Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation

https://doi.org/10.1016/j.cpc.2012.01.006Get rights and content

Abstract

In this paper, four alternating direction implicit (ADI) schemes are presented for solving two-dimensional cubic nonlinear Schrödinger equations. Firstly, we give a Crank–Nicolson ADI scheme and a linearized ADI scheme both with accuracy O(Δt2+h2), with the same method, use fourth-order Padé compact difference approximation for the spatial discretization; two HOC-ADI schemes with accuracy O(Δt2+h4) are given. The two linearized ADI schemes apply extrapolation technique to the real coefficient of the nonlinear term to avoid iterating to solve. Unconditionally stable character is verified by linear Fourier analysis. The solution procedure consists of a number of tridiagonal matrix equations which make the computation cost effective. Numerical experiments are conducted to demonstrate the efficiency and accuracy, and linearized ADI schemes show less computational cost. All schemes given in this paper also can be used for two-dimensional linear Schrödinger equations.

Introduction

In this paper, we consider the following two-dimensional cubic nonlinear Schrödinger equationiut+a2ux2+b2uy2+q|u|2u+v(x,y)u=0,(x,y,t)Ω×(0,T], with initial conditionu(x,y,0)=u0(x,y),(x,y)Ω, and Dirichlet boundary conditionu(x,y,t)=g(x,y,t),(x,y,t)Ω×(0,T], where Ω is a rectangular domain in R2, we suppose Ω=[L1,L2]×[L3,L4], ∂Ω is the boundary of Ω, (0,T] is the time interval, a and b and q are real constants, u0(x,y) and g(x,y,t) are given sufficiently smooth functions, v(x,y) is an arbitrary potential function, u(x,y,t) is an unknown complex function, i=1.

The cubic nonlinear Schrödinger equation with attractive or repulsive nonlinearity and a potential is used for the dynamics of a dilute gas Bose–Einstein condensate (BEC) [1]. So the equation is often referred to as the Gross–Pitaevskii equation. BECs are examples of macroscopic quantum phenomena which display phase coherence [2], [3], [4], [5].

Numerical approximations for Schrödinger equation have drawn much attention. Chang et al. [6] presented two linearized Crank–Nicolson-type schemes for one-dimensional nonlinear Schrödinger equation, and [7], [8] introduced split-step method. Dehghan and Shokri [9] proposed a numerical scheme to solve the two-dimensional Schrödinger equation using collocation points and approximating the solution by multiquadrics and the Thin Plate Splines Radial Basis Function. [10] employed compact boundary value method which has fourth-order accuracy both in space and in time. A symmetrical multi-point difference formula was given in Ref. [11] for time-independent two-dimensional Schrödinger equation, [12] generalized the Numerov method, J. Vigo-Aguiar and H. Ramos [13] presented a variable-step Numerov method. [14] reviewed the multistep methods for the numerical solution of the radial Schrödinger equation.

ADI method [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], which replaces the solution of multidimensional problems by sequences of one-dimensional cases, only needs to solve tridiagonal linear systems, and the resulting schemes are unconditionally stable, and have received much attention in recent years. HOC scheme utilizes grid points only adjacent to the node which the differences are taken about, so the HOC scheme with high order saves much memory. HOC-ADI method has shown its efficiency and accuracy in [22], [23], [24]. [23] introduced HOC-ADI method for parabolic equations, and [20] presented a new ADI method for three-dimensional parabolic equations. For unsteady convection–diffusion equations, an HOC-ADI scheme was given in [15], then a high-order ADI scheme with better phase and amplitude error characteristics was proposed in [16]; Tian and Ge [24] gave a new HOC-ADI scheme. ADI scheme for nonlinear diffusion equations was proposed in [18]. But for Schrödinger equation, there are not so much results; Tian and Yu [22] presented HOC-ADI scheme for two-dimensional unsteady Schrödinger equation.

In this paper, we first propose a Crank–Nicolson ADI scheme and a linearized ADI scheme both with accuracy O(Δt2+h2) for solving two-dimensional cubic nonlinear Schrödinger equation (1). For Crank–Nicolson ADI scheme, Crank–Nicolson method is used for the temporal discretization, it needs a nonlinear iterative algorithm to solve the system of the nonlinear equations. With the same method, by applying extrapolation technique to the coefficient of the nonlinear term, we get a linearized ADI scheme, which avoids the iterative process. While for HOC-ADI schemes, the fourth-order Padé compact difference approximations for the spatial discretization are used, and two HOC-ADI schemes are derived. The solution procedures of the schemes mentioned above consist of a number of tridiagonal matrix equations which can be solved by Thomas algorithm.

We find that the four ADI schemes in this paper also can be used to do with two-dimensional linear Schrödinger equations, we just need to exclude the nonlinear term in the schemes and do some corresponding changes, without the nonlinear term, we donʼt need to iterate to solve.

For numerical experiments, we consider both linear and nonlinear problems, but we focus on nonlinear parts, linear parts show that our schemes also play well for linear problems.

The paper is organized as follows. In Section 2 we give two ADI schemes with accuracy O(Δt2+h2) for the cubic nonlinear Schrödinger equation (1). In Section 3, two schemes given in Section 2 are extended to HOC-ADI schemes by fourth-order Padé approximation for spatial discretization with accuracy O(Δt2+h4). Linear Fourier stability of the schemes is analyzed in Section 4. Numerical experiments for several problems are presented in Section 5. Conclusions are given in Section 6.

Section snippets

Two second-order ADI schemes for the cubic nonlinear Schrödinger equation

In this section, we first derive a Crank–Nicolson ADI scheme, Crank–Nicolson method is used for the temporal discretization, it needs a nonlinear iterative algorithm to solve the system of the nonlinear equations. With the same method, by applying extrapolation technique to the coefficient of the nonlinear term, we get a linearized ADI scheme, which avoids the iterative process, both schemes are O(Δt2+h2).

The domain is divided by a uniform mesh in each direction. Let hx=L2L1Nx+1, xj=L1+jhx,j=0,

Two HOC-ADI schemes for the cubic nonlinear Schrödinger equation

In this section, the two ADI schemes given in Section 2 are extended, with fourth-order Padé compact difference approximation for the spatial discretization, we get two HOC-ADI schemes with accuracy O(Δt2+h4).

Stability analysis

In this section, we consider the stability of the schemes given in this paper using a linear stability analysis method. We just present the analysis process of the linearized HOC-ADI scheme (48), (49) for the cubic nonlinear Schrödinger equation. The other three schemes are about the same.

For the linear stability analysis, we set ωj,kn=c, where c is a constant. The scheme (48), (49) can be rewritten as(1+δx212irxa2δx2)[(1+δy212iryb2δy2)ciΔt2(1+δy212)]uj,kn+1=(1+δx212+irxa2δx2)[(1+δy212+iryb2δ

Numerical experiment

In this section, we give some numerical results for the two-dimensional problems, all the numerical experiments are obtained by Matlab 7.5 on a DELL D630 computer with T7250 CPU and 1 Gbyte of memory.

Problems 1 and 2 are for linear problems, we want to show that our schemes also can be used for linear problems. When applying the four schemes given in this paper for linear problems, we just need to exclude the nonlinear term in the schemes and do some corresponding changes, at this time, ADI-I

Conclusion

In this paper, four alternating direction implicit (ADI) schemes are presented for solving two-dimensional cubic nonlinear Schrödinger equations. According to numerical experiments, the linearized HOC-ADI schemes considered to be the most effective in terms of accuracy and computational cost, use extrapolation technique to the real coefficient q|u|2 of the nonlinear term and apply fourth-order Padé compact difference approximation for the spatial discretization. This method, which only involves

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