The Vlasov-Poisson Equations as the Semiclassical Limit of the Schrödinger-Poisson Equations: A Numerical Study

Transcription

1 The Vlasov-Poisson quations as the Semiclassical Limit of the Schrödinger-Poisson quations: A Numerical Study Shi Jin, Xiaomei Liao and Xu Yang August 3, 7 Abstract In this paper, we numerically study the semiclassical limit of the Schrödinger- Poisson equations as a selection principle for the weak solution of the Vlasov- Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker-Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov-Poisson equations as the semiclassical limit of the Schrödinger-Poisson equations. Introduction We are interested in establishing a selection principle for the weak solution of the following one-dimensional Vlasov-Poisson equations: subject to the initial condition f t + v f (,t) f v V = ρ = b() =, (.) f(,v,t)dv, = V, (.) f(,v,t) t= = f (,v). (.3) In (.)-(.), f(,v,t) is the density of electrons at location traveling with velocity v at time t, ρ is the charge density, b() denotes the fied positively charged Department of Mathematics, University of Wisconsin, Madison, WI 5376, USA. mails: jin@math.wisc.edu, liao@math.wisc.edu, yang@math.wisc.edu. Research supported by NSF grant Nos. DMS-358 and DMS-687.

2 background, (,t) is the electric field and V (,t) is the electric potential. For simplicity in eposition, we assume throughout this paper that b() = and f(,v,t) is -periodic in, i.e. f( +,v,t) = f(,v,t), ( +,t) = (,t) (, t)d =. (.4) In (.3), f (,v) is a non-negative probability measure, i.e. f (,v)ddv =. We first state the definition of a weak solution according to [4]. Definition.. A pair {,f} of a function (,t) and a non-negative measure f(,v,t) is called a weak solution to the problem (.)-(.4), if {,f} is -periodic in and for some T >,. L (Ω T ) BV (Ω T ) where Ω T = [, ] (,T);. f(,v,t) is a probability measure for each t >, f(,v,t)dvd = ; 3. for all φ C (R (,T)), the velocity average of f(,v,t) with φ is a special measure, which is the -derivative a BV function h φ (,t) BV (Ω T ), i.e. φ(,v,t)f(,v,t)dv = h φ ; 4. {,f} satisfies the Poisson equation in the distributional sense, = fdv, and the normalizing condition is compatible with Condition, d = ; 5. {,f} satisfies the Vlasov equation in the weak form: T T ( ) (φ t f + φ vf)dvddt Ē φ v fdv ddt =, (.5) R R R for all test functions φ C (R (,T)), and Ē is given by { (,t), if is approimately continuous at (,t), Ē = [ l(,t) + r (,t)], if has a jump at (,t), (.6) where l (,t) and r (,t) denote respectively the left and right limits of (,t) at a discontinuity point. 6. f is Lipschitz continuous from [,T] to the local negative Sobolev space H L loc (R ) for some L >, and f(,v, ) = f (,v) in H L loc (R ).

3 As a regularization to the Vlasov-Poission equation, consider the Fokker-Planck equation satisfies f η t + v fη η (,t) fη v = f η η v, (.7) V η = ρ η = b() f η (,v,t)dv, where η is the viscous coefficient and the initial condition is η = V η, (.8) f η (,v,t) t= = f (,v). (.9) The eistence of suitable weak solutions to (.)-(.3) was established by Zheng and A. Majda ([8]). Moreover, the non-uniqueness issue of (.)-(.3) was discussed by A. Majda, G. Majda and Zheng in [4], where two eplicit weak solutions with the same charge concentration initial data were constructed. It was shown that the zero smoothing limit of the time reversible particle method converges to the discrete fission weak solution while the zero diffusion limit of the Fokker-Planck equation (.7)-(.8) converges to the continuous fission weak solution. In this paper, we consider another regularization process given by the semi-classical limit of Schrödinger-Poisson equations, and hope to establish a selection principle for the weak solution of the Vlasov-Poisson equations. The convergence of the limit was studied in [8,, 3, 7]. For convenience, we summarize the main results of [7] as below. The Schrödinger-Poisson equations are { iɛψ ɛ t = ɛ ψɛ + V ɛ ()ψ ɛ, V ɛ = ψ ɛ (,t), ɛ = V ɛ, R, t, (.) with the initial condition ψ ɛ t= = ψ ɛ. (.) Definition.. The Wigner transform of ψ ɛ is defined as ([6]) W ɛ (,v,t) = e ivy ψ ɛ ( + ɛy π,t)ψɛ ( ɛy,t)dy (.) R Theorem.3. Let ϕ ɛ () be uniformly bounded in L (R), b() (L L )(R), ψ ɛ = J ɛ ϕ ɛ (), b ɛ = b J ɛ () where the Friedrichs mollifier J ɛ () = ɛ J( ɛ ) and J() = π e. Let W ɛ (,v,t) be the Wigner transform of a solution ψ ɛ defined in (.). Then there is a subsequence of {W ɛ (,v,t)}, which we still denote by {W ɛ (,v,t)} for convenience, and a nonnegative bounded Radon measure f(,v,t) such that W ɛ (,v,t) f(,v,t) as ɛ, where the Wigner measure f(,v,t) is a weak solution of (.)-(.3) in the sense of Definition., with f the Wigner measure of ψ ɛ () and an L > in Condition 6 of Definition.. Here, the test function space for the weak convergence is defined as A = { φ C c (R R v ) : (F v φ)(,ξ) L (R η,c c (R )) } 3

4 with the norm φ(,v) A = R sup (F v φ)(,ξ) dξ, where (F v φ)(,ξ) is the Fourier transformation of φ(,v) with respect to v. Remark.4. The same conclusion of Theorem.3 still holds if one considers the - domain to be [, ] and all the functions are -periodic in, which will allow b() =. We are interested in which weak solution the semiclassical limit selects for the weak solution of the Vlasov-Poisson equations. Our numerical eperiments show that it selects the one obtained by the zero diffusion limit of the Fokker-Planck equations. In addition, we will also check if this limit is sensitive to the numerical regularization of the measure-valued initial data. Our numerical results show that it is not. We are also interested in the justification of the moment system for the multivalued solutions of the uler-poission equations as the correct semiclassical limit of the Schrödinger-Poission equations. It is known that the solution to the Vlasov-Poisson equations with mono-kinetic initial data, like the Liouville or Vlasov equation [9, 5], when projected into the physical space, may become multivalued which cannot be captured by the viscous solution of the single-valued closure the uler-poission equations. A moment system following the work of [9] was introduced in [] for such multivalued solutions. Multivalued solutions to the Vlasov-Poission equations were also studied later in [7, ]. So far little rigorous study on the multi-valued solution to the uler- Poisson equations is available, nor has there been any theoretical or even numerical justification of these multilvalued solutions as the correct semiclassical limit of the Schrödinger-Poission equations. Our numerical results in this paper suggest that the solutions to the moment equations introduced in [] is the semiclassical limit of the Schrödinger-Poission equations. The paper is organized as follows. The numerical schemes to solve the Vlasov- Poisson equations, Fokker-Planck equations, and the Schrödinger-Poisson equations are introduced in Section. The moment method and its discretization by a kinetic scheme for the multi-valued solution of the uler-poission equations is briefly reviewed in Section 3. Four numerical eamples are given in Section 4 to justify our conclusion. In Section 5, we give some concluding remarks. Numerical schemes for three types of equations. A particle method for the D Vlasov-Poisson and Fokker- Planck equations A particle method for the Vlasov-Poisson equations (.)-(.3) was first introduced in [4]. Here we use the slightly modified version in [4] which is summarized below. The particle trajectory equations are { dj = v dt j dv j dt = Ẽ( j,t) 4 j =,,N, (.)

5 with the given initial condition ( j (),v j ()) = ( j,v j), where j is the position of j-th particle, v j is the velocity of j-th particle, N is the total number of particles. In (.), the approimate electric field is Ẽ(,t) = CΣ N j=k δ (, j) Σ N j=α j K δ (, j (t)), (.) where the points j = (j )h with h = /N, j =,,N, are uniformly spaced in [, ]. The weights α j, j =,,N, are determined by the initial data, and C = Σ N j=α j /N is a constant chosen so that the constrain (.4) is always satisfied. The modified kernel K δ is given by y, y δ, K δ (,y) = y (y )/δ, δ y δ, (.3) y, δ y. We approimate the Fokker-Planck equation (.7)-(.9) by a random particle method which combines a first-order splitting procedure with a random walk solution of the diffusion equation contained in (.7). In the time interval [t,t+ t], where t > is some constant, the phase space trajectories are solutions of the approimate particle trajectory equations (.) and (.) with the initial condition given by ( j (t),ṽ j (t)) = ( j (t),v j (t) + G j (, η t)), (.4) where G j (, η t) is a Gaussian random variable with mean and variance η t.. The Time-splitting spectral method for the Schrödinger- Poisson equations To solve the Schrödinger-Poisson system (.)-(.), we use the time-splitting spectral method (SP, []). Namely, the Schrödinger equation is solved in two steps. One solves for one time step, followed by solving iɛψ ɛ t + ɛ ψɛ = (.5) iɛψ ɛ t V ɛ ()ψ ɛ = (.6) again for one time step. q.(.5) will be discretized in space by the spectral method and integrated in time eactly. The OD (.6) will then be solved eactly. Note that if one uses the Strang splitting method (SP), i.e. solves (.5) within half a time step, (.6) within one time step, followed by (.5) within half a time step, then the numerical accuracy in time will be second order. For Poisson equation V ɛ = ρ ɛ, we also use spectral method. Using the fourier transform, we can get V ˆɛ k (t) = ( L πk ) ( ρ ɛ ) k (t), where L is the length of computational domain. And we set V ˆɛ (t) = for every time step. Inverse fourier transform follows after updating ( ρ ɛ ) k (t) and V ˆɛ k(t). 5

6 3 The moment method for multivalued solutions Consider the Schrödinger-Poisson equations (.) with the WKB initial data ψ ɛ = A () ep(i S () ). (3.) ɛ It was proved in [3,, 8, 7] that the weak limit of this initial value problem is the Vlasov-Poisson equations (.)-(.) with the mono-kinetic initial data f (,v) = A δ(v S ). (3.) If one assumes the mono-kinetic ansatz f(,v,t) = ρ(,t)δ(v u(,t)) with ρ(, ) = A, u(, ) = S, then one can close the Vlasov-Poisson equations (.)-(.) by the uler-poisson system ρ + (ρu) =, t t (ρu) + (ρu ) = ρ V, V = ρ. (3.3) This is a weakly hyperbolic system. ven if ρ(, ) and u(, ) are smooth, singularities (shocks, which corresponds to caustics in geometric optics) form in finite time. Beyond the singularity time, multivalued solution should be introduced. In [9], a moment system was introduced for the Vlasov equation with initial data (3.). This idea was etended to the Vlasov-Poisson equations in []. 3. The moment system for multi-valued solutions In this section, we review the moment system, given in [9, ], for multivalued solutions. First define the moments of the Vlasov-Poisson equations (.)-(.): m l = f(,v,t)v l dv, l =,,, K. (3.4) R Then taking moments of (.), one can get the moment equations in the physical space t + =, t + = m V, t N + K = (K )m K V, V = K k= ρ k. (3.5) Suppose the total number of branches is K <, then the solution of (.)-(.) can be epressed as f(,v,t) = Σ K k=ρ k δ(v u k ), (3.6) 6

7 where (ρ k,u k ) is the density and the velocity for the kth-branch. Then one can epress the moments in terms of (ρ k,u k ),k =,,,K: m l = K ρ k u l k, l =,,, K. k= m is called the average density, and the average velocity is defined as ū = m /m (3.7) With (3.6, (3.5) can be closed by epressing m K as a function of m,,m K provided the mapping from {(ρ k,u k ),k =,,K} to (m,m,,m K ) is invertible, thus closing the moment system (3.5). For the eamples we will present later, the value of K is at most 3. We first define the phase indicating functions θ = m m m θ = m 4 m m 4 m m + m 3m m m m 3 + m 3 (3.8) When K = 3, define p = u + u + u 3 = (m 5 m m 5 m m + m 4 m 3 m m 4 m m + m 3 m m m 3)/θ p = u u + u u 3 + u u 3 = (m 5 m 3 m m 5 m m + m 4 m m 4m + m 4 m 3 m m m 3)/θ p 3 = u u u 3 = (m 5 m 3 m m 5 m m 4m + m 4 m 3 m m 3 3 m 4m )/θ Then m 6 can be epressed as m 6 = m 5 p m 4 p + m 3 p 3. So the first si equations of (3.5) are closed. When K =, define p = u + u = (m 3 m m m )/θ p = u u = (m m 3 m )/θ Then m 4 can be epressed as m 4 = m 3 p m p and the first four equations of (3.5) are closed. When K =, the first two equations of (3.5) are reduced to the uler-poisson system (3.3). The number of phases at the point (,t) can be identified using the phase indicating functions via, if θ =, Number of phases =, if θ >,θ =, 3, if θ <. 7

8 3. A kinetic scheme for the multi-phase moment system The second order kinetic scheme is used for the multi-phase system (3.5) (see [9]). Suppose the computational domain of is [a,b]. Let the mesh size of be with = (b a)/m, and the time step be t, then gets where and j := a + j, t n := n t, j =,,,M, n =,,,. Multiplying (.) with v l and integrating it over ( j, j+ ) R v (t n,t n+ ), one (m l ) n+ j (m l ) n j + t (f(l+) j+ f (l) j+ = t f (l+) j (m l ) n j = R tn+ j+ j t n f( j+ ) = l ( mn l + mn+ l n ) j t m n l d, vt,v,t)v l dt dv. We can use piecewise linear construction for (ρ k,u k ) to obtain a second order scheme, { ρ() = (ρ)j + Dρ j ( j ), for u() = ū j + Du j ( j ), j / < < j+/, where ū j is chosen as where and f (l) j+ ū j = u j Dρ jdu j ρ j. With f(,v,t) = K k= ρ kδ(v u k ), then K f (l) j+ = ρ j+ k ()u k () l d t t k= L j+ R j+ j+ L j+/ = j+/ t (ū j + Du j) + + tdu j, (u R j+/ ) R j+/ = j+/ + t. + tdu j+ In more eplicit forms, we have = + + K l k= s= [ Cl s (ū k) s j(du k ) l s j ρ k ()u k () l d, (l s) t (ρ ( k) j ( )l s ( L j+ j ) l s) (l s + ) t (Dρ ( k) j ( )l s+ ( L j+ j ) l s+)] K l [ Cl s (ū k) s j+(du k ) l s j+ (l s) t (ρ ( k) j+ ( R j+ j+ ) l s ( )l s) k= s= (l s + ) t (Dρ ( k) j+ ( R j+ j+ ) l s+ ( )l s+)] 8

9 where we choose the slope limiters in the following way, Dρ j = ( ) sgn(ρ j+ ρ j ) + sgn(ρ j ρ j ) { ρj+ ρ j min, ρ j ρ j, ρ } j, Du j = ( ) sgn(u j+ u j ) + sgn(u j u j ) { min u j+ u j ( Dρ j /6ρ j ), u j u j ( + Dρ j /6ρ j ), t The Poisson equation in (3.5) will be solved by the second order finite difference method. Let Vj n, ρ n j,k, n j be the approimation of V ( j,t n ), ρ k ( j,t n ), ( j,t n ) respectively, then the discretization of V = K k= ρ k is as follows : }. V n j+ V n j + V n j = K k= ρ n j,k and n j = V n j+ V n j. 4 Numerical eamples ample 4.. Solve the Schrödinger-Poisson equations (.) with the initial condition (3.) using the SP method where A () is the square root of a δ-function approimated by the cosine kernel (for a recent work on numerical discretization of the delta-function, see [5]): { ψ ɛ π(.5) = δb c() ep(ic ɛ ), where ( + cos ),.5, δc b() = b b b (4.), otherwise, in which C is a constant, and b = O( ɛ). In numerical implementation, we take = b = O(ɛ), which satisfies the mesh size restriction of SP (SP) methods ([]). We solve the Vlasov-Poisson equations (.)-(.) and the Fokker-Planck equations (.7)-(.8) by the particle methods with the measure initial condition f (,v) = δ(.5)δ(v). The Wigner measure of ψ ɛ as ɛ is f (,v) = δ(.5)δ(v) which allows two weak solutions given by the zero smoothing limit of the time reversible particle methods and the zero diffusion limit of the Fokker-Planck equations respectively ([4]). Since ψ ɛ is uniformly bounded in L (R) (the upper bound is ), Theorem.3 implies that the weak limit of a subsequence of W ɛ (t,,ξ), the Wigner transform of the solution of the Schrödinger-Poisson equations with initial condition ψ ɛ, should converge to a weak solution of the Vlasov-Poisson equations. In Figure, one can see that the semiclassical limit solution agrees with the weak solution given by Fokker-Planck zero diffusion limit, while the particle method gives a different weak solution. Table shows that the convergence rate of the Schrödinger- Poisson solution to the zero diffusion Fokker-Planck equations is of order. in ɛ in L norm and order.69 in L norm. 9

10 Moreover, if we choose another regularization of the δ-function, for instance, the linear kernel approimation below, { ψ ɛ = δb l() ep(ic ɛ ), where δl b() = (.5 ),.5, b b b (4.), otherwise, one can see that from Figure, there is no palpable difference between the numerical solutions computed by the cosine kernel approimation and the linear kernel approimation, which gives some evidence that the selection mechanism provided by the semiclassical limit of the Schrödinger-Poisson equations are insensitive to the regularization of the measure-valued initial data. t=.5 t= Figure : ample 4., comparison of the numerical solutions of the electric field by the Vlasov-Poisson equations and the Fokker-Planck equations, both using the particle method, and the Schrödinger-Poisson equations. number N = N = 89, kernel mollifier parameter δ =., t =.. The viscosity coeffcient for the Fokker-Planck equations is η =.. In the Schrödinger-Poisson equations, we use the re-scaled Planck constant ɛ = 5 5, the constant C =, =, t =.5 89 and δb c given in (4.). In the figures, particle solution refers to the solution of the Vlasov-Poisson equation by the particle method. ɛ ɛ D ɛ D L Table : ample 4., at t =, the L and L errors between the electric field ɛ of the Schrödinger-Poisson equations and the electric field D of the zero diffusion limit of the Fokker-Planck equations. D is approimated by solving the Fokker-Planck equation with the parameters η =., N = N = 6384 and t =.. ɛ is solved by SP using =, t =.5 and 6384 δc b given in (4.).

11 .5.4 t=.5 cosine kernel linear kernel.5.4 t=. cosine kernel linear kernel Figure : ample 4., comparison of the numerical solutions of the electric field of the Schrödinger-Poisson equations with the cosine kernel approimation δb c in (4.) and with the linear kernel approimation δb l in (4.). The re-scaled Planck constant is ɛ = 5 5, the constant C =, = and t = ample 4.. In this eample, we carry out numerical simulations on the three and four particles fission processes. We still use the cosine kernel (4.) to approimate the delta function. These problems are are similar to the eamples used in [4] to show numerical instabilities of the particle method. Three particles fission process: The initial condition for the Schrödinger-Poisson equations: ψ ɛ = 4 δc b (.5) + δc b (.5) + 4 δc b (.75) ep(ic ), (4.3) ɛ the initial condition for the Vlasov-Poisson equations and the Fokker-Planck equations: f (,v) = 4 δ(.5)δ(v) + δ(.5)δ(v) + δ(.75)δ(v). (4.4) 4 Four particles fission process: The initial condition for the Schrödinger-Poisson equations: ψ ɛ = 4 δb c 4 (.j) ep(ic ), (4.5) ɛ j= the initial condition for the Vlasov-Poisson equations and the Fokker-Planck equations: f (,v) = 4 δ(.j)δ(v). (4.6) 4 j=

12 Numerical results are given in Figures 3 and 4, which show that the semiclassical limit solutions agree with the weak solutions given by Fokker-Planck zero diffusion limit. Moreover, with a more refined resolution of the delta-function initial data, we did not observe the numerical instability of the particle method pointed out in [4]. t=.5 t= Figure 3: ample 4., three particles fission process. Comparison of the numerical solutions of the electric field by the Vlasov-Poisson equations and the Fokker-Planck equations, both using the particle method, and the Schrödinger-Poisson equations. The parameters and mesh grids are the same as Figure in ample 4.. In the figures, particle solution refers to the solution of the Vlasov-Poisson equation by the particle method. ample 4.3. In this eample, we study the elastic collision case of three particles. We take the initial condition of the Schrödinger-Poisson equations as: ψ ɛ = 4 δc b (.5) + δc b (.5) + ( 4 δc b (.75) ep cos(π) ), (4.7) πɛ in which δb c is defined in (4.). Then the corresponded initial condition of the Vlasov- Poisson equations and the Fokker-Planck equations is given by: ( f (,v) = 4 δ(.5) + δ(.5) + ) δ(.75) δ(v sin(π)). (4.8) 4 Physically, it describes a process where two particles with momentum /4 at =.5 and =.75 move toward the still one at =.5, and after an elastic collision, they echange momentum and begin to move in the opposite directions. In order to minimize the effect of numerical error in discretizing the delta functions of the particles collision process, we use =, t = and the re-scaled Planck constant ɛ = 5 in the computation of the Schrödinger-Poisson equations. The numerical results of the electric field is given in Figure 5, in which convergence of the solution of the Schrödinger-Poisson toward that of the Vlasov-Poission is shown before and after the collision.

13 ..5 t=.5..5 t= Figure 4: ample 4., four particles fission process. Comparison of the numerical solutions of the electric field by the Vlasov-Poisson equations and the Fokker-Planck equations, both using the particle method, and the Schrödinger-Poisson equations. The parameters and mesh grids are the same as Figure in ample 4.. In the figures, particle solution refers to the solution of the Vlasov-Poisson equation by the particle method t= t= Figure 5: ample 4.3, comparison of the numerical solutions of the electric field by the Vlasov-Poisson equations and the Fokker-Planck equations, both using the particle method, and the Schrödinger-Poisson equations in the three particles elastic collision process. Left: t =. before the first collision; right: t =. after the collision. number N = N = 89, kernel mollifier parameter δ =., t =.. The viscosity coefficient for the Fokker-Planck equations is η =.. In the Schrödinger- Poisson equations, we use the re-scaled Planck constant ɛ = 5. In the figures, particle solution refers to the solution of the Vlasov-Poisson equation by the particle method. 3

14 t= t= Figure 5: (Continued) ample 4.3, left: t =.5 before the second collision; right: t =. after the second collision. Note that the two particles collide at = and = for the second time due to the periodic boundary condition. ample 4.4. Solve the Vlasov-Poisson equations (.)-(.), the Fokker-Planck equations (.7)-(.8), the Schrödinger-Poisson equations (.), and the moment equations (3.5) respectively with initial condition ρ () =, u () = sin(π), f = δ(v u ()) and ψ ɛ () = ep ( cos(π) πɛ ). According to [4], the eact solution of the Vlasov-Poisson equations is given by (a,t) = a + sin(πa) sin(t), u((a,t),t) = sin(πa) cos(t), (4.9) ( ) d(a,t) ρ((a,t)) =, (4.) da ((a,t),t) = sin(πa) sin(t) + a + meas(s + ((a,t))) (4.) where a [, ] is a parameter, meas(a) means the Lebesgue measure of the set A, and S + ((a,t)) = {a : (a,t) (a,t) }. Note that before the singularity, S + ((a,t)) = {a : a a }, and the singularity time is given by t c = inf{t : d(a,t) da =, a,t } After a simple calculation, one can get t c =.598 in this eample. Figure 6 shows that both the zero smoothing limit of the time reversible particle method and the diffusion limit of the Fokker-Planck equation converge to the same weak solution, the one given by the semi-classical limit of the Schrödinger-Poisson equations. We point out that the solution of the Schrödinger-Poisson equations is oscillatory, which cannot be seen in the figure but a zoom-in figure shows it which is omitted here. Table shows the convergence rate of the Schrödinger-Poisson is of order.8 in ɛ in L norm and order.5 in L norm. The convergence rate in the L norm is lower because of the singularities (caustics). Since the semiclassical limit is a weak limit, one cannot epect the same strong convergence in the more oscillatory quantities ρ and u. 4

15 We get the multi-valued solution for q.(.)-(.) by solving (3.5) with the second order kinetic scheme. The mesh size and time steps are = /4, t = /5. The thresholds ɛ and ɛ for the indicator functions θ and θ in (3.8) are ɛ = 4, ɛ = 7. Note that for the moment system (3.5), the numerical solution ρ often displays spurious peaks ([9]), whereas the multivalued velocities u, u, and u 3 can usually be well produced. This is because the computation of the multivalued densities involves inverting a matri of the Vandermonde type, which is ill conditioned near the phase boundaries (i.e. where u 3 or u get close to u ). Therefore, to compute ρ, the strategy proposed by Gosse, Jin and Li (formula (6) in [6]) is used. The numerical results are shown in Fig. 7. In this eample, all methods produce the same weak solutions. In particular, it shows that the moment system gives the weak solution described by the semi-classical limit of the Schrödinger-Poisson equations beyond the caustics. t=.3 t= Figure 6: ample 4.4, comparison of the numerical solutions of the electric field by the Vlasov-Poisson equations and the Fokker-Planck equations, both using the particle method, and by the Schrödinger-Poisson equations. number N = N = 89, kernel mollifier parameter δ =., t =.. The viscosity for the Fokker-Plank equation is η =.. In the Schrödinger-Poisson, the re-scaled Planck constant is ɛ = 5 5, the constant C =, = and t = ɛ ɛ D ɛ D L Table : ample 4.4, at t =.5, the L and L errors between the electric field ɛ of the Schrödinger-Poisson equations and the electric field D of the zero diffusion limit of the Fokker-Planck equations. D is approimated by solving the Fokker-Planck equation with the parameters η =., N = N = 6384 and t =.. ɛ is solved by SP using = and t =

16 5 t=. 5 t=.6 ρ 4.5 ρ eact ρ ρ ρ ρ 3 eact (a) Density t=. t=.6 u eact u u.8.8 u eact u u (b) Velocity t=. t=.6.5. moment eact.5. moment eact (c) lectric field Figure 7: ample 4.4, = /4, t = /5, ɛ = 4, ɛ = 7 by using a second order kinetic scheme for the moment system (3.5) with K = 3. The figure shows, from top to bottom, the comparison of the density, velocity and the electric field between the moment method and the eact solution (solid line). Left: at t =. before the caustic occurs; right: at the critical time t = t c =.6. 6

17 5 t=.3 5 t=.5 ρ 4.5 ρ ρ ρ eact ρ ρ ρ ρ 3 eact (a) Density t=.3 t=.5 u u u u.8 u 3.8 u 3.6 eact.6 eact u u (b) Velocity t=.3 t=.5.5. moment eact.5. moment eact (c) lectric field Figure 7: (Continued) ample 4.4, left: at t =.3 after the caustic occurs; right: at t =.5. 7

18 5 Conclusion Through several numerical eamples, we show that the semiclassical limit of the Schrödinger- Poisson equations provides a selection principle for the weak solution of the Vlasov- Poisson equations, which agrees with the diffusion limit of the Fokker-Planck equation. This limit is not sensitive for the discretization of the measure-valued initial data. Furthermore, we show that the semiclassical limit of the Schrödinger-Poission equations gives the weak solution to the uler-poission equations governed by a moment system introduced in [9, ]. In the future we will study these problems in higher space dimensions. Acknowledgement We thank Prof. Yui Zheng for useful discussion. References [] F. Bouchut, S. Jin and X.T. Li, Numerical Approimations of Pressureless and Isothermal Gas Dynamics, SIAM J. Num. Anal. 4 (3), [] W.Z. Bao, S. Jin and P.A. Markowich, On time-spliting spectral approimations for the Schrödinger equation in the semiclassical regime. J. Comp. Phys. 75 (), [3] W.Z. Bao, S. Jin and P.A. Markowich, Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrodinger quations in the Semi-clasical Regimes. SIAM J. Sci. Comp. 5 (3), 7-64 (elctronic). [4] G.H. Cottet and P.A. Raviart, methods for the D Vlasov-Poisson equations, SIAM J. Num. Anal. (984) [5] B. ngquist, A.K. Tornberg and R. Tsai, Discretization of Dirac Delta Functions in Level Set Methods. J. Comput. Phys. 7 (5), no., 8 5. [6] L. Gosse, S. Jin and X.T. Li, On Two Moment Systems for Computing Multiphase Semiclassical Limits of the Schrödinger quation. Math. Model Methods Appl. Sci. 3, No. (3), [7] L. Gosse and N.J. Mauser, Multiphase semiclassical approimation of an electron in a one-dimensional crystalline lattice C III. From ab initio models to WKB for Schrödinger-Poisson, J. Comp. Phys. (6), [8] P. Gerard, P.A. Markowich, N.J. Mauser, F. Poupaud, Homogenization Limits and Wigner Transforms. Comm. Pure Appl. Math. 5 (4) (997),

19 [9] S. Jin and X.T. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 8 (3), [] X.T. Li, J.G. Wohlbier, S. Jin and J.H. Booske, An ulerian method for computing multi-valued solutions of the uler-poisson equations and applications to wave breaking in klystrons, Phys. Rev.. 7 (4), 65. [] P.-L. Lions and T. Paul, Sur les measure de Wigner. Rev. Mat. Iberoamericana 9 (993), no. 3, [] H. Liu and Z.M. Wang, A field space-based level set method for computing multivalued solutions to D uler-poisson equations, J. Comp. Phys., to appear. [3] P.A. Markowich and N.J. Mauser, The classical limit of a self-consistent quantum- Vlasov equation in 3D. Math. Models and Methods in Applied Sciences, Vol. 3, No. (993), 9-4. [4] A.J. Majda, G. Majda and Y.X. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations: Temporal development and non-unique weak solutions in the single component case. Physica D, 74(994), [5] C. Sparber, P.A. Markowich and N.J. Mauser, Wigner functions vs. WKB methods in multivalued geometrical optics, Asympt. Anal. 33 (3), [6]. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 4 (93), [7] P. Zhang, Y.X. Zheng and N.J. Mauser, The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. on Pure and Applied Mathematics, Vol.LV (), [8] Y.X. Zheng and A. Majda, istence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. on Pure and Applied Math. Vol. XLVII (994),