A numerical study of the Gaussian beam methods

for one-dimensional Schrodinger-Poisson equations ∗

Shi Jin†, Hao Wu‡, and Xu Yang§

March 17, 2009

Abstract

As an important model in quantum semiconductor devices, theSchrodinger-Poisson equations have generated widespread interests inboth analysis and numerical simulations in recent years. In this paper,we present Gaussian beam methods for the numerical simulation of theone-dimensional Schrodinger-Poisson equations. The Gaussian beammethods for high frequency waves outperform the geometrical opticsmethod in that the former are accurate even around caustics. Thepurposes of the paper are first to develop the Gaussian beam methods,based on our previous methods for the linear Schrodinger equation, forthe Schrodinger-Poisson equations, and then check their validity forthis weakly-nonlinear system.

Key words: Schrodinger-Poisson equations, Gaussian beam meth-ods, Vlasov-Poisson equations

1 Introduction

The main purpose of this paper is to extend our Gaussian beam method[22], developed for the linear Schrodinger equation, to the one-dimensional

∗This work was partially supported by NSF grant No. DMS-0608720, NSF FRG grantDMS-0757285, NSFC Projects 10676017, and the National Basic Research Program ofChina under the grant 2005CB321701. SJ was also supported by a Van Vleck Distin-guished Research Prize from University of Wisconsin-Madison.

†Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA(jin@math.wisc.edu)

‡Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China(hwu04@mails.tsinghua.edu.cn)

§Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. Cur-rent address: Program in Applied and Computational Mathematics, Princeton University,Princeton, NJ 08544, USA. (xuyang@math.princeton.edu)

1

nonlinear Schrodinger-Poisson equations

iε∂tΨε = −ε2

2∂xxΨε + V εΨε, x ∈ R, t ≥ 0, (1.1)

∂xxV ε = b(x)− c |Ψε(t, x)|2 , Eε = ∂xV ε, (1.2)

subject to the WKB initial condition

Ψε(0, x) = A0(x)eiS0(x)/ε. (1.3)

Here Ψε = Ψε(t, x) is the highly oscillatory wave function of wave lengthO(ε) (in the so-called semiclassical regime where the re-scaled Plank con-stant ε is small). The electric potential V ε = V ε(t, x) interacts with thewave function Ψε in a self-consistent way through the Schrodinger equation(1.1) and the Poisson equation (1.2). In the poisson equation (1.2), b(x) ≥ 0denotes the fixed positive charged background. The constant c could be ±1,corresponding to focusing (‘+’) or defocusing (‘-’) potential respectively.

The Schrodinger-Poisson equations are a mean-field model for the linearN -particle Schrodinger equation with Coulomb potential [6, 7, 13], which isbased on the Pauli’s exclusion principle and the molecular chaos assumption.It is widely used in quantum semiconductor devices modeling [36] and thequantum transport theory [2, 1].

The direct simulation of the Schrodinger-Poisson equations is expensivesince the wave length O(ε) is extremely small in the semiclassical regime.The standard time-splitting spectral method [3, 4] and its adaptive version[5] need the mesh size to be of O(ε) and the time step to be of O(1) to capturethe correct physical observables. The finite difference methods [34, 35] areeven worse since the mesh size and the time step are restricted to be o(ε).

One efficient alternative approach is to study the semiclassical limit of theSchrodinger-Poisson equations. When taking the rescaled Planck constantε → 0 , one can derive the Vlasov-Poisson equations [28, 33, 48] in the phasespace

∂tf + ξ∂xf − ∂xV ∂ξf = 0 x, ξ ∈ R, t ≥ 0, (1.4)

∂xxV = b(x)− c

∫ ∞

−∞f(t, x, ξ)dξ, E = ∂xV, (1.5)

or the Euler-Poisson equations in the physical space [30]

∂tρ + ∂x(ρu) = 0, x ∈ R, t ≥ 0, (1.6)∂t(ρu) + ∂x(ρu2) = −ρ∂xV, (1.7)

∂xxV = b(x)− cρ. (1.8)

There are many papers discussing mathematical analysis and numericalmethods for those equations [8, 11, 12, 24, 37], such as the existence and

2

uniqueness of suitable weak solution to Vlasov-Poisson equations [9, 19, 29,32, 49] and numerical methods for capturing the multi-valued solutions tothe Euler-Poisson equations [14, 27, 31].

A well-known drawback to the semiclassical approach is that it can notgive accurate solutions around caustics. The Gaussian beam methods, de-veloped for the high frequency linear waves [43, 46, 45, 39, 40, 22, 23, 25, 26]and also in the setting of quantum mechanics [15, 16, 17], on the otherhand, are efficient asymptotic methods that give accurate solutions evenaround caustics ([44]). The key idea of the Gaussian beam methods is tocomplexify the phase function S(t, x) off the beam center. Moreover, theimaginary part of S(t, x) should be chosen delicately so that the solutiondecays exponentially. In this paper, we extend the Gaussian beam methods,proposed previously by the authors [22] for the linear Schrodinger equa-tion, to the weakly nonlinear Schrodinger-Poisson equation (1.1)-(1.2). Theoriginal Gaussian beam methods were developed for linear high frequencywaves, based on the linear superposition principle. It is of great mathe-matical and numerical interests to see if the methods can be extended to(at least weakly-) nonlinear problems. In this paper, we propose a class ofGaussian beam methods, in both Lagrangian and Eulerian frameworks, forthe Schrodinger-Poisson equations, and check their validity for this weaklynonlinear system.

Our numerical studies show that the Gaussian beam methods can indeedbe extended to this one-dimensional, weakly nonlinear system. Indeed, con-vergent results can be observed, even around caustics, for both the focusingand defocusing cases, when ε → 0.

A main feature of our Eulerian Gaussian beam method is that only afew (complex-valued) Liouville equations like (1.4) are solved. Thus thecomputational methods are similar to that of geometrical optics [21, 10, 20].As a result, the local level-set techniques [41, 42, 38] can also be applied tofurther reduce the computational cost.

The paper is organized as follows. In Section 2 we systematically presentthe Gaussian Beam methods, in both Lagrangian and Eulerian frameworks,for solving (1.1)-(1.2). We conduct numerical experiments in Section 3 toverify the validity and convergence of the numerical methods, and give someconclusive remarks in Section 4.

2 The Gaussian beam method

In this section, we introduce the Gaussian beam method in details for solv-ing the Schrodinger-Poisson system (1.1)-(1.2). The main procedure is tosolve the Poisson equation (1.2) and the Schrodinger equation (1.1) itera-tively. Suppose the solution Ψε(tn, x) at time tn are given, then the solutionΨε(tn+1, x) at time tn+1 = tn + ∆t is computed as follows:

3

• Step 1. We solve the Poisson equation (1.2) first to get the potentialV ε,n(x) by some Poisson solvers.

• Step 2. With V ,ε,n(x) given in step 1, Ψε(tn+1, x) is computed by solv-ing the Schrodinger equation (1.1) using the Gaussian beam methodon a fixed time interval ∆t.

The Poisson solvers and the Gaussian beam method are given in the follow-ing Section 2.1 and 2.2 respectively.

2.1 Poisson solvers

We use the finite difference method to solve the Poisson equation (1.2)equipped with general boundary conditions (Dirichlet, Neumann etc). Sup-pose the domain of x is [xl, xr] and the mesh point is xj = xl + j∆x, j =1, · · · , N where ∆x is the mesh size and xr = xl + (N + 1)∆x, then the po-tential V ε,n(x) and the electric field En,ε(x) are solved by the linear system:

V ε,nj+1 − 2V ε,n

j + V ε,nj−1

∆x2= bj − c

∣∣∣Ψn,εj

∣∣∣2, En,ε

j =V ε,n

j+1 − V ε,nj−1

2∆x,

where V ε,nj , En,ε

j , bj , Ψn,εj are the approximation of V ε(tn, xj), Eε(tn, xj),

b(xj), Ψε(tn, xj) respectively.If the periodic boundary condition is considered for (1.2), the pseudo

spectral method will be applied for faster performance. By using the fourier

transform F , we can get F(V ε,n)k = (xr − xl

2πk)2

(F(b− c |Ψn,ε|2)k

), and

we set F(V ε,n)0 = 0 for each n. Inverse fourier transform follows afterupdating F(b−c |Ψn,ε|2)k and F(V ε,n)k. Remark that the periodic boundarycondition forces the following compatibility constraint:

∫ xr

xl

b(x)− c |Ψε(t, x)|2 dx = 0,

which explains why we could set F(V ε,n)0 = 0 for each n.

2.2 Gaussian beam method

In this subsection we describe how to solve (1.1) on a fixed time interval ∆tby the Gaussian beam methods [22] and discuss the issue of constructingthe potential V ε,n+1(x) from the summation of the Gaussian beams. Sincethe potential V ε,n(x) is given for computing Ψε(tn+1, x), we denote it as Vfor simplicity.

4

2.2.1 Lagrangian formulation

The Lagrangian Gaussian beam ansatz is

ϕεla(t, x, y0) = A(t, y)eiT (t,x,y)/ε,

where y = y(t, y0) and T (t, x, y) is given by

T (t, x, y) = S(t, y) + p(t, y)(x− y) +12M(t, y)(x− y)2,

here S, p ∈ R, A,M ∈ C. The time evolution equations of these quantitiescan be written as a set of ODEs ([22])

dy

dt= p, (2.1)

dp

dt= −∂yV, (2.2)

dM

dt= −M2 − ∂yyV, (2.3)

dS

dt=

12|p|2 − V, (2.4)

dA

dt= −1

2(Tr(M)

)A. (2.5)

The Lagrangian Gaussian beam summation solution to the Schrodingerequation (1.1) at time tn+1 is constructed as

Φεla(t

n+1, x) =∫

R

(1

2πε

) 12

rθ(x− y(tn+1, yn))ϕεla(t

n+1, x, yn)dyn. (2.6)

where yn = y(tn) and rθ ∈ C∞0 (R), rθ ≥ 0 is a truncation function with

rθ ≡ 1 in a ball of radius θ > 0 about the origin. The initial conditions att = 0 come from the approximation of the initial condition (1.3) ([18]):

y(tn, y0) = y0, (2.7)p(tn, y0) = ∂xS0(y0), (2.8)

M(tn, y0) = ∂xxS0(y0) + iI, (2.9)S(tn, y0) = S0(y0), (2.10)A(tn, y0) = A0(y0). (2.11)

The discrete form of (2.6) is given as

Φεla(t

n+1, x) =Nyn∑

j=1

(1

2πε

) 12

rθ(x− y(tn+1, yjn))ϕε

la(tn+1, x, yj

n)∆yn, (2.12)

where yjn’s are the Lagrangian mesh points, and Nyn is the number of the

beams at time tn.

5

For the evolution from tn+1 to tn+2, we need to construct V ε,n+1, ∂yVε,n+1

and ∂2yV ε,n+1 from the beam summation solution (2.6) which is done by the

following procedure:

• Step 1, compute (2.6) on a fixed gird of x, then solve V ε,n+1(x) andEn+1,ε(x) = ∂xV ε,n+1 from (1.2) by some Poisson solver given in Sec-tion 2.1. Interpolate V ε,n+1(x) and ∂xV ε,n+1 to get their values onthe Lagrangian mesh points yj

n. Note that V ε,n+1(x) and ∂xV ε,n+1

are the integration functions of the density |Ψε|2 which implies thatthey are oscillatory but with very small oscillatory magnitudes (forexample, see Figure 6 of [19]). This feature implies that if one imple-ments the interpolation on a coarse mesh gird, although the numericalresult could not capture the small scale oscillations, it provides goodapproximation for the envelope of the solution, which in fact is veryclose to the true solution due to the oscillation amplitude is very small.

• Step 2, directly compute ∂yyVε,n+1 for each yj

n using the Poisson equa-tion (1.2), i.e.

∂yyVε,n+1 = b(y)− c |Φε

la|2 .

The reason we have to compute ∂yyVε,n+1 directly instead of using

interpolation is that, the oscillatory magnitude of ∂yyVε,n+1 is compa-

rable to that of the density |Ψε|2.Note that we have two sets of meshes here: one is the Eulerian mesh for x,and the other is the Lagrangian mesh for y. The values exchanged betweenthese two meshes are through high order interpolation. This inevitablycomplicates the algorithms and the notations which could be avoided by theEulerian formulation below.

2.2.2 Eulerian formulation

For the Eulerian formulation of the Gaussian beam approximation, we have

ϕεeu(t, x, y, ξ) = A(t, y, ξ)eiT (t,x,y,ξ)/ε,

withT (t, x, y, ξ) = S(t, y, ξ) + ξ(x− y) +

12M(t, y, ξ)(x− y)2.

Define the linear Liouville operator as

L = ∂t + ξ∂y − ∂yV ∂ξ,

then the Eulerian Gaussian beam method is constructed by solving the fol-lowing Liouville equations:

Lφ = 0, (2.13)

LS =12|ξ|2 − V, (2.14)

Lf = 0. (2.15)

6

where φ ∈ C, and S, f ∈ R. M and amplitude A are computed by

M(t, y, ξ) = −∂yφ(∂ξφ)−1,

A(t, y, ξ) = (det(∂ξφ)−1f)1/2.

The Eulerian Gaussian beam summation solution to the Schrodingerequation(1.1) is constructed via

Φεeu(tn+1, x) =

∫

R

∫

R

(1

2πε

) 12

rθ(x− y)ϕεeu(tn, x, y, ξ)δ(Re[φ])dξdy, (2.16)

in which δ is the Dirac delta function.According to [22], the initial data for (2.13)-(2.15) are given by

φ(0, y, ξ) = −iy + (ξ − ∂yS0(y)), (2.17)S(0, y, ξ) = S0(y), (2.18)f(0, y, ξ) = A2

0(y). (2.19)

The construction of V ε,n+1, ∂yVε,n+1 and ∂yyV

ε,n+1 from (2.16) is sim-ple in the Eulerian framework since every quantity is computed in a fixedgrid. All we need to do is to take the same mesh for x as that for y andsolve (1.2) by some Poisson solver given in Section 2.1. For the numericalcomputation of (2.16), the discretized delta function integration method [47]is recommended since it avoids the difficulty of computing singular integrals[22].

3 The numerical examples

In this section, we will present both focusing and defocusing numerical re-sults of the Schrodinger-Poisson equations (1.1)-(1.2) by using the Gaussianbeam method proposed in section 2. In our computations, the initial con-dition is always chosen in the WKB form (1.3). We compute the referencesolution Ψε using the Strang splitting spectral method [3, 4, 5] with meshsize ∆x and time step ∆t small enough. We always take a large computa-tional domain such that the periodic boundary condition does not introducea significant error to the whole problem. To diminish the cut-off error, thetruncation parameter θ appears in (2.16) is picked fairly large as we dis-cussed in [22]. We will denote solutions obtained by (2.16) as Φε

GB.

Example 1. Consider the 1D Schrodinger-Poisson equation on computa-tional domain [−0.5, 0.5] with a focusing potential

Vxx =√

2π

10− |ψ(x)|2 .

7

(ε,Ny) ( 1256 , 128) ( 1

1024 , 256) ( 14096 , 512)

l1 error 1.12× 10−2 3.93× 10−3 9.22× 10−4

l2 error 4.09× 10−2 1.47× 10−2 3.80× 10−3

l∞ error 3.09× 10−1 1.09× 10−1 3.09× 10−2

Table 1: the l1, l2 and l∞ errors of the wave amplitude for ε =1

256 , 11024 , 1

4096 for Example 1.

(ε,Ny) ( 1256 , 128) ( 1

1024 , 256) ( 14096 , 512)

l1 error 8.16× 10−3 2.60× 10−3 8.35× 10−4

l2 error 3.20× 10−2 9.24× 10−3 2.94× 10−3

l∞ error 1.74× 10−1 5.30× 10−2 1.95× 10−2

Table 2: the l1, l2 and l∞ errors of the wave amplitude for ε =1

256 , 11024 , 1

4096 for Example 2.

The initial conditions are given by

A0(x) = e−25x2, S0(x) =

1π

cos(x).

At time t = 0.4, we output the l1, l2 and l∞ error of the wave amplitudein Table 1. We can see the convergence rate in ε is of order 0.9006 in thel1 norm. Here the number of Gaussian beams Ny is optimized with ε, seediscussion in [22]. In Figure 1, we plot the wave amplitude and absoluteerror for different ε.

Example 2. Consider the 1D defocusing Schrodinger-Poisson equation

Vxx = −√

2π

10+ |ψ(x)|2 .

with same boundary condition and initial data as in Example 1. At timet = 0.4, the l1, l2 and l∞ error of the wave amplitude are also given in Table2. We can see the convergence rate in ε is of order 0.8221 in the l1 norm.The wave amplitude and absolute error for different ε are plotted in Figure2.

Example 3. Consider the same defocusing potential as in Example 2. Theinitial conditions are changed to be

A0(x) = e−25x2, S0(x) = 0.

In this example, we examine the convergence when there is no caustics. This,along with the above two caustic examples, will show that the Gaussian

8

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(a) ε =1

256

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(b) ε =1

1024

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(c) ε =1

4096

Figure 1: Example 1, the Schrodinger-Poisson solution |Ψε| versus the Gaus-sian beams solution |Φε

eu| for ε = 1256 , 1

1024 , 14096 . The left figures are the

comparisons of the wave amplitude at t = 0.4; the right figures plot theerrors ||Φε

GB| − |Ψε||.

9

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(a) ε =1

256

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(b) ε =1

1024

−0.1 −0.05 0 0.05 0.1

0

0.5

1

1.5

2

2.5

x

|Ψε|

|ΦεGB

|

−0.1 −0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

||Ψε|−|ΦεGB

||

(c) ε =1

4096

Figure 2: Example 2, the Schrodinger-Poisson solution |Ψε| versus the Gaus-sian beams solution |Φε

eu| for ε = 1256 , 1

1024 , 14096 . The left figures are the

comparisons of the wave amplitude at t = 0.4; the right figures plot theerrors ||Φε

GB| − |Ψε||.

10

(ε,Ny) ( 1256 , 16) ( 1

1024 , 32) ( 14096 , 64)

l1 error 3.31× 10−2 9.40× 10−3 2.45× 10−3

l2 error 4.26× 10−2 1.25× 10−2 3.30× 10−3

l∞ error 1.05× 10−1 3.28× 10−2 8.92× 10−3

Table 3: the l1, l2 and l∞ errors of the wave amplitude for ε =1

256 , 11024 , 1

4096 for Example 3.

beam method we propose here gives satisfactory results from the numericalpoint of view. We evolve the solution to time t = 0.5, and the l1, l2 and l∞

errors of the wave amplitude are given in Table 3. The convergence rate inε is of order 0.9389 in the l1 norm. In Figure 3, the wave amplitude andabsolute error are plotted for different ε. We remark that since the solutionis not as oscillatory as in the last two examples, only a very small numberof beams are needed to get the accurate solution.

4 Conclusion

In this paper, we extended the Gaussian beam methods, in both Lagrangianand Eulerian framework, to the one dimensional Schrodinger-Poisson equa-tions. Using the method introduced in [22], the Schodinger equation (1.1)can be directly simulated in each time step. For the poisson equation (1.2),the potential function and its derivatives only need to be constructed atmesh points. Such a setup makes the Gaussian beam method valid and effi-cient for solving the one-dimensional Schrodinger-Poisson equations. Severalexamples have been given to confirm the convergence and accuracy.

It will be of interest to study the method in higher space dimension,which will be the subject of our future study.

Acknowledgement

Xu Yang is grateful to Prof. James Rossmanith for useful discussions onthe compatibility condition of the force when using the spectral method tosolve the Poisson equation.

References

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11

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

x

|Ψε|

|ΦεGB

|

−0.5 0 0.50

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4096

Figure 3: Example 3, the Schrodinger-Poisson solution |Ψε| versus the Gaus-sian beams solution |Φε

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12

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