1. Introduction. In this paper, we construct and study a numerical scheme for the Liouville equation in d-dimension:

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1 SIAM J. NUMER. ANAL. Vol. 44 No. 5 pp c 6 Society for Industrial and Applied Mathematics A HAMILTONIAN-PRESERVING SCHEME FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH PARTIAL TRANSMISSIONS AND REFLECTIONS SHI JIN AND XIN WEN Abstract. We construct a class of Hamiltonian-preserving numerical schemes for the Liouville equation of geometrical optics with partial transmissions and reflections. This equation arises in the high frequency limit of the linear wave equation with a discontinuous inde of refraction. In our previous work [Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds J. Comput. Phys. 4 6 pp ] we introduced the Hamiltonian-preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces. These schemes are etended in this paper to the general case of partial transmissions and reflections. The key idea is to build into the numerical flu the behavior of waves at the interface namely partial transmissions and reflections that satisfy Snell s law of refraction with the correct transmission and reflection coefficients. This scheme allows a hyperbolic stability condition under which positivity and stabilities in both l and l norms are established. Numerical eperiments are carried out to study the numerical accuracy. Key words. geometrical optics Liouville equation transmission and reflection Hamiltonianpreserving schemes AMS subject classifications. 35L45 65M6 7H99 DOI..37/ Introduction. In this paper we construct and study a numerical scheme for the Liouville equation in d-dimension:. f t H v f H v f = t > v R d where the Hamiltonian H possesses the form. H v = c v = c v v v d with c being the local wave speed of the medium /c is the inde of refraction; ft v is the density distribution of particles depending on position time t and the slowness vector v. We are concerned with the case when c W with isolated discontinuities due to different media. The discontinuity in c corresponds to an interface and as a consequence waves crossing this interface will undergo transmissions and reflections. Received by the editors May 5; accepted for publication in revised form March 3 6; published electronically September 9 6. This research was supported in part by NSF grant DMS-358 NSFC under project 8 the Basic Research Projects of Tsinghua University under Project JC and the Knowledge Innovation Project of the Chinese Academy of Sciences K35D and K55F. Department of Mathematics University of Wisconsin Madison WI 5376 and Department of Mathematical Sciences Tsinghua University Beijing 84 P.R. China jin@math.wisc.edu. This author s research was also supported in part by the Institute for Mathematics and its Applications IMA under a New Direction Visiting Professorship. Institute of Computational Mathematics Chinese Academy of Science P. O. Bo 79 Beijing 8 China wenin@amss.ac.cn. 8

2 8 SHI JIN AND XIN WEN The bicharacteristics of this Liouville equation. satisfy the Hamiltonian system.3 d dt = c v v dv dt = c v. In classical mechanics the Hamiltonian. of a particle remains a constant along the particle trajectory even when it is being transmitted or reflected by the interface. This Liouville equation arises in the phase space description of geometrical optics. It is the high frequency limit of the wave equation.4 u tt c Δu = t > R d. In the past numerous numerical methods have been proposed for the wave equation.4 with discontinuous coefficients c; see [3] and references therein. However our interest is in the high frequency waves for which many current numerical methods such as the phase space based level set methods are based on the Liouville equation. with smooth c; see [8 5 34]. Semiclassical limits of wave equations with transmissions and reflections at the interfaces were studied in [ 33 39]. A Liouville equation based level set method for the wave front but with only reflection was introduced in [9]. In our previous work [8] two classes of numerical schemes that are suitable for the Liouville equation. with a discontinuous local wave speed c were constructed. The designing principle there was to build the behavior of waves at the interface either cross over with a changed velocity according to a constant Hamiltonian or be reflected with a negative velocity or momentum into the numerical flu; see also earlier works [36 7]. These schemes were called Hamiltonian-preserving schemes. By providing an interface condition it connects the two domains of Liouville equation with smooth coefficients. This gives a physically relevant selection criterion for a unique solution to the governing equation which is linearly hyperbolic with singular discontinuous or measure-valued coefficients. For a plane wave hitting a flat interface it selects the solution at the interface governed by Snell s law of refraction when the wave length is much shorter than the width of the interface while both lengths go to zero. Nevertheless this is not the only physically relevant possibility to choose a solution across the interface. When the wave length is much longer than the width of the interface while both lengths go to zero the waves can be partially transmitted and reflected and the transmission and reflection coefficients can be analytically computed [33]. The goal of this paper is to construct the numerical scheme which is suitable to deal with partial transmissions and reflections with computable transmission and reflection coefficients. As in [8] we still use the Hamiltonian-preserving principle to determine the transmitted velocity across the interface. The new contribution of this paper is to incorporate the transmission and reflection coefficients into the numerical flu in order to treat partial transmissions and reflections. This new eplicit scheme like those in [7 8] allows a typical hyperbolic stability condition Δt = OΔ Δv under which we also establish the positivity and l and l stability theory for the scheme. In geometrical optics applications one has to solve the Liouville equation like. with measure-valued initial data.5 f v =ρ δv u ;

3 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 83 see for eample [38 4 5]. 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 [ ]. Direct numerical methods DNM for the Liouville equation with measure-valued initial data.5 which approimate the initial delta function first then evolve the Liouville equation could suffer from a poor numerical resolution due to the numerical approimation of the initial data of delta function as well as numerical dissipation [4]. The level set method proposed in [4 5] decomposes the density distribution f into the bounded level set functions obeying the same Liouville equation which greatly enhances the numerical resolution. One only involves numerically the delta function at the output time when the moments which has delta functions in their integrands need to be evaluated numerically. However the etension of this density distribution decomposing approach to the case of partial transmission and reflection is not straightforward. In particular as the number of transmissions and reflections increase in time so does the number of needed level set functions satisfying.. This difficulty was already pointed out in [9]. In this paper when dealing with the measure-valued initial data.5 we will just use the DNM. This does not offer the same resolution as those in [8]. It remains an open question on how to etend the decomposition idea of [4 5] to the case of partial transmissions and reflections. This paper is organized as follows. In section we present the behavior of waves at an interface which guides the designing of our scheme. We also give an interface condition.5 which allows us to define the analytic solution to the Liouville equation. with singular coefficients. We present the scheme in d in section 3 and study its positivity and stability in both l and l norms. We etend the scheme to the two space dimension in section 4 in the simple case of an interface aligning with the grids. Numerical eamples are given in section 5 to verify the accuracy of the scheme. We make some concluding remarks in section 6.. The behavior of waves at an interface... Transmissions and reflections at the interface. In geometrical optics when a wave moves with its density distribution governed by the Liouville equation. its Hamiltonian H = c v should be preserved across the interface. c v = c v where the superscripts ± indicate the right and left limits of the quantity at the interface. The wave can be partly reflected and partly transmitted. The condition. can be used to determine the particle velocity on one side of the interface from its value on the other side. When a plane wave hits a flat interface this condition is equivalent to Snell s law of refraction [8]:. and the reflection law sin θ i c = sin θ t c.3 θ r = θ i

4 84 SHI JIN AND XIN WEN ξ η ξ η θ r θt θ i c c ξ η Fig... Wave transmission and reflection at an interface. where θ i θ t and θ r stand for angles of incident and transmitted and reflected waves; see Figure.. The reflection coefficient is given by c α R cos θ i c cos θ t.4 = c cos θ i c cos θ t while the transmission coefficient is α T = α R ; see for eample [ 33 39]. We will discuss this behavior in more detail in D and D respectively. The D case is simpler. Consider the case when at an interface the characteristic on the left of the interface is given by ξ >. Then with probability c α R c = c c the wave is reflected by the interface with a new velocity ξ and with probability α T = α R it will cross the interface with the new velocity ξ = c c ξ determined by.. The D case when an incident wave hits a vertical interface see Figure.. Let = y v =ξη. Assume that the incident wave has a velocity ξ η to the left side of the interface with ξ >. Since the interface is vertical.3 implies that η is not changed when the wave crosses the interface. There are two possibilities: [ ] c c ξ c c η >. In this case the wave can partially transmit and partially be reflected. With probability α R = c γ c γ the wave is reflected with a new velocity ξ η c γ c γ where γ = cosθ t = ξ γ ξ ξ = cosθ i =. η ξ η With probability α T = α R it will be transmitted with the new velocity ξ η where [ ξ = c c ξ ] η c c

5 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 85 is obtained using.. [ ] c <c c and c ξ c c η <. In this case it is impossible for the wave to transmit so the wave will be completely reflected with velocity ξ η. If ξ < similar behavior can also be analyzed using the constant Hamiltonian condition.... The interface condition for density distribution. The solution to the Liouville equation. which is linearly hyperbolic can be solved by the method of characteristics. Namely the density distribution f remains a constant along a bicharacteristic. However with partial transmissions and reflections this is no longer true since f needs to be determined from two bicharacteristics one accounting for the transmission and the other for reflection. Therefore we use the following condition at the interface:.5 ft v =α T ft v α R ft v where v is defined from v through the constant Hamiltonian condition. α T and α R are the transmission and reflection coefficients which add up to and vary with v ecept in the D case. This is the main idea of this paper and will be used in constructing the numerical flu across the interface in the net section. As will be seen in the net section our scheme incorporates the interface condition into the numerical flu. For hyperbolic systems with discontinuous coefficients renormalized solution was introduced by DiPerna and Lions [] and further etended in [7 8 3] for uniqueness and stability. The renormalized solution idea cannot be applied here since the coefficients can be measure-valued. Our approach here is to use the interface condition.5 to connect two domains in which the Liouville equation has smooth Hamiltonians. Concretely we define the solution for. when the local wave speed has discontinuities as follows. Definition.. The analytic solution for the Liouville equation. when the local wave speed c has discontinuities is constructed by method of characteristics away from the interface plus the interface condition.5. Below we justify the well-posedness of the initial value problem for the simple case of a step function c with a vertical interface. The more general situation remains to be worked out and will be deferred to a future work. Consider the simple case that the local wave speed c R d is piecewise constant as follows: { c <.6 c = c > where we assume c <c. We will also eclude some singular points working in the domain defined by.7 Ω= { v R d v R d \{} } \{ v = v =}. We have the following theorem. Theorem.. Assume the initial data f v has a compact support in v. With the solution defined in Definition. the initial value problem to.8 f t H v f H v f = t > v Ω

6 86 SHI JIN AND XIN WEN with H given by. c given by.6 and Ω given by.7 is well-posed in l and l norms. Proof. The proof is based on eplicit construction of the analytical solution ft v. The l stability follows from the maimum principle while the key for the l stability is to prove that the Liouville theorem volume preserving for a Hamiltonian flow holds at the interface for partial tranmissions and reflections. To make the following description easier we define a function etended from the local wave speed.6 c < c c v = >.9 c =v < c =v > which is defined on the whole definition domain Ω. The values of c v on = however are not crucial as long as they are positive. Split the domain Ω into two parts Ω = Ω Ω with { Ω = v Ω c v v v T > or c v v } v T = { Ω = v Ω c v v } v T < or = where Ω consists of those points whose positions are not on the interface and when tracing back along the bicharacteristics will not hit the interface within time T ecept possibly the end point. We further split domain Ω Ω as Ω =Ω Ω Ω =Ω Ω with Ω = { v Ω < } Ω = { v Ω > } Ω = { v Ω c v v v T Ω = { v Ω c v v v T For v Ω one has. ft v =f c v. ft v =f v Tv c v v Tv c } > } <. v Ω v Ω. Define a subset of Ω { Ω s = v Ω c } v v vd. For v Ω one has. ft v =f R v R v Ω s ft v =α T v T f T v T α R v R f R v R v Ω \Ω s.3

7 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 87 where α T vα R v denote the transmission and reflection coefficients determined by the incident wave slowness vector v with condition α T vα R v =. In geometrical optics the transmission coefficient also satisfies α T v T =α T v R for the slowness vectors v T v R appearing in.3 thus it holds that α T v T α R v R =. This contributes to the maimum principle of the solution for.. The positions and slowness vectors T v T R v R can be eplicitly epressed by v as follows [ĉ ] v vt = v vd v T v > ĉ T v Ti = v i i =...d T = ĉ T v T v T v ĉ ĉv Ti = i v i ĉ T v i v v v ĉ ĉv v R = v v Ri = v i i =...d R = ĉv T v v ĉv Ri = i v i ĉv i v v T v ĉv T i =...d i =...d where ĉ ĉ T are given by ĉ = c ĉ T = c for v Ω ĉ = c ĉ T = c for v Ω. Since the solution ft v can be eplicitly epressed as... and.3 we have proved the eistence and uniqueness of the solution for the initial value problem in Theorem.. The l stability follows easily from the maimum principle and linearity of the Liouville equation. In the following we prove the l -stability of the solution for this initial value problem. Define the l -norm of the solution as f = ft v ddv. Ω Due to the linearity of the Liouville equation one only needs to prove that when the initial value is bounded in l -norm then the solution remains bounded in l -norm at later time. Assume f v eists we now investigate the relation between ft v and f v. Define the sets Ω 3 { = v Ω y v v Ω s.t. = y cy Ω 3 { = v Ω y v v Ω s.t. = y cy v T { Ω 4s = v Ω < c v c v T c Ω 4 =Ω\{Ω 3 Ω 3 Ω 4s}. } v T } v v v d }

8 88 SHI JIN AND XIN WEN One has ft v = ft v ddv ft v ddv. Ω ft v ddv Ω s Ω ft v ddv. Ω \Ω s For the first part in. since the map v c v v Tv is volumepreserving. gives ft v ddv = f c v. Ω Ω v Tv ddv = f v ddv. In the same way the second part in. holds.3 ft v ddv = f v ddv. Ω Ω 3 To calculate the last two parts in. we need to investigate the Jacobians of the maps T v T v and R v R v. From.4. these two maps can be eplicitly written out. The nonzero elements in the two Jacobian matrices include T i T v v Ti R i R v i v Ri v T i Ti v T... i v T i =...d v R i Ri v R... i v R i =...d Ω 3 v Td i v T... v i v Ti v Rd i v R... i v Td i v Rd i =...d i =...d i =...d from which only the diagonal elements influence the Jacobians. They are ĉ = T i Ti = v v T = v i v Ti = ĉ T v v T i =...d ĉt v T ĉ v i =...d

9 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 89 = R v = v R i = Ri v i = v Ri i =...d i =...d. Thus it is verified that the two maps T v T v and R v R v are volume-preserving. For the third part in. from. one has.4 ft v ddv = Ω s f R v R ddv = Ω s f v ddv. Ω 4s For the fourth part in. from.3 one has ft v ddv = α T v T f T v T ddv Ω \Ω s Ω \Ω s α R v R f R v R ddv Ω \Ω s = α T v f v ddv α R v f v ddv Ω 4 Ω 4.5 = f v ddv. Ω 4 Together with and.5 one gets ft v = f v ddv f v ddv Ω 3 Ω 3 f v ddv f v ddv Ω 4s Ω 4 = f v. This is the l -stability in fact l preservation of the solution for the initial value problem in Theorem.. Remark.. In [] a classical-classical coupling model that connects two domains of classical mechanics with constant potentials with a classical domain [a b]inbetween where the potential is variable was introduced where the interface conditions at a and b were given. When a = b their interface conditions reduce to The scheme in D. 3.. The numerical flu. We now describe our finite difference scheme for the D Liouville equation 3. f t csignξf c ξ f ξ =. We employ a uniform mesh with grid points at i i =...N in the - direction and ξ j j =...M in the ξ-direction. The cells are centered at iξ j i =...Nj =...M with i = i i and ξ j = ξ j ξ j. The uniform mesh size is denoted by Δ = i i Δξ = ξ j ξ j. We also assume a uniform time step Δt and the discrete time is given by = t <t < <t L = T.

10 8 SHI JIN AND XIN WEN We introduce the mesh ratios λ t average of f is defined by = Δt Δ λt ξ = Δt Δξ assumed to be fied. The cell f ij = i ξj f ξ tdξd. ΔΔξ i ξ j We assume the local wave speed is Lipschitz continuous ecept at its isolated discontinuous points. Assume that the discontinuous points of the wave speed c are located at the grid points. Let the left and right limits of c atpoint i/ be c i and c respectively. Note that if c is continuous at i j/ then c i approimate c by a piecewise linear function = c.we i c c j / c j/ c j / j /. Δ We also define the average wave speed as c i = c c. We will adopt i i the flu splitting technique used in [36 7 8]. The semidiscrete scheme with time continuous reads 3. f ij t c isignξ j Δ c c f f i j i j i i ΔΔξ ξ j f ij f ij = where the numerical flues f ij are defined using the upwind discretization. Since the characteristics of the Liouville equation may be 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 at each cell i jf i j interface. We will use.5 to define these flues. Assume c is discontinuous at i Consider the case ξ j >. Using upwind scheme f = f i ij. However by.5 j f = i j αt ft ξ α R ft ξ i i while ξ is obtained from ξ = ξ j from.. Since ξ may not be a grid point we have to define it approimately. One can first locate the two cell centers that bound this velocity and then use a linear interpolation to evaluate the needed numerical flu at ξ. The case of ξ j < is treated similarly. The detailed algorithm to generate the numerical flu is given below. Algorithm I if ξ j > f = f i ij j ξ = c i c ξ j i if ξ k ξ <ξ k for some k c c α R i = i c c α T = α R i i f = ξk ξ i j αt Δξ f ik ξ ξ k f ik α R f ik Δξ

11 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 8 end if ξ j < f i j = f ij where ξ k = ξ k ξ = c i c ξ j i if ξ k ξ <ξ k for some k c c α R i = i c c α T = α R i i f = ξk ξ i j αt Δξ f ik ξ ξ k f ik α R f ik Δξ where ξ k = ξ k end 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 the van Leer or minmod slope limiter [3] into the above algorithm. This can be achieved by replacing f ik with f ik Δ s ik and replacing f ik with f ik Δ s ik in the above algorithm for all 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 ξ j > and c <c for all i the other cases can be treated similarly with the same i i conclusion. The scheme reads f n ij f n ij Δt c i f ij d f i k d f i k α R f ik Δ c i c i Δ ξ j f ij f ij Δξ = where d d α R are nonnegative and d d = α T = α R. We omit the superscript n of f. The above scheme can be rewritten as c c f n ij = c i λ t i i ξ j λ t ξ f ij c i λ t d f i k d f i k α R f ik Δ c c i i 3.3 ξ j λ t Δ ξf ij. Now we investigate the positivity of scheme 3.3. This is to prove that if fij n for all i j then this is also true for f n. Clearly one just needs to show that all of the coefficients before f n are nonnegative. A sufficient condition for this is clearly c c c i λ t i i ξ j λ t ξ Δ

12 8 SHI JIN AND XIN WEN or 3.4 Δt ma ij c i Δ c i c i Δ ξ j Δξ. c i The quantity c i Δ now represents the wave speed gradient at its smooth point which has a finite upper bound since c W. In addition typically f has a compact support so in practical computation ξ is confined in a bounded set. Thus our scheme allows a time step Δt = OΔ Δξ. According to the study in [35] our second order scheme which incorporates a slope limiter into the first order scheme is positive under the half CFL condition namely the constant on the right-hand side of 3.4 is /. The above conclusion is drawn on the forward Euler time discretization. One can draw the same conclusion for the second order TVD Runge Kutta time discretization [4]. The l -contracting property of this scheme: f n f follows easily because the coefficients in 3.3 are positive and the sum of them is The l -stability of the scheme. In this section we prove the l -stability of the scheme with the first order numerical flu and the forward Euler method in time. For simplicity we consider the case when the wave speed has only one discontinuity at grid point m with c > c and c > at smooth m m points. The other cases namely when c or the wave speed having several discontinuity points with increased or decreased jumps can be discussed similarly. Denote λ c c /c <. m m We consider the general case that ξ < ξ M >. For this case as adopted in [5 8] the computational domain should eclude a set O ξ = { ξ R ξ = } which causes singularity in the velocity field. For eample we can eclude the following inde set: { D o = i j ξ j < Δξ } from the computational domain. Since c has a discontinuity we also define an inde set D 4 l = {i j i m ξ j <λ c ξ }. As mentioned in [8] Dl 4 represents the area where waves come from outside of the domain [ N ] [ξ ξ 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 E d = {i j i =...Nj =...M}\ { D o Dl 4 } 3.5. As a result of ecluding the inde set Do from the computational domain the computational domain is split into two nonoverlapping parts: E d = {i j E d ξ j > } {i j E d ξ j < } E d E d.

13 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 83 In [8] we analyzed the l -stability of the scheme on E d and E d separately. Here we will conduct the analysis on the full phase space E d since transmission and reflection waves coeist at the interface. We define the l -norm of a numerical solution u ij in the set E d to be f = 3.6 f ij N d ij E d with N d being the number of elements in E d. Given the initial data fij i j E d. Denote the numerical solution at time T to be fij L i j E d. To prove the l -stability we need to show that f L C f. Due to the linearity of the scheme the equation for the error between the analytical and the numerical solutions is the same as 3.3 so in this section f ij will denote the error. We assume there is no error at the boundary thus fij n = at the boundary. If the l -norm of the error introduced at each time step in the incoming boundary cells is ensured to be o part of f n our following analysis still applies. Now denote A i = c c 3.7. Δ i i Since c is Lipschitz continuous at its smooth part there eists an A u > such that A i <A u i. Assume also that there is an C m > such that c i >C m i. The finite difference scheme is given as follows: When ξ j > if i m 3.8 f n ij = A i ξ j λ t ξ c i λ t fij A i ξ j λ t ξf ij c i λ t f i j 3.9 f n mj = A m ξ j λ t ξ c m λ t fmj A m ξ j λ t ξf mj When ξ j < 3 if i m c m λ t d j f mk d j f mk α R f mk f n ij = A i ξ j λ t ξ c i λ t fij A i ξ j λ t ξf ij c i λ t f ij f n mj = A m ξ j λ t ξ c m λ t fmj A m ξ j λ t ξf mj c m λ t d j f mk d j f mk α R f mk where d j d j and d j d j = α T = α R =. In 3.9 k is determined by ξ k λ c ξ j <ξ k and ξ k = ξ k. In 3. k is determined by ξ k ξj λ c <ξ k and ξ k = ξ k. When summing up all absolute values of f n ij in one typically gets the following inequality: f n 3. α ij f N ij n d ij E d

14 84 SHI JIN AND XIN WEN where the coefficients α ij are positive. One can check that under the CFL condition 3.4 α ij A u Δt ecept for possibly i j D m D m where Dm = {i j E d i = m } D m = {i j E d i = m}. We net derive the bounds for M M defined as M = ma α mj M = ma α mj. mj D m mj D m Define the set S m j = { } j ξj < ξ j ξ j λ c < Δξ for m j D m. Let the number of elements in S m j be N m j. One can check that N m j λ c because every two elements j j S m j satisfy ξ j λ c ξ j λ c Δξ On the other hand one can easily check from 3.9 and 3. for m j D m that α mj < c m λ t c m λ t λ c α T α R c m λ t =α T c m c m λ t OΔ so for sufficiently small Δ M can be bounded by M < α T c m c m λ t. Similarly one can prove for sufficiently small Δ M is also bounded by M < α T c m c m λ t. Denote M =α T c m c m λ t. From 3. f n < A u Δt f n M N d mj D m 3.3 Consecutively using 3.3 gives f L < A u Δt L f M L 3.4 M N d N d n= L n= mj D m f n mj M mj D m f n mj. λ c. f N mj. n d mj D m f n mj Define 3.5 L S = f n mj L S = n= mj D n= mj D m m f n mj.

15 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 85 These two terms can be proved in the same way as in [9] to get 3.6 S S <C T N d f where 3.7 Au C T ep N C m C m λ t. Combing 3.4 and 3.6 f L < A u Δt L { f C T M f } = ep A u T [C T M ] f C f where C ep A u T [C T M ]. Thus we prove the following theorem. Theorem 3.. Let c W have a discontinuity at one point and be bounded below from zero c >C m >. Assume f has a finite l -norm defined 3.6 with a compact support in ξ. Then under the hyperbolic CFL condition 3.4 the solution yielded by the scheme is stable in l -norm: f L <C f. 4. The scheme in two space dimension. Consider the D Liouville equation 4. f t c yξ ξ η f c yη ξ η f y c ξ η f ξ c y ξ η f η =. We employ a uniform mesh with grid points at i y j ξ k η l in each direction. The cells are centered at i y j ξ k η l with i = i i y j = y j y j ξ k = ξ k ξ k η l = η l η l. The mesh size is denoted by Δ = i i Δy = y j y j Δξ = ξ k ξ k Δη = η l η l.we define the cell average of f as f ijkl = i yj ξk ηl f y ξ η tdηdξdyd. ΔΔyΔξΔη i y j ξ k η 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 potential interface values are identical. We also define the average wave speed in a cell by averaging the four wave speed values at the cell interface: c ij = 4 c i j c i j c c. ij ij The D Liouville equation 4. can be semidiscretized as c ij ξ k f ijkl t f Δ i ξk f i η jkl jkl l

16 86 SHI JIN AND XIN WEN c ij η l f Δy ij ξk f ij η kl kl l c i j c i j ξk ΔΔξ η l f ijk l f ijk l c c ij ij ξk ΔyΔη η l f ijkl f ijkl = where the interface values f ijk l f ijkl are provided by the upwind approimation and the split interface values f should be i jklf i jklf ij klf ij kl obtained using a similar but slightly different algorithm for the D case. For eample to evaluate f ± we can etend Algorithm I as i jkl Algorithm I in D if ξ k > f = f i ijkl ξ jkl k = ξ k C C i if j ξ k i j η l > C i j ξ = C i j C i j C i j ξ k C i j C i j if ξ k ξ <ξ k for some k γ ξ k = γ ξ = k η l α R = c γ c γ i i c γ i c γ i η l ξ ξ η l α T = α R f = ξk i jkl αt ξ Δξ end else f = f i jkl ijk l end if ξ k < f = f i ijkl ξ jkl k = ξ k C C i if j ξ k i j η l > C i j ξ = C i j C i j C i j ξ k f ijk l ξ ξ k f ijk Δξ l C i j C i j η l if ξ k ξ <ξ k for some k γ ξ = γ ξ ξ k = η l ξ k η l α R f ijkl

17 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 87 α R = c γ c γ i i c γ i c γ i f i jkl = α T ξk ξ Δξ α T = α R f ijk l ξ ξ k f ijk Δξ l α R f ijkl end else f = f i jkl ijk l where ξ k = ξ k end The flu f ± can be constructed similarly. ij kl As introduced in section the essential difference between the D and D flu definitions is that in the D case the phenomenon that a wave is completely reflected at the interface does occur while in D the transmission and reflection waves always coeist at the interface. Since the wave speed c W this scheme similar to the D scheme is also subject to a hyperbolic CFL condition under which the scheme is positive. 5. Numerical eamples. In this section we present numerical eamples to demonstrate the validity of the proposed scheme and to study the numerical accuracy. In the numerical computations the second order TVD Runge Kutta time discretization [4] is used. We use the second order scheme with the van Leer slope limiter in constructing the numerical flues ecept for Eample 5.. Eample 5.. A D problem with eact L -solution. Consider the D Liouville equation 5. f t csignξf c ξ f ξ = with a discontinuous wave speed given by {.6 < c =. >. The initial data is given by 5. <ξ > 4ξ < f ξ = >ξ < ξ < otherwise. In this eample the reflection and transmission coefficients α R α T at the interface are α R = 4 αt = 3. The eact solution for f at t = is given by 4

18 88 SHI JIN AND XIN WEN Fig. 5.. Eample 5. the nonzero part of the eact solution f ξ depicted on the 4 4 mesh. The horizontal ais is the position the vertical ais is the slowness. α T <<.. <ξ< ; <<. <ξ<. ; <<.8. <ξ<;.4 << <ξ<.6 ; f ξ =.6 << 3 3. <ξ<; α R.6 <<.6 <ξ< ; 3 3. otherwise 5.3 as shown in Figure 5.. We are also interested in computing the moments of f which include the density ρ t = f ξ tdξ and the averaged slowness u t = / f ξ tξdξ ρ t.

19 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 89 At t = the eact density is.. <<.8;.5α T 3.6 α R. ρ =. <<.; α T 3 α T 3 3. α R.6 3. α R.6.6 <<.4;.6.4 << otherwise. 5.4 The averaged slowness only has definition in [.6.8] since the density is zero outside this interval. The eact averaged slowness in [.6.8] is [. ]. <<.8; u = ρ.5α T [ 3.6 ] α R [. ] [. ] α T 9 α T 9 [ ] 3. αr 4 [ ] 3. αr 4 <<.; [.6 ].6 <<.4; [.6 ] 4 [.6 ].4 <<. 5.5 We choose the time step as Δt = Δξ. The computational domain is chosen as [ ξ] [.5.5] [.6.6]. Table 5. compares the l -error of the numerical solutions for f ρ on [.5.5] and u on [.6.8] computed with different meshes respectively. The convergence rate of f in the l -norm is shown to be about.74. This agrees with the well-established theory [3 4] that the l -error by finite difference scheme for a discontinuous solution of a linear hyperbolic equation is at most half order. The convergence rate of ρ and u are shown to be about.74 and.98 respectively since the solutions also contain discontinuities away from the interface. Figure 5. shows the numerical density ρ and averaged slowness u computed with a 4 4 cell along with the eact solutions in the physical space. Eample 5.. Computing the physical observables of a D problem with measurevalued solution. Consider the D Liouville equation 5. where the wave speed is a

20 8 SHI JIN AND XIN WEN Table 5. l error of the numerical solutions with different meshes. meshes f ρ u ρ u Fig. 5.. Eample 5. the density ρ and averaged slowness u at t =. Solid line: the eact solution; o : the numerical solutions using the 4 4 mesh. Left: the density ρ; Right: the averaged slowness u. well-shaped function {.6.4 <<.4 c = else and the initial data is a delta-function 5.6 with 5.7 w = f ξ = δξ w.5.6; < ; <<.6;.5.6. Figure 5. plots w in dashed lines. In this eample we are interested in the approimation of the density ρ t = f ξ tdξ

21 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS Fig Eample 5. slowness. Dashed line: the initial slowness w; Solid line: the slowness at t =. The horizontal ais is the position the vertical ais is the slowness. and the averaged slowness u t = f ξ tξdξ f ξ tdξ. In the computation we first approimate the delta function initial data 5.6 by a discrete delta function [6]: 5.8 δ β = β β β β >. If ξ j w i <β set fij = ξ β j w i β and fij = otherwise. The choice of the discrete delta function support size β will be made more precise later. We then use the Hamiltonian-preserving scheme to solve the Liouville equation 5.. Then the moments are recovered by ρ n i = j fijδξ n u n i = j fijξ n j Δξ / ρ n i. With partial transmissions and reflections the eact multivalued slowness at t = is depicted as the solid line in Figure 5.3. In this eample the reflection and transmission coefficients α R α T at the wave speed interface are α R = 6 αt = 5. At t = the eact density and averaged 6

22 8 SHI JIN AND XIN WEN slowness are given by.6 <<.4; α R.4 <<.4 /3; α R.6α T.4 /3 <<.4; α R α T /.6.4 <<.; 5.9 ρ = α T /.3. <<.; α R α T /.6. <<.4; α R.6α T.4 <<.4/3; α R.4/3 <<.4;.4 <<.6; and.5.6 <<.4;.5 α R Υ..4 <<.4 3 ;.5 α R Υ..36α T Υ <<.6; u = ρ Υ.6 α R Υ..36α T Υ <<.4; α T.36 Υ Υ αr Υ.8.4 <<.; α T.36 Υ αt.36 Υ <<.; αt.36 Υ Υ αr Υ.8. <<.4; Υ.6 α R Υ..36α T Υ <<.6;.5α R Υ..36α T Υ <<.4 3 ;.5α R Υ <<.4;.4 <<.6; 5. with Υ = The time step is chosen as Δt = Δξ. We will give respectively the numerical results computed by the first order Hamiltonian-preserving method and the second order method using the van Leer slope limiter. The choice of β in the first and second order methods are different. In the first order method we use a linear relation between β and the mesh size Δξ: β =Δξ. In the second order method this choice does not guarantee the numerical convergence rather β must decay to zero slower than Δξ. Our numerical eperiments indicate that β Δξ will be appropriate. Table 5. presents the l -error of ρ and u computed with several different meshes on the domain [.6.6] [..] by using the first order method. It can be observed that the l -convergence order of the numerical solutions is about / order. Tables 5.3 and 5.4 present the same errors computed by the second order method with two sets of β s. Clearly the second order methods give more accurate solutions than the first order method. In comparison between the results by the second order methods with different choices of β one sees that a smaller β gives more accurate

23 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 83 numerical solutions but might cause mild oscillations than a larger one. We do not have a rigorous analysis on the relation between β and Δξ to provide the optimal results by a second order method. Table 5. l error of the numerical moments with different meshes β =Δξ first order method. meshes ρ 3.35E-.438E-.685E-.45E- u.48e E- 6.6E E- Table 5.3 l error of the numerical moments with different meshes β =5Δξ 7Δξ Δξ 4Δξ for the four meshes second order method. meshes ρ.8969e- 9.8E E E- u 6.79E- 3.7E-.96E-.536E- Figure 5.4 shows the numerical solutions of ρ and u using the mesh by the first order method along with the eact solutions. The numerical solution captures the correct dynamics and discontinuities but the resolution is poor even on such a fine mesh. In contrast Figure 5.5 shows the computed densities ρ using the mesh by the second order method with different β s. The results have much higher resolution across the discontinuities than the first order method. However the numerical density by using β = 4Δξ ehibits some small oscillations near the discontinuities between while the use of a larger β = 4Δξ creates no oscillations at the epense of a slight accuracy or resolution loss. These results show that although the second order method can give more accurate solutions than the first order method there is a support size parameter β that needs to be properly chosen in order to compromise between convergence and accuracy of the numerical solution. It is not clear how to choose β a priori. In the future we will study the feasibility of introducing the decomposition technique proposed in [5] into such a problem with measure-valued data which could avoid such an inconvenience as well as improve the numerical accuracy and resolution. Eample 5.3. Computing the physical observables of a D problem with a L solution. Consider the D Liouville equation 4. with a discontinuous wave speed { y> c y = y< and a smooth initial data f y ξ η = y. ξ η. ep πc 3 c 4 c c c 3 c 4 where c =.3c 3 =.5c = c 4 =.5. In this eample we aim at computing the density which is the zeroth moment of the density distribution 5. ρ y t = f y ξ η tdξdη.

24 84 SHI JIN AND XIN WEN Table 5.4 l error of the numerical moments with different meshes β = 5Δξ Δξ 3Δξ 4Δξ for the four meshes second order method. meshes ρ 4.379E-.464E- 9.73E E- u.3585e E-.988E-.857E ρ u Fig Eample 5. density ρ and averaged slowness u at t =. Solid line: the eact solution; : numerical solutions by first order method using the mesh. Left: ρ; Right: u ρ ρ.5.5 Fig Eample 5. density ρ at t =. Solid line: the eact solution; : numerical solutions by second order method using the mesh. Left: β = 4Δξ; Right: β = 4Δξ. The computational domain is chosen to be [ y ξ η] [..] [..] [..] [..]. The reflection and transmission coefficients α R α T at the interface are given by.4. The eact solution of ρ is obtained by first solving for f y ξ η t analytically and then evaluating the integral 5. on a very fine mesh in the ξη space. The time step is chosen as Δt = 3Δ. Figures 5.6 and 5.8 show respectively the numerical density ρ at t =..5 using different meshes along with the eact solution. Figures 5.7 and 5.9 show respectively the numerical density ρ on = at

25 A SCHEME FOR TRANSMISSIONS AND REFLECTIONS 85 ρ. ρ y..... y.. ρ. ρ y..... y.. Fig Eample 5.3 density ρ at t =.. Upper left: the eact solution; Upper right: 3 4 mesh; Lower left: mesh; Lower right: mesh ρ y Fig Eample 5.3 density ρ along =at t =.. Solid line: eact solution; o : 3 4 mesh; * : mesh; : mesh. t =..5 using different meshes along with the eact solution. Table 5.5 presents the l errors of ρ computed with different meshes in phase space at t =..5. The convergence rate is slightly higher than first order which does not suffer from the accuracy degeneration of an usual finite difference method for solving the discontinuous solution of a linear hyperbolic equation which is at most / order stated by the well-established theory [3 4]. This is because the only discontinuity in the solutions is at the interface which has been taken care of by the Hamiltonian-preserving mechanism and no linear discontinuity travels to the downstream direction like in the D case. Table 5.5 l error of ρ using different meshes. meshes t =..4556E E-4.75E-4 t = E E E-4

26 86 SHI JIN AND XIN WEN.. ρ.5 ρ y..... y.... ρ.5 ρ y..... y.. Fig Eample 5.3 density ρ at t =.5. Upper left: eact solution; Upper right: 3 4 mesh; Lower left: mesh; Lower right: mesh ρ y Fig Eample 5.3 density ρ along =at t =.5. Solid line: eact solution; o : 3 4 mesh; * : mesh; : mesh. 6. Conclusion. In this paper we etended our previous work [8] to the Liouville equation of geometrical optics with partial transmissions and reflections. Such problems arise in geometrical optics through inhomogeneous media. While still utilizing the constant Hamiltonian structure in constructing the numerical flu we also account for the transmission and reflection coefficients in the numerical flu. By doing so the numerical flu automatically absorbs the interface condition. This gives an eplicit scheme for the time dependent Liouville equation with discontinuous indices of refraction that can capture correctly the partial transmissions and reflections across the interface. This scheme is subject to a hyperbolic CFL condition under which the scheme is positive and stable in both l and l norms. Numerical eperiments are carried out to study the numerical accuracy. We only etended a finite difference version of the Hamiltonian-preserving scheme developed in [8]. The finite volume version of the method in [8] can also be etended in a similar fashion but will not be given here. In the future we will consider analytical issues such as the well-posedness of the problem in a more general contet than that discussed in this paper and the convergence of the numerical scheme. We will also investigate its applications to more

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