Elsevier

Applied Mathematics and Computation

Volume 297, 15 March 2017, Pages 131-144
Applied Mathematics and Computation

An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation

https://doi.org/10.1016/j.amc.2016.10.037Get rights and content

Abstract

We propose an efficient time-splitting compact finite difference method for Gross–Pitaevskii equation (GPE). In our method, we solve the GPE in time with time-splitting technique and in space by the compact finite difference method. To find the numerical solution of the resulting discretized system in one-dimension (1D), two-dimensions (2D) and three-dimensions (3D), we apply the fast discrete Sine transform in 1D, 2D and 3D respectively and get an efficient solver for the discretized system in 1D, 2D and 3D, respectively. Our numerical algorithm at every time step does not need linear-algebraic-equations-solver, whose computation cost will be much higher when the spatial dimension increases. The method also has the merit that it is unconditionally stable and conservative. Moreover the method can achieve spectral-like accuracy in space when high-order compact finite difference method is applied. Extensive numerical tests for the GPE in 1D, 2D and 3D are presented to demonstrate the power and accuracy of the proposed numerical method.

Introduction

Gross–Pitaevskii equation (GPE) is one of the most important mathematical models in Bose–Einstein condensation [21]. It is one of nonlinear Schrödinger-type equations (NLSE) as well. The NLSE is a classical field equation and has many principal applications such as study of the propagation of light in nonlinear optical fibres and planar waveguides [19], [24] and investigation of Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps [21].

Numerous numerical methods are proposed for solving NLSE in the past years [9], [15], [31], [32]. For example, Weideman and Herbst proposed a split-step Fourier pseudospectral method [33]. Sulem studied several numerical scheme including spectral method to study the singular solutions to the two-dimensional cubic NLSE [25]. Taha and Ablowitz compared eight numerical methods for one-dimensional NLSE in [26]. For more about the numerical methods for the NLSE and GPE, see the review papers by Bao et al. [2], [3], [4] and by Fairweather and Khebchareon [16], respectively.

Compact finite difference method has also been proposed to study Schrödinger equation [8], [20], [34], [35], nonlinear Klein–Gordon equation [13], and Gross–Pitaevskii equations [29]. It has been known for almost fifty years [6]. Its implementation as finite difference schemes on approximating partial differential equations began in the early 1970s for some fluid mechanics problems [5], [17], [27]. Since that time, it has become popular and has been applied into dealing with many physical problems such as wall-bounded flows described by the Navier–Stokes equations [1], [23], supersonic boundary-layer flow in large-eddy simulation, and also the scattering of electromagnetic waves [7], [18], [22], [23], [28], [36].

Despite its popularity, compact finite difference method can incur higher computation cost when it is applied in solving the higher-dimensional partial differential equations. This is because it usually need larger linear-algebraic-equations-solver in higher-dimensions for the discretized system. We note that recent appearance of faster and more powerful computers as well as the development of fast algorithms for linear-algebraic-equations-solver may remove some burden of it.

In this paper, we show that compact finite difference scheme may not need to seek any linear-algebraic-equations-solver for the discretized system. As an example, we propose an efficient time-splitting compact finite difference method for the GPE. In the method, we numerically solve the GPE in time with splitting technique and in space by the compact finite difference method. The time-splitting method is a quite efficient and popular numerical method for time-dependant partial differential equation. It has been applied to solve parabolic equations [10], [11], [12], advection–diffusion equation [12], transport equation [14], and Schrödinger-type equation [31]. For the resulting discretized system in one-dimensional, two-dimensional and three-dimensional space, we show that our numerical algorithm at every time step does not need linear-algebraic-equations-solver. Instead we apply the fast discrete Sine transform in 1D, 2D and 3D, respectively, and get an efficient solver for the resulting discretized system in 1D, 2D and 3D, respectively. Except for its fast implementation, the method also has the merit that it is unconditionally stable and conservative. Moreover the method can achieve spectral-like accuracy in space when high-order compact finite difference method is applied.

The paper is organized as follows. In Section 2, we introduce the time-dependent GPE. In Section 3, we propose a time-splitting compact finite difference method for the GPE. Detailed numerical algorithms in 1D, 2D and 3D are provided respectively. In Section 4, Extensive numerical tests for the GPE in 1D, 2D and 3D are presented to demonstrate the power and accuracy of the proposed numerical method. Finally, some conclusions and remarks are drawn in Section 5.

Section snippets

The Gross–Pitaevskii equation

For N bosons in an external trap potential at very low temperatures, the condensate can be described using mean-field theory. All the particles in the condensate have the same wave function ψ(x, t), the condensate is then described by the following time-dependent GPE [2], [21] : iψt=(22m+V+g|ψ|2)ψ,with m is the atom mass. Usually the external trap potential V is the harmonic trapping potential m2(ωx2x2+ωy2y2+ωz2z2) with ωx, ωy and ωz being the trap frequency in x-direction, y-direction

A time splitting compact finite difference method

In this section, we present a unconditionally stable and compact finite difference method for the GPE (2.5). In numerical computation, we always truncate the problem into a bounded domain Ω with homogeneous Dirichlet boundary conditions: iψt=[12+V+β|ψ|2]ψ,ψ(x,t)=0,xΓ=Ω,t0,ψ(x,0)=ψ0(x),xΩ.where ψ=ψ(x,t),φ=φ(x,t). In 1D, x=(x) and Ω=(a,b). In 2D, x=(x,y) and Ω=(a,b)×(c,d). In 3D, x=(x,y,z) and Ω=(a,b)×(c,d)×(e,f). Constants |a|, b, |c|, d, |e| and f are chosen sufficiently large.

Numerical results

In this section, we present one-dimensional, two-dimensional and three-dimensional numerical tests for the proposed splitting compact finite difference (SCFD) method. In all of our numerical computation, we take the time step t=0.0001. In addition, we also compare our proposed SCFD method with splitting Fourier spectral (SFS) method, which is perhaps one of the most popular methods for solving time-dependant GPE [31], [33]. The basic steps of the SCFD method are similar to those of the SFS

Conclusions and remarks

We propose an efficient time-splitting compact finite difference method for GPE. Our extensive numerical tests for the GPE in 1D, 2D and 3D have shown spectral-like accuracy of the proposed numerical method. The GPE is solved in time with time-splitting technique and in space by compact finite difference method. We employ the discrete Sine transform in 1D, 2D and 3D, respectively, and get an efficient solver for the discretized system in 1D, 2D and 3D, respectively. Our numerical algorithm at

Acknowledgments

The research of H. Wang is supported in part by the Natural Science Foundation of China under grant nos. 11261065 and 91430103, and by Ministry of Education Program for New Century Excellent Talents in Chinese Universities under grant no. NCET-13-0995. We also thank the reviewers and editors for their critical comments on how to improve this manuscript.

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