topography which can capture the steady state solutions corresponding to a constant energy of the shallow water equations when the water velocity is zero. As a consequence, these new schemes allow a typical hyperbolic stability condition Dt = O(, Dv). We etend both classes of the Hamiltonian-preserving schemes developed in [] here. One (called Scheme I) is based on a finite difference approach, and involves interpolation in the slowness space. The second (called Scheme II) uses a finite volume approach, and numerical quadrature rule in the slowness space is needed. These new schemes allow a typical hyperbolic stability condition Dt = O(, Dv). We will also establish the ð:3þ

3 67 S. Jin,. Wen / Journal of Computational Physics (6) positivity and stability theory for both schemes. It is proved that Scheme I is positive, l contracting, and l stable under a hyperbolic stability condition, while Scheme II is positive, l stable and l contracting under the same stability condition. By building in the wave behavior at the interface, we have also provided a selection principle to pick up a unique solution to this linear hyperbolic equation with singular coefficients. For a plane wave hitting a flat interface, we show that it selects the solution that describes the interface condition in geometrical optics governed by SnellÕs law of refraction when the wave length is much shorter than the width of the interface while both lengths go to zero. In geometrical optics applications, one has to solve the Liouville equation like (.) with measure-valued initial data f ð; v; Þ ¼q ðþdðv u ðþþ; ð:5þ see for eample [,,38]. The solution at later time remains measure-valued (with finite or even infinite number of concentrations corresponding to multivalued solutions in the physical space). Computation of multivalued solutions in geometrical optics and more generally in nonlinear PDEs has been a very active area of recent research, see [ 5,8,,,3,,6 8,,3,3,37,]. Numerical methods for the Liouville equation with measure-valued initial data (.5) could easily suffer from poor resolution due to the numerical approimation of the initial data as well as numerical dissipation. The level set method proposed in [,] decomposes f into / and w i (i =,...,d) where / and w i solve the same Liouville equation with initial data /ð; v; Þ ¼q ðþ; w i ð; v; Þ ¼v i u i ðþ; ð:6þ respectively. (We remark here that the common zeroes of w i give the multivalued slowness, see [8,3,,].) This allows the numerical computations for bounded rather than measure-valued solution of the Liouville equation, which greatly enhances the numerical resolution (see []). The moments can be recovered through Z Z qð; tþ ¼ f ð; v; tþ dv ¼ /ð; v; tþp d i¼ dðw iþ dv; ð:7þ uð; tþ ¼ Z Z f ð; v; tþv dv ¼ /ð; v; tþvp d i¼ qð; tþ dðw iþ dv=qð; tþ. ð:8þ Thus one only involves numerically the delta-function at the output time! Numerical computations of multivalued solution for smooth c() using this technique were given in []. In this paper, we will also give numerical eamples using this technique with a discontinuous c(). The more general case with partial transmissions and reflections will be studied in a forthcoming paper [6]. This paper is organized as follows. In Section, we first show that the usual finite difference scheme to solve the Liouville equation with a discontinuous wave speed suffers from the severe stability constraint. We then present the design principle of our Hamiltonian-preserving scheme by describing the behavior of waves at an interface. We present Scheme I in one space dimension in Section 3 and study its positivity and stability in both l and l norms. Scheme II in one space dimension is presented and studied in Section. We etend these schemes to two space dimension in Section 5 in the simple case of interface aligning with the grids and a plane wave. Numerical eamples, with analytical solutions constructed, are given in Section 6 to verify the accuracy of the schemes. For comparison, we also present numerical solutions by methods ignoring or smearing the discontinuity of c. We make some concluding remarks in Section 7.. The design principle of the Hamiltonian-preserving scheme.. Deficiency of the usual finite difference schemes Consider the numerical solution of the Liouville equation in one physical space dimension f t þ cðþsignðnþf c jnjf n ¼ with a discontinuous wave speed c(). ð:þ

4 We employ a uniform mesh with grid points at iþ, i =,...,N, in the -direction and n jþ, j =,...,M in the n-direction. The cells are centered at ( i,n j ), i =,...,N, j =,...,M with i ¼ ð iþ þ i Þ and n j ¼ ðn jþ þ n j Þ. The mesh size is denoted by ¼ iþ i, Dn ¼ n jþ n j. We also assume a uniform time step Dt and the discrete time is given by = t < t << t L = T. We introduce mesh ratios k t ¼ Dt, k t n ¼ Dt, assumed to be fied. We define the cell average of f as Dn f ij ¼ Z iþ Z njþ f ð; n; tþ dn d. Dn i S. Jin,. Wen / Journal of Computational Physics (6) n j A typical semi-discrete finite difference method for this equation is o t f ij þ c i signðn j Þ f iþ ;j f i ;j Dc i jn j j f i;jþ f i;j ¼ ; ð:þ Dn where the numerical flues f iþ ;j, f i;jþ are defined by the upwind scheme, and Dc i is some numerical approimation of c at = i. Such a discretization suffers from two problems: If an eplicit time discretization is used, the CFL condition for this scheme requires the time step to satisfy 3 jdc c i j ma jn j j i Dt ma i þ j 5 6. ð:3þ Dn Since the wave speed c() is discontinuous at some points, ma i Dc i = O(/), so the CFL condition (.3) requires Dt = O( Dn), which is too epensive for a hyperbolic problem. The above discretization in general does not preserve a constant Hamiltonian H = c n across the discontinuities of c, thus may not produce numerical solutions consistent with, for eample, SnellÕs law of refraction... Behavior of waves at the interface When a wave moves with its density distribution governed by the Liouville equation (.), the Hamiltonian H = c v should be preserved across the interface: c þ jv þ j¼c jv j; ð:þ where the superscripts ± indicate the right and left limits of the quantity at the interface. We will discuss the wave behavior in one and two space dimensions, respectively. One space dimension. The D case is simple. Consider the case when, at an interface, the characteristic on the left of the interface is given by n >. Then the particle definitely crosses the interface and n þ ¼ c n. c þ Two space dimension, when an incident plane wave hits an interface that aligns with the grid. In the D case, =(,y), v =(n,g). Consider the case that the interface is a line parallel to the y-ais. The incident wave has slowness (n,g ) to the left side of the interface, with n >. Since the interface is vertical, (.3) implies that g will not change across the interface, while n has three possibilities: () c > c +. In this case, the local wave speed decreases, so the wave will cross the interface and increase its n value in order to maintain a constant Hamiltonian. (.) implies sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ c ¼ ðn Þ þ c ðg c þ c Þ. þ () c < c + and ð c Þ ðn Þ þ½ð c Þ Šðg Þ >. In this case the wave can also cross the interface with a c þ c þ reduced n value. (.) still gives

5 676 S. Jin,. Wen / Journal of Computational Physics (6) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ c ¼ ðn Þ þ c ðg c þ c Þ. þ (3) c < c + and ð c Þ ðn Þ þ½ð c Þ Šðg Þ <. In this case, there is no possibility for the wave to cross the c þ c þ interface, so the wave will be reflected with slowness ( n,g ). If n <, similar behavior can also be analyzed using the constant Hamiltonian condition (.). Remark.. In general, one cannot define a unique weak solution to a linear hyperbolic equation with singular (discontinuous or measure-valued) coefficients. By using the wave behavior described above, we give a selection criterion for a unique solution. This solution is the one when the wave length of the incident wave is much smaller than the width of the interface, both of which go to zero. It is equivalent to SnellÕs law of refraction: sin h i c ¼ sin h t c ; ð:5þ þ where h i and h t stand for angles of incident and transmitted waves, respectively. This is to say: g g þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. ð:6þ c ðn Þ þðg Þ c þ ðn þ Þ þðg þ Þ If c = c(), then (.3) implies that g þ ¼ g. ð:7þ Clearly (.6) and (.7) imply (.). Of course this is not the only physically relevant way to choose a solution. In particular, this principle ecludes the more general case that allows partial reflections and transmissions. It applies to the case when the wave length of the incident wave is much shorter than the width of the interface as both lengths go to zero. The more general case of partial transmissions and reflections is a topic of a forthcoming paper [6]. The main ingredient in the well-balanced kinetic scheme by Perthame and Semioni [33] for the shallow water equations with topography was to build in the Hamiltonian-preserving mechanism into the numerical flu in order to preserve the steady state solution of the shallow water equations when the water velocity is zero. This is achieved using the fact that the density distribution f remains unchanged along the characteristic, thus f ðt; þ ; n þ Þ¼fðt; ; n Þ ð:8þ at a discontinuous point of c(), where for eample, n + is defined through the constant Hamiltonian condition (.). In this paper, we use this mechanism for the numerical approimation to the Liouville equation (.) with a discontinuous wave speed. This approimation, by its design, maintains a constant Hamiltonian modulus the numerical approimation error across the interface. In [] we introduced two Hamiltonian-preserving schemes for the Liouville equation arising from the semiclassical limit of the linear Schrödinger equation by incorporating this particle behavior into the numerical flu. 3. Scheme I: a finite difference approach 3.. A Hamiltonian-preserving numerical flu We now describe our first finite difference scheme (called Scheme I) for the Liouville equation with a discontinuous local wave speed. Assume that the discontinuous points of wave speed c are located at the grid points. Let the left and right limits of c() at point i+/ be c þ and c, respectively. Note that if c is continuous at iþ iþ i+/, then c þ ¼ c iþ iþ. We approimate c by a piecewise linear function

6 S. Jin,. Wen / Journal of Computational Physics (6) cðþ c þ i = þ c iþ= cþ i = ð i = Þ. We also define the averaged wave speed as c i ¼ [33]. The semi-discrete scheme (with continuous time) reads ðf ij Þ t þ c i signðn j Þ f þ f iþ ;j i ;j c iþ c þ i Dn c þ i þc iþ. We will adopt the flu splitting technique used in jn j j f i;jþ f i;j ¼ ; where the numerical flues f i;jþ is defined using the upwind discretization. Since the characteristics of the Liouville equation maybe different on the two sides of the interface, the corresponding numerical flues should also be different. The essential part of our algorithm is to define the split numerical flues f iþ ;j, f þ at each cell i ;j interface. We will use (.8) to define these flues. Assume c is discontinuous at i+/. Consider the case n j >. Using upwind scheme, f ¼ f iþ ij. However, ;j f þ iþ=;j ¼ f ðþ iþ= ; nþ j Þ¼fð iþ= ; n j Þ while n j is obtained from n þ j ¼ n j from (.). Since n j may not be a grid point, we have to define it approimately. The first approach is to locate the two cell centers that bound n j, then use a linear interpolation to evaluate the needed numerical flu at n. The case of n j < is treated similarly. The detailed algorithm to generate the numerical flu is given below. Algorithm I if n j > f ¼ f iþ ij, ;j n c þ iþ ¼ c j iþ n if n j < f þ if n k 6 n < n kþ for some k then f þ ¼ n kþ n iþ ;j Dn ¼ f iþ iþ;j, ;j n þ c iþ ¼ c þ j iþ n f i;k þ n n k f Dn i;kþ, if n k 6 n þ < n kþ for some k then f f Dn iþ;k þ nþ n k f Dn iþ;kþ. iþ ;j ¼ n kþ n þ The above algorithm for evaluating numerical flues is of first order. One can obtain a second order flu by incorporating the slope limiter, such as van Leer or minmod slope limiter [9], into the above algorithm. This can be achieved by replacing f ik with f ik þ s ik, and replacing f i+,k with f iþ;k s iþ;k in the above algorithm for all the possible inde k, where s ik is the slope limiter in the -direction. After the spatial discretization is specified, one can use any time discretization for the time derivative. 3.. Positivity and l contraction Since the eact solution of the Liouville equation is positive when the initial profile is, it is important that the numerical solution inherits this property. We only consider the scheme using the first order numerical flu, and the forward Euler method in time. Without loss of generality, we consider the case n j > and c < c þ for all i (the other cases can be treated iþ i similarly with the same conclusion). The scheme reads f nþ ij f n ij Dt f ij ðd f i ;k þ d f i ;kþ Þ þ c i c iþ c i þ n j f ij f i;j Dn ¼ ; ð3:þ

7 678 S. Jin,. Wen / Journal of Computational Physics (6) where d, d are non-negative and d + d =. We omit the superscript n of f. The above scheme can be rewritten as c c þ c c þ f nþ ij ¼ c i k t iþ jn j jk t n Af ij þ c i k t ðd iþ f i ;k þ d f i ;kþ Þþ i jn j jk t n f i;j. ð3:þ Now we investigate the positivity of scheme (3.). This is to prove that if fij n P for all (i,j), then this is also true for f n+. Clearly, one just needs to show that all coefficients for f n are non-negative. A sufficient condition for this is clearly c c þ c i k t iþ i jn j jk t n P ; or Dt ma i;j 6 c i þ c iþ c þ i jn j j 3 Dn ð3:3þ This CFL condition is similar to the CFL condition (.3) of the usual finite difference scheme ecept that jc iþ c þ i j the quantity now represents the wave speed gradient at its smooth point, which has a finite upper bound. Thus our scheme allows a time step Dt = O(, Dn), a significant improvement over a standard discretization. According to the study in [3], our second order scheme, which incorporates slope limiter into the first order scheme, is positive under the half CFL condition, namely, the constant on the right-hand side of (3.3) is /. The above conclusion is analyzed based on forward Euler time discretization. One can draw the same conclusion for the second order TVD Runge Kutta time discretization []. The l -contracting property of this scheme follows easily, because the coefficients in (3.) are positive and the sum of them is The l -stability of Scheme I In this section, we prove the l -stability of Scheme I (with the first order numerical flu and the forward Euler method in time). The proof is similar to that in [5] with difference in details due to different particle behaviors at the interface. For simplicity, we consider the case when the wave speed has only one discontinuity at grid point mþ with c > c þ and c mþ mþ, () > at smooth points. The other cases, namely, when c () 6, or when the wave speed has several discontinuous points with increased or decreased jumps, can be discussed similarly. Denote k c c þ =c <. mþ mþ We consider the general case that n <,n M >. For this case, the study in [] suggests that the computational domain should eclude a set O n ¼fð; nþ R jn ¼ g which causes singularity in the velocity field. For eample, we can eclude the following inde set D o ¼ ði; jþ jn j j < Dn ; from the computational domain. Since c() has a discontinuity, we also define an inde set D l ¼fði; jþj i 6 m ; n j < k c n g. Due to the slowness change across the wave speed jump at mþ, D l represents the area where waves come from outside of the domain [, N ] [n,n M ]. In order to implement our scheme conveniently, this inde set is also ecluded from the computational domain. Thus the computational domain is chosen as

8 E d ¼fði; jþji ¼ ;...; N; j ¼ ;...; MgnfD o [ D l g. ð3:þ A sketch of E d and D l is shown in Fig. in Section.. As a result of ecluding the inde set D o from the computational domain, the computational domain is split into two independent parts E d ¼fði; jþ E d jn j > g[fði; jþ E d jn j < g E þ d [ E d. The l -stability study of Scheme I can be carried out in these two domains, respectively. In the following we prove the l -stability of Scheme I in the domain E d. The study in the domain Eþ d can be made similarly. We define the l -norm of a numerical solution u ij in the set E d to be jf j ¼ N jf ij j d ði;jþe d with N d being the number of elements in E d. Given the initial data fij ; ði; jþ E d. Denote the numerical solution at time T to be fij L, ði; jþ E d. To prove the l -stability, we need to show that f L 6 C f. Due to the linearity of the scheme, the equation for the error between the analytical and the numerical solution is the same as (3.), so in this section, f ij will denote the error. We assume there is no error at the boundary, thus fij n ¼ at the boundary. If the l -norm of the error introduced at each time step in incoming boundary cells is ensured to be o() part of f n, our following analysis still applies. Now denote A i ¼ c c þ iþ. ð3:5þ i Assume an upper bound for the wave speed slope is A u, A i < A u "i. These notations will be used below as well as in the stability proof of Scheme II. One also has jc i c i j < A u 8i. Assume the wave speed has a lower bound C m, c i > C m >"i. When n j <, Scheme I is given by () if i = m, f nþ ij ¼ð A i jn j jk t n c ik t Þf ij þ A i jn j jk t n f i;jþ þ c i k t f iþ;j; ð3:6þ () f nþ mj ¼ð A m jn j jk t n c mk t Þf mj þ A m jn j jk t n f m;jþ þ c m k t ðd jkf mþ;k þ d j;kþ f mþ;kþ Þ; ð3:7þ where 6 d jk 6 andd jk + d j,k + =.In(3.7) k is determined by n k 6 n j k c < n kþ. When summing up all absolute values of fij nþ in (3.6) and (3.7), one typically gets the following inequality: jf nþ j 6 N d ði;jþe d a ij jf n ij j; where the coefficients a ij are positive. One can check that, under the CFL condition (3.3), a ij 6 + A u Dt ecept for possibly ði; jþ D mþ defined as D mþ ¼fði; jþ E d ji ¼ m þ g. We net derive the bound for M defined as M ¼ ma a mþ;j. ðmþ;jþd mþ S. Jin,. Wen / Journal of Computational Physics (6) ð3:8þ Define the set S mþ j ¼ j jn j < ; n j n k j < Dn c for ðm þ ; jþ D mþ.

9 68 S. Jin,. Wen / Journal of Computational Physics (6) Let the number of elements in S mþ j be N mþ j. One can check that N mþ j 6 k c þ because every two elements j, j Sm j satisfy j n j k c nj k c j P Dn k c. On the other hand, one can easily check from (3.6) and (3.7), for ðm þ ; jþ D mþ, a mþ;j < c mþ k t þ c mk t ð k c þ Þ ¼ þðc m þ c mþ Þk t þ OðÞ; so for sufficiently small, M can be bounded by M < þ ðc m þ c mþ Þk t. Denote M ¼ ðc m þ c mþ Þk t. From (3.8), jf nþ j < ð þ A u DtÞjf n j þ M jf n mþ;j j. ð3:9þ N d ðmþ;jþd mþ We now establish the following theorem: Theorem 3.. Under the CFL condition (3.3), the scheme (3.6), (3.7) is l -stable jf L j < Cjf j. Proof. From (3.9), 8 < jf L j < ð þ A u DtÞ L jf j þ M : It remains to estimate 8 S ¼ L < : n¼ ðmþ;jþd mþ N d L n¼ Define the set S r ¼fði; jþj i > mþ ; ðm þ ; jþ D mþ g ðmþ;jþd mþ 39 = jf n mþ;j j 5 ;. ð3:þ 9 = jf n mþ;j j ;. ð3:þ "(i,j) S r, due to the zero boundary condition and the upwind nature of the scheme, one has f n ij ¼ b ijn f ; ði; jþ S r ð3:þ ðp;qþs r;ppi with b ijn P. Notice D mþ S r, S L ðp;qþs r n¼ where we have defined F ðp; qþ ¼ L n¼ ðmþ;jþd mþ ðmþ;jþd mþ b mþ;jn Ajf j The net step is to estimate these coefficients. Define b ij ¼ n¼ then (3.) gives F ðp; qþ ¼ ðp;qþs r F ðp; qþjf j; ð3:3þ b mþ;jn ; ðp; qþ S r. ð3:þ b ijn ; ði; jþ; ðp; qþ S r; p P i; ðmþ;jþd mþ L n¼ b mþ;jn 6 ðmþ;jþd mþ b mþ;j ; ðp; qþ S r.

10 We first evaluate P ðmþ;jþd b ij mþ Under the CFL condition (3.3), c ij, cij, ci 3 (3.6), it can be directly computed ðmþ;jþd mþ b pj < n¼ when i = p. Denote c ij ¼ A ijn j jk t n c ik t, cij ¼ A ijn j jk t n, ci 3 ¼ c ik t. are positive. One also has cij þ cij 6 C mk t. From scheme ð C m k t Þn ¼ C m k t. ð3:5þ We now study the relation between P ðmþ;jþd b ij and P ðmþ;jþd b iþ;j when i < p. From scheme (3.6), mþ mþ b ij;nþ; ¼ c ij bijn þ c ij bi;jþ;n þ c i 3 biþ;jn. ð3:6þ Summing up j in (3.6) gives ¼ ðmþ;jþd mþ b ij;nþ; ðmþ;jþd mþ ðc ij þ ci;j Þb ijn þ c i 3 < ð c i 3 þ A uk t n DnÞ then a sum for n from to in (3.7) gives ðc i 3 A uk t n DnÞ b ij ðmþ;jþd mþ S. Jin,. Wen / Journal of Computational Physics (6) ðmþ;jþd mþ < ci 3 ðmþ;jþd mþ b iþ;j ; b ijn ðmþ;jþd mþ þ c i 3 b iþ;jn ðmþ;jþd mþ b iþ;jn ; ð3:7þ so ðmþ;jþd mþ b ij < c i 3 c i 3 A uk t n Dn ðmþ;jþd mþ b iþ;j < þ A u þ oðþ C m ðmþ;jþd mþ b iþ;j. Thus for sufficiently small, one has b ij < þ A u C m ðmþ;jþd mþ ðmþ;jþd mþ b iþ;j ; i < p. ð3:8þ We now can evaluate F(p,q) for (p,q) S r. From the definition of S r, when (p,q) S r, one has p P m +. F ðp; qþ 6 ðmþ;jþd mþ < ep A u ð N Þ C m b mþj < þ A u C m ðmþ;jþd mþ ðmþ;jþd mþ b mþj < < þ A p m u C m b pj < ep A u C m ð N Þ C m k t C T. ðmþ;jþd mþ b pj ð3:9þ Therefore, from (3.3) one gets S 6 F ðp; qþjf j < C T jf j 6 C T ðp;qþs r ðp;qþs r ðp;qþe d jf j¼c T N d jf j. ð3:þ Combing (3.) and (3.), jf L j < ð þ A u DtÞ L fjf j þ C T M jf j g < epða u T Þ½ þ C T M Šjf j Cjf j ; where C ep(a u T )[ + C TM ]. Thus Theorem 3. is proved. h One can prove the similar conclusion for inde set E þ d. Remark 3.. Theorem 3. holds for any l initial data. The corresponding result in the case of semiclassical limit of Schrödinger equation [5] ecludes the case of measure-valued initial data.

11 68 S. Jin,. Wen / Journal of Computational Physics (6) Scheme II: a finite volume approach.. A Hamiltonian-preserving numerical flu In this section, we derive another flu based on the finite volume approach which results in an l -contracting scheme. We call this scheme as Scheme II. Assuming the mesh grid is such that n j and n jþ do not have opposite sign. By integrating the Liouville equation (.) over the cell [ i /, i+/ ] [n j /,n j+/ ], one gets the following semi-discrete flu splitting scheme: c i þ ðf ij Þ t þ signðn jþ c ðc f iþ iþ ;j cþ f þ iþ i i ;jþ Dn jn jþ jf i;jþ jn j jf i;j ¼. In the finite volume approach, the numerical flues are defined as integrals of solution along the cell interface c iþ which depend on the sign of n j and c þ i c iþ. To illustrate the basic idea, we assume n j >, c þ i <. In this case f ¼ Z njþ f ; n; t dn; iþ ;j iþ Dn n j Z iþ f i;jþ ¼ c c þ c f ; n jþ ; t d. iþ i i Note that f(,n,t) may be discontinuous at the grid point ¼ iþ and n ¼ n jþ. By using condition (.8): f þ ¼ Z njþ f þ ; n; t dn ¼ Z njþ f ; n; t dn; ð:þ iþ ;j Dn iþ iþ n Dn j n j where f is defined as c þ! f n; t ¼ f iþ; iþ iþ; n; t. c iþ Using change of variable on (.) leads to f þ ¼ Z njþ c þ! n c Z f iþ ; iþ iþ ;j iþ Dn ; t dn ¼ c þ iþ n jþ =c n c c þ iþ f n; t dn. ð:3þ Dn j iþ iþ c þ iþ n j =c iþ; iþ The integral in (.3) will be approimated by a quadrature rule. Since the end point c þ n iþ jþ =c in (.3) may iþ not be a grid point in the n-direction, special care needs to be taken at both ends of the interval h.. i c þ n iþ j c ; c þ n iþ iþ jþ c. ð:þ iþ We propose the following evaluation of the split flues f in (.). iþ ;j Algorithm II if n j > f iþ ;j ¼ f ij, n þ c ¼ iþ c iþ þ n j ; n c ¼ iþ n c jþ iþ if n k 6 n < n 6 n kþ for some k f þ ¼ f iþ ;j ik else n k 6 n < n kþ < < n kþs < n 6 n kþsþ for some k, s n kþ n f Dn ik þ f i;kþ þþf i;kþs þ n n kþs f Dn i;kþs f þ iþ end c iþ ¼ ;j c þ iþ ð:þ

12 S. Jin,. Wen / Journal of Computational Physics (6) if n j < f þ iþ ;j ¼ f iþ;j, n þ c ¼ iþ n c þ j iþ ; n þ c ¼ iþ n c þ jþ iþ if n k 6 n þ < nþ 6 n kþ for some k f ¼ f iþ ;j iþ;k else n k 6 n þ < n kþ < < n kþs < nþ 6 n kþsþ for some k, s n kþ n þ f Dn iþ;k þ f iþ;kþ þþf iþ;kþs þ n þ n kþs f Dn iþ;kþs f iþ end end cþ iþ ¼ ;j c iþ Remark.. The above algorithm uses a first order quadrature rule at the ends of the interval (.), thus it is of first order even if the slope limiters in -direction are incorporated into the algorithm. One can also use a second order quadrature rule at the ends of intervals (.). But the resulting second order scheme is no longer l -contracting, which is the property of Scheme II, as will be proved in the net subsection. One can still prove that this scheme is l -stable, similar to the property of Scheme I. Compared with Scheme I, this scheme is second order accurate and l -stable, but more comple to implement. We will not present the detail of this numerical scheme in this paper... The l -contraction, l -stabilities and positivity of Scheme II In this subsection, we study the l and l stability of Scheme II. Its positivity is obvious under the CFL condition (3.3). Theorem.. If the forward Euler time discretization is used, then the flu given by Algorithm I yields the scheme (.) which is l -contracting and l -stable. Proof. In this proof we only discuss the case when the wave speed has one discontinuity at grid point mþ with c > c þ and c mþ mþ, () > at smooth points. The other situations can be discussed similarly. We consider the general case that n <,n M >. We assume the mesh is such that is a grid point in n- direction. In this case, the inde set D o ¼ ði; jþ jn j j < Dn that needs to be ecluded from the computational domain is null. As such, the cell interface {(,n) n = } is actually the computational domain boundary where appropriate boundary conditions should be imposed []. As discussed in Section 3.3, the computational domain is chosen as E d ¼fði; jþji ¼ ;...; N; j ¼ ;...; MgnD l ; where ( c þ ) D l ¼ ði; jþ mþ i 6 m ; n j < c n. mþ Define some subsets of E d D þ m ¼ ðm; jþjn j P Dn ; D þ mþ ¼ ðm þ ; jþ n j P Dn ;

13 68 S. Jin,. Wen / Journal of Computational Physics (6) ( c þ ) D m ¼ ðm; jþ mþ n 6 n c j mþ 6 Dn ; D mþ ¼ ðm þ ; jþ n j 6 Dn. These domains are shown in Fig.. Recall the definition of A i in (3.5). Our scheme (.) with Algorithm II can be made precise as () if n j >,i 6¼ m +, f nþ ij ¼ A i n j k t c n iþk t f ij þ A i n jþ k t n f i;jþ þ c þ i k t f i ;j; ð:5þ () if n j <,i 6¼ m, f nþ ij ¼ A i jn j jk t n cþ i k t (3) if n j >, f nþ mþ;j ¼ A mþn j k t n c mþ 3 k t () if n j <, f nþ mj ¼ A m jn j jk t n cþ m k t where we omit the superscript n on the right-hand side. f ij þ A i jn jþ jk t n f i;jþ þ c iþ k t f iþ;j; ð:6þ f mþ;j þ A mþ n jþ k t f mþ;jþ þ c þ n mþk t f þ mþ ;j; ð:7þ f mj þ A m jn jþ jk t n f m;jþ þ c mþk t f mþ ;j; ð:8þ By summing up (.5) (.8) for (i,j) E d, one typically gets the following epression: jf nþ ij j 6 a ij jf ij jþ c þ k t mþ jf þ mþ ;jjþ c mþ k t jf mþ ;jj I þ I þ I 3. ð:9þ ði;jþe d ði;jþe d ðmþ;jþd þ mþ ðm;jþd m ξ M D m + + D m+ E d D m D m+ D l ξ m+/ N Fig.. Sketch of the inde sets D þ m, Dþ mþ, D m, D mþ, D l.

14 As in the proof of stability of Scheme I, we assume that f satisfies the zero boundary condition. In this situation, the coefficients a ij in (.9) satisfy a ij 6 ; ði; jþ E d nfd þ m [ D mþg; ð:þ a ij 6 c mþk t ; ði; jþ Dþ m ; ð:þ a ij 6 c þ mþ k t ; ði; jþ D mþ. ð:þ We now study the relation between I and P ðm;jþd þ m jc mþ k t f mjj. Let p Mþ ¼ c þ n mþ Mþ c mþ ; S. Jin,. Wen / Journal of Computational Physics (6) and assume n k < p Mþ 6 n kþ 6 n Mþ. Assume n J ¼ for some J, since k t c mþ I 6 k jf mj jþ p Nþ n k jf mk j 6 j¼j Dn ðm;jþd þ m jf mj j; ðm;jþd þ m thus I 6 c mþk t f mj. ð:3þ Similarly, one gets I 3 6 c þ mþk t f mþ;j. ð:þ ðmþ;jþd mþ Combining (.9) (.) gives jf nþ ij j 6 jf n ij j. ð:5þ ði;jþe d ði;jþe d This is the l -contracting property of Scheme II. Net we prove the l -stability. Observing that the coefficients on the right-hand side of (.5) (.8) are positive, it remains to estimate the sum of these coefficients (SC). In (.5), the SC is SC ¼ þðc þ c i iþ Þk t þ A idnk t n < þ A udt. ð:6þ In (.6), the SC is SC ¼ þðc c þ iþ i Þk t A idnk t n < þ A udt. ð:7þ Now we derive the SC in (.8). Denote n ¼ c mþ c þ mþ c n j ; n ¼ mþ c þ n jþ. mþ ð:8þ The condition c þ < c gives n mþ mþ n > Dn. Therefore, it is impossible that n k 6 n < n 6 n kþ for any k. Assume n k 6 n < n kþ < < n kþs < n 6 n kþsþ with s P. In this case c þ ( f ¼ mþ n mþ kþ n f ;j c mþ;k þ f mþ;kþ þþf mþ;kþs þ n n ) kþs f mþ;kþs. ð:9þ Dn Dn mþ

15 686 S. Jin,. Wen / Journal of Computational Physics (6) Substituting (.9) into (.8) yields the evaluation " c þ SC 3 ¼ A m Dnk t n cþ k t m þ c mþ k t mþ n kþ n þ n kþ 3 n kþ þþ n n!# kþs Dn Dn Dn c mþ ¼ A m Dnk t n cþ k t m þ c mþ k t < þ A udt. ð:þ Now we consider case (.7). Denote n ¼ c þ mþ c mþ c þ n j ; n ¼ mþ n c jþ. mþ In this case, we know n n < Dn. So there are two cases n k 6 n < n 6 n kþ n kþ < n 6 n kþ 3 corresponding, respectively, to f þ ¼ f mþ ;j mk ð:þ or n k 6 n < ð:þ or f þ mþ ;j ¼ c mþ c þ mþ ( n kþ n f mk þ n n ) kþ f m;kþ Dn Dn. ð:3þ Similar to the deduction of (.), one can check, for both cases, that SC ¼ þ A mþ Dnk t n c k t mþ 3 þ cþ mþk t < þ A udt. ð:þ Combining (.6), (.7), (.) and (.), one gets jf nþ j < ð þ A u DtÞjf n j ; thus jf L j < ð þ A u DtÞ L jf j < e AuT jf j. This is the l -stability property of Scheme II. h ð:5þ 5. The schemes in two space dimension Consider the Liouville equation in two space dimension: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð; yþn cð; yþg f t þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f y c n þ g f n c y n þ g f g ¼. n þ g n þ g ð5:þ We employ a uniform mesh with grid points at iþ, y jþ, n kþ, g lþ in each direction. The cells are centered at ( i,y j,n k,g l ) with i ¼ ð iþ þ i Þ, y j ¼ ðy jþ þ y j Þ, n k ¼ ðn kþ þ n k Þ, g l ¼ ðg lþ þ g l Þ. The mesh size is denoted by ¼ iþ i, Dy ¼ y jþ y j, Dn ¼ n kþ n k, Dg ¼ g lþ g l. We define the cell average of f as Z iþ Z yjþ Z nkþ Z glþ f ijkl ¼ f ð; y; n; g; tþ dg dn dy d. Dy DnDg i y j n k g l Similar to the D case, we approimate c(,y) by a piecewise bilinear function, and for convenience, we always provide two interface values of c at each cell interface. When c is smooth at a cell interface, the two interface values are identical. We also define the averaged wave speed in a cell by averaging the four cell interface values c þ þ i c ij ¼ ;j c þ c þ iþ i;j c i;jþ.

16 S. Jin,. Wen / Journal of Computational Physics (6) The D Liouville equation (5.) can be semi-discretized as c ij n ðf ijkl Þ t þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi k f þ c ij g f þ q l n k þ iþ ;jkl g i ;jkl l Dy c iþ ;j cþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ;j n k Dn þ g l f ij;kþ ;l f ij;k ;l ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n k þ g l c i;jþ f þ f i;jþ ;kl i;j ;kl c þ i;j Dy Dg qffiffiffiffiffiffiffiffiffiffiffiffiffiffi n k þ g l f ijk;lþ f ijk;l ¼ ; where the numerical flues f ij;kþ ;l, f ijk;lþ are provided by the upwind approimation, and the split flues values f iþ ;jkl, f þ i ;jkl, fi;jþ ;kl, f þ should be obtained using similar but slightly different algorithm for the D case, i;j ;kl since the particle behavior at the interface is different in d from that in d, as described in Section. For eample, to evaluate f we can etend Algorithm I as iþ ;jkl Algorithm I in D if n k > f ¼ f iþ ijkl, ;jkl! C þ iþ if ;j n C iþ ;j k þ þ C iþ ;j 3 ;j! 5g C iþ l > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n C þ ;j! þ 3 iþ ¼ ;j n C u k þ iþ t 5g C iþ l C iþ ;j if n k 6 n < n k +for some k then f þ ¼ n kþ n iþ ;jkl Dn f ij;k ;l þ n n k f Dn ij;k þ;l else f þ iþ ;jkl ¼ f iþ;j;k ;l where n k = n k end if n k < f þ ¼ f iþ iþ;jkl, ;jkl! C iþ if ;j n C þ iþ ;j k þ C iþ ;j 3 ;j! 5g C þ l > iþ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ;j! 3 n þ C u iþ ¼ ;j n C k þ iþ t 5g C þ l iþ C þ iþ ;j if n k 6 n + < n k +for some k then f ¼ n kþ n þ iþ ;jkl Dn else f iþ ;jkl ¼ f i;j;k ;l where n k = n k end f iþ;j;k ;l þ nþ n k f Dn iþ;j;k þ;l

17 688 S. Jin,. Wen / Journal of Computational Physics (6) The flu f can be constructed similarly. i;jþ ;kl The d version of Scheme II can be constructed similarly. As introduced in Section., the essential difference between D and D flu definition is that in D case, the phenomenon that a wave is reflected at the interface does occur. While in D, a wave is always transmitted across an interface with a change of slowness. Since the gradient of the wave speed at its smooth points are bounded by an upper bound, this scheme, similar to the D scheme, is also subject to a hyperbolic CFL condition under which the scheme is positive, and Hamiltonian preserving. 6. Numerical eamples In this section, we present numerical eamples to demonstrate the validity of the proposed schemes and to study their accuracy. In the numerical computations the second order TVD Runge Kutta time discretization [] is used. In Eample 6., we compare the results of Schemes I and II. In other eamples, we present the numerical results using Scheme I. Eample 6.. An D problem with eact L -solution. Consider the D Liouville equation f t þ cðþsignðnþf c jnjf n ¼ with a discontinuous wave speed given by :6; < ; cðþ ¼ :5; >. The initial data are given by 8 >< ; < ; n > ; f ð; n; Þ ¼ ; > ; n < ; >: ; otherwise; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n < ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n < ; as shown in the upper part in Fig. which depicts the non-zero part of f(,n,). The eact solution at t = is given by 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; < < :5; < n < : ð: :6Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >< ; < < :5; ðþ:5þ < n < ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ð; n; Þ ¼ ; : < < ; < n < ð :6Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; :6 < < ; ð þ : : :5Þ < n < ; >: ; otherwise; ð6:þ ð6:þ ð6:3þ as shown in the lower left part in Fig.. The numerical solution computed with a cell on the domain [.5,.5] [.5,.5] using Scheme I is shown in the lower right part in Fig.. The time step is chosen as Dt ¼ Dn. It shows a good agreement with the eact solution. Table compares the l -error of the numerical solutions computed by Scheme I using 5 5, and cells, respectively. This comparison shows that the convergence rate of the numerical solution in

18 S. Jin,. Wen / Journal of Computational Physics (6) Fig.. Eample 6., solution in the phase space. Upper: non-zero part of the initial data; lower left: non-zero part of eact solution f(,n,); lower right: the part of numerical solution f(,n,) >.5 computed by the mesh. The horizontal ais is the position, the vertical ais is the slowness. Table l error of numerical solutions for f on different meshes Meshes l -norm is about.68. This agrees with the well established theory [7,], that the l -error by finite difference scheme for a discontinuous solution of a linear hyperbolic equation is at most halfth order. Eample 6.. Computing the physical observables of an D problem with measure-valued solution. Consider the D Liouville equation (6.), where the wave speed is 8 e >< ; 6 ; þ þ ; < < ; e cðþ ¼ þ :5 ; < < ; e >: :5; P ; e the initial data are given by f ð; n; Þ ¼dðn wðþþ ð6:þ with 8 :8; 6 :5; >< :8 :8 ð þ :5Þ ; :5 < 6 ; ð:5þ wðþ ¼ ð6:5þ :8 þ :8 ð :5Þ ; < < :5; ð:5þ >: :8; P :5. Fig. 3 plots w() in a dashed line.

19 69 S. Jin,. Wen / Journal of Computational Physics (6) In this eample we are interested in the approimation of the moments, such as the density Z qð; tþ ¼ f ð; n; tþ dn; and the averaged slowness R f ð; n; tþn dn uð; tþ ¼ R. f ð; n; tþ dn These quantities are computed by decomposition techniques described in Section. We first solve the level set function w and modified density function / which satisfy the Liouville equation (6.) with initial data n w() and, respectively. Then the desired physical observables q and u are computed from the numerical singular integrals (.7), (.8), which are computed by the technique described in []. The eact slowness and corresponding density at t = are given in Appendi A. Fig. 3 shows the eact multivalued slowness in solid line. Firstly, we give the results using the standard finite difference method (SFDM) by either ignoring or smoothing out the wave speed discontinuity. In the first case, denoted by SFDMI, one uses the same wave speed approimation as in the Hamiltonian-preserving schemes, ecept that a standard upwind flu is used to approimate the space derivative. Such a method has a hyperbolic CFL condition. In the second approach, denoted by SFDMS, one smoothes out the wave speed discontinuity throughout several grid points, then uses the standard upwind scheme for the space derivative. Fig. presents the numerical densities given by these two approaches using 8 mesh. For SFDMS, we choose the transition zone width to be 5, and connect the discontinuous wave speed by a linear function through the transition zone. We take Dt ¼ 5Dn. One can observe that SFDMI gives a wrong solution, while the SFDMS gives correct but smeared solution across the interface. Compared with the results by Schemes I and II, as shown in Fig. 5, SFDMS gives poorer numerical resolution with a much smaller CFL number. In addition, for SFDMS, one has to choose the width of the transition zone properly in order to guarantee the correct solution. A too narrow transition zone leads to a more severe CFL condition and also may produce incorrect results as in SFDMI, while an appropriately wider transition zone relaes the CFL condition and may allow a convergent solution, but leads to more smeared numerical solution across the interface. We then present the results computed by our Hamiltonian-preserving schemes. In the computation, the time step is chosen as Dt ¼ 3Dn. We first present numerical results without treating n = as the domain boundary and performing the delta function integrals (.7), (.8) without separating n > and n <. This is feasible for this eample because the zero points of the level set function w are away from n =.Fig. 5 shows Fig. 3. Eample 6., slowness. Dashed line: initial slowness w(); solid line: eact slowness at t =. The horizontal ais is position, the vertical ais is slowness quantity.

20 S. Jin,. Wen / Journal of Computational Physics (6) ρ.5 ρ Fig.. Eample 6., density q(, t) at t =. Solid line: the eact solution; ÔÕ: the numerical solutions by the SFDM using 8 mesh. Left: SFDMI; Right: SFDMS. the numerical density and averaged slowness on different meshes using Scheme I plotted against the eact solutions. Table presents the l -errors of numerical densities q computed with several different meshes on the domain [.5,.5] [.,.]. Table 3 presents the l -errors of numerical averaged slowness u. R E in the tables denotes the estimated convergence rate. It can be observed that the l -convergence rate of the numerical solutions for q and u are about first order. In comparison, Scheme II generally has larger numerical errors than Scheme I in this test. We net present numerical results by treating n = as the domain boundary. We use the meshes on the domain [.5,.5] [.,.] such that is the a grid point in n coordinate. We impose the outflow boundary condition at the domain boundary including the mesh interface n = as done in [], and perform the delta function integrals (.7), (.8) on n > and n < separately. Fig. 6 shows the calculated density using Schemes I and II together with the eact density on 3 mesh. It is observed that the Scheme I gives more accurate numerical solutions than Scheme II near the wave speed jump =. This is reasonable since Scheme I uses second order interpolation in constructing the Hamiltonian-preserving numerical flues while Scheme II only uses first order integration rule, thus Scheme I behaves better in preserving Hamiltonian across the wave speed discontinuity than Scheme II. Table presents the l -errors of numerical densities q computed with several different meshes. Table 5 presents the l -errors of numerical averaged slowness u. These results are more accurate than those given in Tables and 3 due to the imposing boundary condition at n = as well as performing the delta function integrals on n > and n < separately. It should be remarked here that the l -convergence rate of the numerical solutions reported in [] is only halfth order due to the discontinuities that eist in the level set function w which influence the accuracy when evaluating the delta function integrals (.7), (.8) to obtain the moments. In the D Liouville equation of

21 69 S. Jin,. Wen / Journal of Computational Physics (6) ρ ρ u u Fig. 5. Eample 6., density q(, t) and averaged slowness u(, t) at t =. Solid line: the eact solutions; ÔÕ: the numerical solutions without imposing the boundary condition at n =. Upper: density; lower: averaged slowness. Left: 8 mesh; Right: 3 mesh. Table l error of numerical densities q on different meshes, without imposing the boundary condition at n = Meshes R E Scheme I 3.7E.79E 6.89E.5 Scheme II 3.5E.E 6.53E.3 Table 3 l error of numerical averaged slowness u on different meshes, without imposing the boundary condition at n = Meshes R E Scheme I.66E.9E 8.956E 3. Scheme II.9E.5E E 3.3 geometrical optics, there are also such discontinuities for the level set function due to the change of particle slowness at the interface. But such discontinuities typically form the line parallel to the n-ais in the domain n > and n <, respectively, since the velocity of particles are only dependent on the local wave speed and the sign of particle slowness, thus they do not influence the accuracy when evaluating the integrals (.7), (.8) of the delta function along n direction at most part of physical domain. Thus the accuracy loss in the moment evaluations does not occur here. Eample 6.3. Computing the physical observables of a D problem with a measure-valued solution. Consider the D Liouville equation (5.) with a discontinuous wave speed given by

22 S. Jin,. Wen / Journal of Computational Physics (6) ρ ρ Fig. 6. Eample 6., density q(, t) at t =. Solid line: the eact solution; ÔÕ: the numerical solutions on 3 mesh by imposing the boundary condition at n =. Left: Scheme I; Right: Scheme II. Table l error of numerical densities q on different meshes, imposing the boundary condition at n = Meshes R E Scheme I.3968E.56E.3879E.7 Scheme II.579E 5.53E.59E. Table 5 l error of numerical averaged slowness u on different meshes, imposing the boundary condition at n = Meshes R E Scheme I.879E.3353E.E 3.8 Scheme II.9E.E.67E 3. ( p ffiffiffiffiffiffi :8 ; > ; y > ; cð; yþ ¼ p ffiffiffiffiffiffi :6 ; else and a delta function initial data f ð; y; n; g; Þ ¼qð; y; Þdðn pð; yþþdðg qð; yþþ; where > :; y > :; qð; y; Þ ¼ ; else; pð; yþ qð; yþ ¼:6.