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1 Journal of Comutational Physics 9 () Contents lists available at ScienceDirect Journal of Comutational Physics journal homeage: A level set method for the semiclassical limit of the Schrödinger equation with discontinuous otentials Dongming Wei a, *, Shi Jin a, Richard Tsai b, Xu Yang c a Deartment of Mathematics, University of Wisconsin, Madison, WI 5376, USA b Deartment of Mathematics, University of Teas at Austin, Austin, TX 77, USA c Program in Alied and Comutational Mathematics, Princeton University, Princeton, NJ 544, USA article info abstract Article history: Received January Received in revised form 3 May Acceted 4 June Available online 3 June Keywords: Schrödinger equation Level set Interface We roose a level set method for the semiclassical limit of the Schrödinger equation with discontinuous otentials. The discontinuities in the otential corresonds to otential barriers, at which incoming waves can be artially transmitted and reflected. Previously such a roblem was handled by Jin and Wen using the Liouville equation which arises as the semiclassical limit of the Schrödinger equation with an interface condition to account for artial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (6) ). However, the initial data are Dirac-delta functions which are difficult to aroimate numerically with a high accuracy. In this aer, we etend the level set method introduced in (S. Jin, H. Liu, S. Osher, R. Tsai, J. Com. Phys. (5) 497 5) for this roblem. Instead of directly discretizing the Delta functions, our roosed method decomoses the initial data into finite sums of smooth functions that remain smooth in finite time along the hase flow, and hence can be solved much more easily using conventional high order discretization schemes. Two ideas are introduced here: () The solutions of the roblems involving artial transmissions and artial reflections are decomosed into a finite sum of solutions solving roblems involving only comlete transmissions and those involving only comlete reflections. For roblems involving only comlete transmission or comlete reflection, the method of JLOT alies and is used in our simulations; () A reinitialization technique is introduced so that waves coming from multile transmissions and reflections can be combined seamlessly as new initial value roblems. This is imlemented by rewriting the sum of several delta functions as one delta function with a suitable weight, which can be easily imlemented numerically. We carry out numerical eeriments in both one and two sace dimensions to verify this new algorithm. Ó Elsevier Inc. All rights reserved.. Introduction In this aer, we roose a numerical scheme for the Liouville equation f t þ H n r f H r n f ¼ ; t > ; ; n R d ; ð:þ whose solutions are delta functions of variable weight concentrating on the bi-characteristics stris of the equation. The function f(t,,n) is the density distribution of articles deending on osition, time t and velocity n. The Hamiltonian H takes the form * Corresonding author. Tel.: ; fa: addresses: dwei@math.wisc.edu (D. Wei), jin@math.wisc.edu (S. Jin), ytsai@math.uteas.edu (R. Tsai), uyang@math.rinceton.edu (X. Yang). -999/$ - see front matter Ó Elsevier Inc. All rights reserved. doi:.6/j.jc..6.6

2 D. Wei et al. / Journal of Comutational Physics 9 () Hð; nþ ¼ jnj þ VðÞ; ð:þ with V() corresonding to the otential function. In articular, we are concerned with the case when the otential V() W, has jum discontinuities along a smooth hyersurface; such tye of otentials corresond hysically to roblems with otential barriers. Eq. (.) rovides a hase sace descrition of the semiclassical limit [6,] of the Schödinger equation: ih@ t w h ¼ h Dwh þ VðÞw h ; R d ; ð:3þ where w h is the comle-valued wave function, h the reduced Planck constant. In this setting, one tyically considers the Schrödinger equation (.3) with the WKB initial data assuming the form wð; Þ ¼A ðþ eðis ðþ=hþ; ð:4þ with smooth S. In the semiclassical limit?, this corresonds to mono-kinetic initial data for the Liouville equation (.): f ð; ; nþ ¼jA ðþj dðn r S ðþþ: ð:5þ Consequently, the solutions of the initial value roblem (.) and (.5) remain as delta functions concentrating on the bicharacteristics. When otential barriers are resent in the roblem, the otential function V has jum discontinuities along the barrier. Waves traveling into a otential barrier tyically undergo artial transmissions and reflections; i.e. a roortion of an incident wave is transmitted through the discontinuity of the otential while the remaining ortion is reflected. [,,5]. In this aer, we shall use the terms comlete transmission or comlete reflection when each incident ray is either transmitted through or reflected off the interface, and we shall refer to the discontinuities in the otential function as the interface. It is ossible to solve the Liouville equation (.) with delta function initial data (.5) by relacing the initial data with aroimate delta functions. We shall refer to this tye of methods as the direct methods. However, this tye of methods usually roduces oor quality aroimations as the delta functions are quickly smeared out due to numerical dissiation, and that the large gradient of the numerical aroimations of the delta functions result in very large error constants. To avoid comuting directly the delta function solutions, one can track the bi-characteristic stris of the Liouville equation either elicitly [5] or imlicitly [3,3,3,9,4,9], and evolve various hysical quantities defined on the bi-characteristics along the way. In the contet of solving the Schrödinger equation, the method introduced in [9] decomoses the article density function f into w and / j (j =,...,d): f ð; n; tþ ¼wð; n; tþp d j¼ dð/ jð; n; tþþ; ð:6þ where w and / j solve the same Liouville equation (.) with initial data wð; n; Þ ¼q ðþ; / j ð; n; Þ ¼ðn u ðþþ j ; ð:7þ resectively. Here (n u ()) j denotes the jth comonent of the vector. In this setu, the common zeros of / j defines the bicharacteristics and w tracks the density on the bi-characteristics. We shall refer to such methods as decomosition methods. The decomosition methods allow for numerical comutations of bounded solutions rather than the measure-valued solution of the Liouville equation with singular initial data (.5). Physical observables of the system (such as the density q and momentum qu) can be comuted assively via simle integrals in the hase dimensions qð; tþ ¼ wð; n; tþp d i¼ dð/ iþdn; ð:þ uð; tþ ¼ wð; n; tþnp d i¼ dð/ iþdn=qð; tþ: ð:9þ For roblems involving comlete transmissions and reflections, interface conditions can be formulated to cature such henomena. The level set method roosed in [] uses such idea to cature the reflections of wave fronts for the wave equations. It was also mentioned in [3] that an interface condition is needed to incororate Snell s Law of Refraction into the transmission of wave fronts. In [4,5] a transmission and reflection interface condition was introduced for the Liouville equation (.) and a corresonding Hamiltonian reserving scheme was develoed for comlete transmissions and reflections for roblems containing otential barriers or discontinuous wave seeds. Then the decomosition method of [9] was used to offer a more accurate numerical aroimation. However, when one has to deal with artial transmissions and reflections using such an interface condition, as was done in [6] (and subsequent etensions to quantum barriers [ ] and wave diffractions [7,]), the direct method still works while the decomosition method requires additional level set functions to be added each time a ray in the incoming wave slits into a reflected one and a transmitted one [6]. This can be easily understood from the Lagrangian oint of view. For smooth otentials V the solution to the Liouville equation (.) can be defined by the method of characteristics defined by the Hamiltonian system

3 744 D. Wei et al. / Journal of Comutational Physics 9 () ¼ ¼ r H: However, at discontinuity of V, in order to define a hysically relevant solution to the initial value roblem of (.), a article (or a characteristic defined by (.)) is slit into two articles with weights corresonding to the transmission and reflection coefficients [,7]. Since each level set Liouville equation (.) is the hase reresentation of the article trajectory determined by (.), when a article slits at the interface, one has to add another level set function to describe such a article or ray slitting. With multile artial transmissions and reflections, the total number of level set functions will increase eonentially in time [6]. In this aer, we etend the decomosition method so that artial transmissions and reflections that occur in the roblems with otential barriers can be catured with a fied number of level set functions, indeendent of how many times artial transmissions and reflections occurs at oints on the interface. In order to achieve this, we introduce two new ideas. First, we decomose the roblem with artial transmission and reflection into the sum of roblems with only comlete transmissions and reflections, so the decomosition method of Jin Liu Osher Tsai can be used here as in [4,5]. Since this decomosition is valid only if the waves or articles hit the interface at most once, we need to utilize a reinitialization technique, which is the second new idea of the aer. After article transmissions and reflections f no longer has the form of (.6). Rather it is a suerosition of several functions of the form (.6). In order to continue to use the decomosition, we need to rewrite f in the form of (3.). A simle numerical rocedure is introduced for this urose. This enables us to handle multile transmissions and reflections for a long time. Details of the decomosition is resented in Section. The level set algorithm for the decomosition method is resented in Section 3; this is followed by details about the reinitialization ste in Section 4. In Section 5, we resent some eamles comuted using the roosed numerical methods. The aer is concluded in Section 6.. Decomosition of the interface roblem Consider a oint on the interface. Since the interface is a hyersurface in the hysical domain, we may consider the left and the right side of the interface at this oint. Denote it by and +. The transmission-reflection interface condition for the Liouville equation (.) at ( +, n), for hase directions ointing into the right of the interface, can be written in a general form as f ðt; þ ; nþ ¼a T f ðt; ; n T Þþa R f ðt; þ ; n R Þ; where n T and n R are two functions deending on the interface normal at and the hase variable n; a T and a R are transmission and reflection coefficients that may deend on n T and n R. This boundary condition dictates that the the density of articles at ( +,n) are the sum of what is being transmitted, f(t,,n T ), and what is being reflected f(t, +,n R ) from aroriate hases and sides of the interface. Since the Liouville equations and the Schrödinger equation are translation and rotation invariant, we may assume that a oint of interest on the interface is at the origin of the satial domain and the -ais is arallel to the normal of the interface at least locally. Thus, without loss of generality, we discuss the decomosition idea in the following model roblem in two dimensions (one in sace and one in hase). Thus, we assume that the interface is at = and the otential takes on the values V + and V from the right and the left, resectively. At ( +,n), n >, the transmission comes from the left side of the interface with a hase gradient which has the same sign as n and satisfies the condition for the conservation of Hamiltonian: ð:þ n þ V þ ¼ n T þ V ; n T > : ð:þ The reflection comes from the same side of the interface as + with a hase gradient which has the oosite sign from n and satisfies n þ V þ ¼ n R þ V þ ; n R < : ð:3þ To simlify the notation, we introduce two new notations, n + and n, where n + and n have the same sign and they satisfy the relationshi ðnþ Þ þ V þ ¼ ðn Þ þ V : ð:4þ Then at ( +,n + ), n + >,(.) and (.3) can be rewritten as n T = n and n R = n +. Similarly, one can derive the condition for (,n ), n 6. Combining the two ossible cases of and +, we t f þ nf V f n ¼ ; t > ; ; >< f ð; ; nþ ¼f ð; nþ; f ðt; þ ; n þ Þ¼a R ðn þ Þf ðt; þ ; n þ Þþa T ðn Þf ðt; ð:5þ ; n Þ; n þ P ; f ðt; ; n Þ¼a R ðn Þf ðt; ; n Þþa T ðn þ Þf ðt; þ ; n þ Þ; n 6 : ð:þ

4 D. Wei et al. / Journal of Comutational Physics 9 () n + and n, having the same sign, satisfy the conservation of Hamiltonian ðn Þ þ V ¼ ðnþ Þ þ V þ : ð:6þ The well-osedness of (.5) was shown in [6]. For now, we assume that the characteristics of the Liouville equation, defined by (.), emanating from the suort of f (,n) at time, intersect the interface at most once in the interval [,s ] for some s >. Let X(s) denote all (,n) such that they can be traced backward from time s to time along the characteristics of the Liouville equation without intersecting with the interface =. We can then easily write down the solution for (.5) as follows: If (,n) X(s), then f ðs; ; nþ ¼f ð ; n Þ; here (,n) and (,n ) are resectively the coordinates of the unique characteristics at time s and at time. If (,n) X c, then f ðs; ; nþ ¼a R ðn þ Þf ðt c ; þ ; n þ Þþa T ðn Þf ðt c ; ; n Þ¼a R ðn þ Þf ð R ; nr Þþa Tðn Þf ð T ; nt Þ: ð:7þ ð:þ Here t c is the time at which the characteristics emanating from ð R ; nr Þ and from ðt ; nt Þ at time arrive at the interface =. From this time on, these two rays travels along the same ath and arrive at (,n) at time s. See Fig.. The solution at oints lying in the left side of the interface, i.e. (,n): 6, is a sum of values that are convected along the characteristics that came from the same side and remain in the same side, maybe due to reflection, and those that were transmitted from the other side. Hence, the density function f can be decomosed into the sum of the solutions of three interface roblems of the same Liouville equation, but with only either comlete transmissions or reflection. This fact is summarized in the following theorem. We shall refer to such solutions as the generalized characteristics solutions. Theorem.. Let f be the generalized characteristic solution of (.5). Define Lf f þ nf V f n, and consider the following three initial value roblems with interface conditions: >< Lf R ¼ ; f R ð; ; nþ ¼f Rð; nþ :¼ f ð; nþ; f R ðt; þ ; n þ Þ¼a R ðn þ Þf R ðt; þ ; n þ Þ; n þ P ; ð:9þ f R ðt; ; n Þ¼a R ðn Þf R ðt; ; n Þ; n 6 ; >< Lf T ¼ ; f T ð; ; nþ ¼f Tð; nþ :¼ I f<gf ð; nþ; f T ðt; þ ; n þ Þ¼a T ðn Þf T ðt; ; n Þ; n þ P ; ð:þ f T ðt; ; n Þ¼a T ðn þ Þf T ðt; þ ; n þ Þ; n 6 ; and >< Lf T ¼ ; f T ð; ; nþ ¼f Tð; nþ :¼ I fpgf ð; nþ; f T ðt; þ ; n þ Þ¼a T ðn Þf T ðt; ; n Þ; n þ P ; f T ðt; ; n Þ¼a T ðn þ Þf T ðt; þ ; n þ Þ; n 6 ; ð:þ Fig.. Illustration of the characteristic solution.

5 7444 D. Wei et al. / Journal of Comutational Physics 9 () where n + and n satisfy condition (.6) with the same sign. If the characteristics of the Liouville equation, defined by (.), emanating from the suort of f (,n) at time, intersect the interface at most once in the interval [,s ] for some s >. Then f ¼ f R þ I fpg f T þ I f<g f T ; 6 t 6 s : ð:þ Proof. Let 6 t 6 s. We decomose the n lane into two arts: X(t) and X c (t). According to the definition of the solution to (.5), f ðt; ; nþ ¼I X f ð ; n ÞþI X ca R ðn þ Þf ð R ; nr ÞþI X ca Tðn Þf ð T ; nt Þ: ð:3þ Similarly, f R ðt; ; nþ ¼I X f ð ; n ÞþI X ca R ðn þ Þf ð R ; nr Þ; ð:4þ f T ðt; ; nþ ¼I X I f <gf ð ; n ÞþI X ca T ðn ÞI f T <g f ð T ; nt Þ; ð:5þ f T ðt; ; nþ ¼I X I f Pgf ð ; n ÞþI X ca T ðn ÞI f T Pg f ð T ; nt Þ: ð:6þ By the definitions of X, for all (,n) X, one can trace backward in time along the trajectory of (.) to (,n ) without hitting the interface =. Therefore, < if and only if <. Thus I X I f <g ¼ I X Tf<g ; I XI f Pg ¼ I X TfPg : Similarly, definitions of X c and T imly that Hence I X c I f T <g ¼ I X c T fpg ; I X c I f T Pg ¼ I X c T f<g : f R ðt; ; nþþi fpg f T ðt; ; nþþi f<g f T ðt; ; nþ ¼I X f ð ; n ÞþI X ca R ðn þ Þf ð R ; nr ÞþI fpgi X Tf<g f ð ; n Þ þ I fpg I X c T fpg a Tðn Þf ð T ; nt ÞþI f<gi X TfPg f ð ; n Þ þ I f<g I X c T f<g a Tðn Þf ð T ; nt Þ ¼ I X f ð ; n ÞþI X ca R ðn þ Þf ð R ; nr ÞþI ca X Tðn Þf ð T ; nt Þ¼fðt; ; nþ: Theorem. shows that an interface roblem with artial transmissions and reflections can be decomose into the interface roblems with comlete transmissions and reflections, in a time interval in which the characteristics hit the interface at most once. Based on this result, we can use the following strategy to obtain the solution of (.5): for 6 t 6 s, we solve roblems (.9), (.) and (.), then f ðt; ; nþ ¼f R ðt; ; nþþi fpg f T ðt; ; nþþi f<g f T ðt; ; nþ; 6 t 6 s : Using f(s,,n) as the initial data, we redo the revious ste to get solution of (.5) on [s,s + s ], where s > such that no article trajectory will hit the interface more than once in the time eriod [s,s + s ]. One can reeat this rocess to obtain the solution for any time interval [,K]. Remark.. If there are N interfaces which divide R into N + arts denoted by A, A,..., A N+, resectively, then the solution f will be f ¼ f R þ XNþ I fr na i g f T i ; i¼ where each f T i has the initial data f T i ð; ; nþ ¼f T i ð; nþ :¼ I A i f ð; nþ: Similarly, in the multi-dimensional case, if the interfaces divide R d into N + arts denoted by A, A,..., A N+, resectively, then the solution f will be f ¼ f R þ XNþ I fr d na i g f T i ; i¼ where each f T i has the initial data f T i ð; ; nþ ¼f T i ð; nþ :¼ I A i f ð; nþ:

6 D. Wei et al. / Journal of Comutational Physics 9 () For general initial data or general geometry, it is not easy to determine s, s,...here we roose a reinitialization after each time ste Dt so f remains the form of (.6). This will be addressed in Section The level set decomosition Consider the d function initial data (.5), namely, f ð; ; nþ ¼f ð; nþ ¼q ðþdðn u ðþþ: ð3:þ Corresondingly, the initial data for f R, f T and f T, which only involve comlete transmission or reflection, are of the mono-kinetic form (3.). According to [4,5], they can be solved by the decomosition method of [9]. More secifically, f R = w R d(/ R ) where / R and w R satisfy >< L/ R ¼ ; / R ðþ ¼n u ðþ; / R ðt; þ ; n þ Þ¼/ R ðt; þ ; n þ Þ; n þ P ; ð3:þ / R ðt; ; n Þ¼/ R ðt; ; n Þ; n 6 ; >< Lw R ¼ ; w R ðþ ¼q ðþ; w R ðt; þ ; n þ Þ¼a R ðn þ Þw R ðt; þ ; n þ Þ; n þ P ; ð3:3þ w R ðt; ; n Þ¼a R ðn Þw R ðt; ; n Þ; n 6 ; Similarly, f T = w T d(/ T ) and f T = w T d(/ T ) where L/ T ¼ ; / >< T ðþ ¼n u ðþ; / T ðt; þ ; n þ Þ¼/ T ðt; ; n Þ; n þ P ; / T ðt; ; n Þ¼/ T ðt; þ ; n þ Þ; n 6 ; Lw T ¼ ; w T ðþ ¼I f<gq ðþ; >< w T ðt; þ ; n þ Þ¼a T ðn þ Þw T ðt; ; n Þ; n þ P ; w T ðt; ; n Þ¼a T ðn Þw T ðt; þ ; n þ Þ; n 6 ; L/ T ¼ ; / >< T ðþ ¼n u ðþ; / T ðt; þ ; n þ Þ¼/ T ðt; ; n Þ; n þ P ; / T ðt; ; n Þ¼/ T ðt; þ ; n þ Þ; n 6 ; Lw T ¼ ; w T ðþ ¼I fpgq ðþ; >< w T ðt; þ ; n þ Þ¼a T ðn þ Þw T ðt; ; n Þ; n þ P ; w T ðt; ; n Þ¼a T ðn Þw T ðt; þ ; n þ Þ; n 6 ; At time s, one can sum these solutions to obtain ð3:4þ ð3:5þ ð3:6þ ð3:7þ f ðs Þ¼f R þ I fpg f T þ I f<g f T ¼ w R dð/ R ÞþI fpg w T dð/ T ÞþI f<g w T dð/ T Þ: ð3:þ 4. Reinitialization Clearly, in one sace dimension, the maimal value of s deends on the distance between the discontinuities as well as the derivative of the otential in each smooth region. For general initial data and general iecewise smooth interfaces in higher sace dimensions, it is not easy to determine s. In this section we introduce a reinitialization rocedure which can be carried out at each time ste, after the decomosition ste roosed in the revious section. With this decomosition reinitialization rocess no knowledge of s is needed. When the discontinuity set of the otential is singular or the wave comes at critical angle, diffraction haens, which introduces a net order term and is out of the scoe of this aer. We refer readers to [,7,]. As discussed in the revious section, even though f has the form (3.), at time s it is a sum of more than one delta functions. In fact, it may be the sum of more than three delta functions shown in (.6), since / R;T ;T may have multile zeroes, corresonding to multihased velocities [,6]. Clearly, to continue the decomosition of Section 3, we need to reinitialize f(s,,n) so it becomes the form in (.6). In other words, we want to find / and w such that f ð; n; s Þ¼w R ð; n; s Þd/ R ð; n; s Þ þ IfPg w T ð; n; s Þd/ T ð; n; s Þ þ If<g w T ð; n; s Þd/ T ð; n; s Þ ¼ wð; n; s Þdð/ð; n; s ÞÞ: ð4:þ The following theorems rovide a generic strategy on how this can be done.

7 7446 D. Wei et al. / Journal of Comutational Physics 9 () Theorem 4. (-Dimension). Assume g j () are continuous functions with N j distinct zeros ji ; i ¼ ;...; N j ; j ¼ ;...; N. Assume f j () are bounded continuous functions. Then there eists an > and a function j defined by ; jj < ; jðþ ¼ ð4:þ ; jj P ; such that X N j¼ f j ðþdðg j ðþþ ¼ f ðþdðgðþþ ð4:3þ in the distributional sense. Here f ðþ ¼ XN j¼ and g() is defined by f j ðþjðg j ðþþ gðþ ¼sgnðg l ðþþ minðjg j ðþjþ; j ð4:4þ ð4:5þ where l is the inde such that jg l ðþj 6 jg j ðþj; j. Proof. Since g j () are continuous with distinct zeros, there eists an g > small enough, such that ji g; ji þ g ; i ¼ ;...; Nj ; j ¼ ;...; N are disjoint intervals and ma 6j6N;6i6N j ma jg j ðþj ð ji g; ji þgþ! < min 6j6N;6i6N j! min jg m ðþj ; ð ji g; ji þgþ;m j Let ¼ ma 6j6N;6i6N j! ma ðjg j ðþjþ ð ji g; ji þgþ ; ð4:6þ then on each interval ð ji g; ji þ gþ, f ðþ ¼f j ðþ and gðþ ¼g j ðþ: Furthermore, one can find a ositive number h, such that [ gðþ > h > ; R n ð ji g; ji þ gþ: 6j6N;6i6N j ð4:7þ Therefore, u C c ðrþ, uðþf ðþdðgðþþd ¼ XN j¼ X N j i¼ ji þg uðþf ðþdðgðþþd: ji g ð4:þ Hence, for all test functions u C c ðrþ, uðþ XN f j ðþdðg j ðþþd j¼ ¼ XN uðþf j ðþdðg j ðþþd ¼ XN X N j j¼ j¼ i¼ ¼ XN X N j ji þg uðþf ðþdðgðþþd j¼ i¼ ji g ¼ uðþf ðþdðgðþþd: This comletes the roof. h ji þg uðþf j ðþdðg j ðþþd ji g ð4:9þ

8 D. Wei et al. / Journal of Comutational Physics 9 () Theorem 4. (Multi-Dimension). Assume g j () are C continuous functions such that R R d dðg jðþþd < for j =,..., N. Denote the zeros sets of g j () byx j. Assume that MðX j \ X j Þ¼; j ; j. Suose there eists a constant C > such that jdg j ðþj > C; j X j : ð4:þ Let f j () be bounded continuous functions. Define the function j by ; jj < ; j ðþ ¼ ; jj P : Then f ðþdðgðþþ * XN j¼ in the distributional sense. Here ð4:þ f j ðþdðg j ðþþ;! ð4:þ f ðþ ¼ XN f j ðþj ðg j ðþþ; j¼ and g() is defined by gðþ ¼g k ðþ; ð4:3þ ð4:4þ where k is the inde such that jg k ðþj 6 jg j ðþj; j. Proof. For every fied test function uðþ C c ðrd Þ, for every fied g >, one can choose h small enough such that Rd uðþ XN f j ðþdðg j ðþþd XN uðþf j ðþdðg j ðþþd < g; j¼ j¼ X h j ð4:5þ where X h j is a subset of X j with MðX h j Þ < and distðxh j ; X h j Þ > h >, " j, j. Following the same idea of the roof of Theorem 4., one can further choose a. > small enough, such that if < <. then X N uðþf j ðþdðg j ðþþd ¼ uðþf ðþdðgðþþd: j¼ X h [X h j j ð4:6þ We chose h > small enough so that! X N uðþ jf j ðþj dðgðþþd < R ns d X h g: j j¼ ð4:7þ Combining (4.5), (4.6) and (4.7), we obtain that for every fied uðþ C c ðrd Þ, and for every fied g >, there eists. >, such that if < <., then Rd uðþ XN f j ðþdðg j ðþþd uðþf ðþdðgðþþd < g: ð4:þ R d j¼ Corollary 4.. If d P and g j () are iecewise C continuous functions, then the conclusion of Theorem 4. is still true. Formally, the comutational comleity of the reinitialization rocess (even it is erformed after each time ste) is the same as of solving (.9), (.) and (.), which is also the same as solving the Liouville equation (.5) directly. All of them have order O(D Dn ). Furthermore, all of the functions / R,T,T and w R,T,T only need to be solved locally around the zero level sets of / R,T,T. Therefore, the entire algorithm can be imlemented using the local level set methods. 5. Numerical eamles In this section, we give several numerical eamles. In each eamle, we comute the density and momentum which are given by q ¼ f ðt; ; nþdn; qu ¼ nf ðt; ; nþdn: We use the uwind scheme to comute all the one-dimensional eamles with a minmod sloe limiter and the twodimensional eamle with no sloe limiter.

9 744 D. Wei et al. / Journal of Comutational Physics 9 () When comuting the hysical observables, we use the following discretized delta function [4], ( jj ð þ cos d b ðþ ¼ Þ; j j 6 ; b b b ; otherwise; where the arameter b is taken as b ¼ :5 D ffiffiffiffiffiffi. ð5:þ Eamle 5. (Plane waves). We consider (.5) with the following arameters: 6 ; V ¼ :45 > ; and the initial conditions: f ð; nþ ¼q ðþdðn u ðþþ; q ðþ ¼I f6g ; u ðþ ¼:5: By the conservation of Hamiltonian, :5 ¼ :45 þ ðnþ Þ ; one has n + =. Therefore, a R ¼ ðn n þ Þ ðn þ n þ Þ ¼ :; a T ¼ 4n n þ ðn þ n þ Þ ¼ :9: One could solve this roblem analytically, and the solution at the time t = is ð5:þ f ðt; ; nþ ¼I f6g dðn :5Þþ:I f :566g dðn þ :5Þþ:35I f66:4g dðn :4Þ: The analytical density and momentum are given by 6 :5; :5 6 :5; >< : :5 < 6 ; >< :494 :5 < 6 ; q ¼ qu ¼ :35 < 6 :4; :494 < 6 :4; :4 < : :4 < : The errors and comarison figures are given in Table and Fig.. The convergence orders for the density and momentum are 99 and. As we may see from this eamle, sometimes even though the incident and transmitted angles are fairly large, the reflection coefficient is still too small to be noticeable; on the other hand, the choice of the coefficients makes no difference in testing the method, therefore to better illustrate the erformance of this numerical method, we will use artificial a R, a T in the numerical eamles thereafter. Eamle 5. (Harmonic oscillator). We consider (.5) with the following arameters: ( V ¼ = 6 ; a R ¼ :; a T ¼ : = þ :45 > : and the initial conditions: f ð; nþ ¼q ðþdðn u ðþþ; q ðþ ¼eð ð þ :3Þ Þ; u ðþ ¼:5: The reference solution is comuted in fine mesh and using small time stes. The comarison is given in Fig. 3. Eamle 5.3 (Reinitialization). We consider (.5) with the following arameters: V ¼ 6 ; a R ¼ :; a T ¼ : þ :45 > : and the initial conditions: Table Eamle 5., the l errors of the density and momentum. D...5 kq err k k(qu) err k

10 D. Wei et al. / Journal of Comutational Physics 9 () t= t=. density..6. Δ=.5 Δξ=.5 momentum.3.. Δ=.5 Δξ= Fig.. Eamle 5., the comarison of density and momentum between the analytical solution and numerical solution. density t= reference solution Δ=.5 Δξ=.5 Δ=. Δξ=. Δ=. Δξ=. momentum reference solution Δ=.5 Δξ=.5 Δ=. Δξ=. Δ=. Δξ=. t= Fig. 3. Eamle 5., the comarison of density and momentum between the reference solution and numerical solution. f ð; nþ ¼q ðþdðn u ðþþ; q ðþ ¼I f6g ; u ðþ ¼:5: In this eamle, the articles will hit the interface frequently due to the strong harmonic otential in the domain { > }. We assume the seed of the article becomes zero at = turn, then by the conservation of Hamiltonian, :5 ¼ :45 þ ðnþ Þ ¼ :45 þ 4 turn ; one has n + =, turn =., which imlies a lower bound for the reinitialization time is s = (actually the second hitting time is t = /). We comare the numerical solution with analytical solution at t = and t =. When reinitialization, we let j be the following cutoff function ( ffiffiffiffiffiffi ; jj 6 :5 D ; jðþ ¼ ; jj > :5 D ffiffiffiffiffiffi : Following [7], one can find the analytical solution. The analytical solution at t = is given by f ð; ; nþ ¼I f6g dðn :5Þþ:I f :566g dðn þ :5Þþ:I f66:g : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d n :4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 5 þ :I f: sin 6<:g : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d nþ :4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 : ð5:3þ 5

11 745 D. Wei et al. / Journal of Comutational Physics 9 () The analytical density and momentum at t = are given by 6 :5; :5 6 :5; : :5 < 6 ; >< >< q ¼ ffiffiffiffiffiffiffiffiffiffiffi :4 :5 < 6 ; < 6 : sin ; 5 qu ¼ :4 < 6 : sin ; ffiffiffiffiffiffiffiffiffiffiffi : sin < 6 :; 5 : sin < : > :: The analytical solution at t = is given by f ð; ; nþ ¼I f6g dðn :5Þþ:I f 66g dðn þ :5Þþ:64I f=4 6<g dðn þ :5Þþ:I f66:g : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d n :4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 þ :I f6<:g 5 : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d nþ :4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 5 þ :6I f66: sin ð =Þg : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d n :4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 : ð5:4þ 5 The analytical density and momentum at t = are given by 6 ; : < 6 =4 ; >< :4 =4 < 6 ; q ¼ ffiffiffiffiffiffiffiffiffiffiffi : < 6 : sin ð =Þ; 5 ffiffiffiffiffiffiffiffiffiffiffi 5 : sin ð =Þ < 6 :; > :: :5 6 ; >< :4 < 6 =4 ; qu ¼ : =4 4 < 6 ; : < 6 : sin ð =Þ; : sin ð =Þ < : Since the density blows u at =., we aly the following formula to calculate f(,,n) and then the hysical observables f ð; ; nþ ¼ðw R I fpg w T Þdð/ R ÞþI fpg w T dð/ T Þþw T dð/ T Þ: ð5:5þ This formula is equivalent to (3.). The advantage of (5.5) is that it singles out the ure reflection art and thus decreases the effect of the very large q on the momentum near.. The comarison of the numerical solution and the analytical solution is given in Figs We now comare with the direct method, in which one discretizes the d function initial data and solves the Liouville equation directly. Using the same discretized delta function (5.) with the same arameter b ¼ :5 D ffiffiffiffiffiffi, we comute the solution of this eamle by the direct method. The comarison of the results at time t = and t = is given in Figs. 7 and, which shows that the decomosition method roosed in this aer gives a much more accurate solution esecially for longer time..5 d=., dξ=. t= t= d=., dξ=. density.5 momentum Fig. 4. Eamle 5.3, the comarison of density and momentum between the reference solution and numerical solution at t =.

12 D. Wei et al. / Journal of Comutational Physics 9 () d=., dξ=. t= t= d=., dξ=. 3.5 density momentum Fig. 5. Eamle 5.3, the comarison of density and momentum between the reference solution and numerical solution at t =, reinitialization was erformed at t = t= d=., dξ=. t= d=., dξ=. 3.5 density moment Fig. 6. Eamle 5.3, the comarison of density and momentum between the reference solution and numerical solution at t = with a finer grid and b ¼ :5 ffiffiffiffiffiffi D, reinitialization was erformed at t =. Eamle 5.4 (d eamle). We consider a two-dimensional interface roblem in the domain [,] [,]. The otential well is given by ð; yþ X; V ¼ a ð; yþ X c R ¼ :; a T ¼ :; : where X ={(,y)jy >, y <, y + < }. The initial condition is f ð; nþ ¼q ðþdðn u ðþþ; q ðþ ¼q ð; yþ ¼I fjþ:5j<:5; y>:g ; u ¼ð; ffiffiffi Þ: See Fig. 9 for the illustration of the interface and the initial data. We comare the numerical solution with analytical solution at t =. The analytical solution of the density at t = can be obtained by the method of generalized characteristics, and it takes the following form: qð; yþ ¼I reg dðu ð; ffiffiffi ÞÞ þ :Ireg dðu ð ffiffiffi : ; ÞÞ þ ffiffiffi I reg3 dðu ðsinð=þ; cos =ÞÞ 3 þ :6 : cosð=þ ffiffiffi I reg4 dðu ðsinð=þ; cos =ÞÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I reg5 dðu ðsinð=þ; 3 cos ð=þ 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 cos ð=þ Þ;

13 745 D. Wei et al. / Journal of Comutational Physics 9 () d=., dξ=. direct Liouville t= t= d=., dξ=. direct Liouville density.5 momentum Fig. 7. Eamle 5.3, the comarison of density and momentum between the solutions obtained by our method and by the direct method d=., dξ=. direct Liouville t= t= d=., dξ=. direct Liouville density momentum Fig.. Eamle 5.3, the comarison of density and momentum between the solutions obtained by our method and by the direct method...6 Incident wave. y. Ω Fig. 9. Illustration of the otential well and initial conditions in Eamle 5.4.

14 D. Wei et al. / Journal of Comutational Physics 9 () where reg ¼fð; yþj y :5 > ; :55 < < :45:g; reg ¼fð; yþj y :5 > ; :5 < y < :5:g; reg3 ¼fð; yþj y :5 < ; y > :4; y < :5 ð þ :45Þ= tanð=þ; y > :5 ð þ :55Þ= tanð=þg; reg4 ¼fð; yþj y :5 < ; y > :4; y > :4 þð Þ= tanð=þ; y < :4 þð Þ= tanð=þg; ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 cos reg5 ¼ ð; yþj y :5 < ; y < :4; y < :4 ð Þ ð=þ ; sinð=þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 4 cos y > :4 ð Þ ð=þ ; sinð=þ ¼ :45 tanð=þ :45; ¼ :35 tanð=þ :55: The comarison of the numerical solution and the eact solution was shown in Fig., where the densities after the third transmission and reflection are ignored since their magnitudes are already very small. The absolute value of the difference between the numerical solution and eact solution was shown in Fig.. The mesh size is D =.65,D y =.65 and the time ste is taken to be Dt =.5. In Table, the l errors are given for different D = Dy. The convergence order is 64. Fig.. Eamle 5.4, the comarison of density between the numerical solution (left) and (right) at t =. Fig.. Eamle 5.4, the difference between the numerical solution and at t = (left), a bird s eye view of the difference (right).

15 7454 D. Wei et al. / Journal of Comutational Physics 9 () Table Eamle 5.4, the l errors of the density. D = Dy kq err k Fig.. Eamle 5.4, a bird s eye view of the density. Left: numerical solution. Right:. Fig. 3. Eamle 5.4, a bird s eye view of the density with interface lotted, to better illustrate the result, density was taken to be.5 times the original one. Left: numerical solution. Right:. Fig. gives a bird s eye view of the solution. Fig. 3 gives a bird s eye view of the solution with interface lotted. To better illustrate the result, the density in Fig. 3 was taken to be.5 times the original one. 6. Concluding remarks In this aer, we roosed a level set method for the semiclassical limit of the Schrödinger equation with discontinuous otentials that corresond to otential barriers. At a otential barrier, waves can be artially transmitted and reflected. We combine the method of Jin Wen [4,6] using an interface condition for the Liouville equation to account for artial transmissions and reflections, and the level set decomosition method of [9], in order to have a level set method for artial transmissions and reflections with a higher numerical accuracy. We introduced two new ideas here. First, we decomose the solutions involving artial transmissions and reflections into a finite sum of solutions solving roblems involving only comlete transmissions and those involving only comlete reflections, since for roblems involving only comlete transmission or comlete reflection, the method of [9] can be alied for a higher numerical accuracy. This decomosition is only valid

16 D. Wei et al. / Journal of Comutational Physics 9 () when waves or articles hit the interface at most once. For more general roblems, a reinitialization technique is introduced so that waves coming from multile transmissions and reflections can be combined seamlessly as new initial value roblems. This is imlemented by rewriting the sum of several delta functions as one delta function with a suitable weight, which can be easily imlemented numerically. Both one and two sace dimension roblems were used to demonstrate the validity of this new numerical method. One can etend the method to higher order accuracy by combining the technique roosed here and high order methods for comuting hysical observables, e.g., []. Acknowledgment This work was artially suorted by NSF Grant No. DMS-67, and NSF FRG grant DMS SJ was also suorted by a Van Vleck Distinguished Research Prize from University of Wisconsin-Madison. 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