An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations
Introduction
The Ginzburg–Landau–Schrödinger equation (GLSE) describes a large variety of physical phenomena, including nonlinear waves' propagation, phase transitions, superconductivity, superfluidity and Bose–Einstein condensation [1], [12], [13], [17], [22], [26]. It takes the form along with given initial data and the following zero/nonzero far-field boundary condition Here . α, β and γ are constant (). is a complex function. and are some known functions in .
The GLSE (1.1) is one of the most well-studied equations in nonlinear science. For example, when , , and (ε is a positive constant), it collapses into the Ginzburg–Landau equation (GLE), which is well known for modelling superconductivity [10], [11], [12], [13], [17]; when , and is so-called harmonic potential, the GLSE reduces to the well-known nonlinear Schrödinger equation (NLSE) – Gross–Pitaevskii equation (GPE) for modelling Bose–Einstein condensates confined in the harmonic trap at extremely low temperature [4].
Usually, standard finite difference methods or finite element methods have been applied to solve the GLE numerically [10], [11]. While numerous numerical methods, for example, split-step Fourier-spectral method [24], Fourier-pseudospectral method [21], conservative finite difference methods [7], [19] and finite element methods [14], [15] have been proposed for studying the numerical solutions of the standard NLSE in the past years.
For the NLSE which models the Bose–Einstein condensation or the well-known GPE, Perez-Garcia et al. studied several numerical methods in one dimension [16]; Cerimele et al. proposed an explicit finite difference method [8]; Tian et al. investigated explicit symplectic schemes for the Gross–Pitaevskii equation in a rotational frame [20]; and Dion et al. developed a Galerkin-spectral method based on Hermit functions [9]. Recently, Bao et al. studied a time-splitting Fourier-spectral method [2], [3] and a time-splitting Sine-spectral [5] method for the GPE with zero far-field conditions and successfully applied them to simulate dynamics of Bose–Einstein condensates [4]. Unfortunately, both the time-splitting Fourier-spectral method and time-splitting Sine-spectral method are difficult to be applied to approximate GLSE (1.1) with nonzero far-field conditions.
Quite recently, to study the vortex dynamics of GLSE in 2D with the nonzero far-field conditions, Bao et al. proposed an efficient and unconditionally stable numerical method for the GLSE in 2D [25], [26]. The key features of this method are based on [25]: (i) the application of a time-splitting technique for decoupling the nonlinearity in the GLSE; (ii) the adoption of polar coordinates to match the oscillatory nature in the transverse direction of the far-field conditions; and (iii) the utilization of Fourier pseudospectral discretization in the transverse direction and a second order finite difference discretization or finite element discretization in the radial direction. However, the method's numerical accuracy in space remains to be second order.
In [6], Bao presented a time-splitting Chebyshev–Tau spectral method for the GLSE in 1D. Although it can deal with nonzero far-field conditions, it is expensive to implement in one dimension and the computation cost is in one dimension because its numerical implementation has not been optimized. It would be more expensive to extend the method to solve two-dimensional or three-dimensional GLSE numerically. In this paper, we proposed a new efficient time-splitting Chebyshev–Tau spectral method to solve the GLSE in one dimension (1D), two dimensions (2D) and three dimensions (3D) with the zero/nonzero far-field conditions. In 1D, we first split the GLSE in time into two parts, a nonlinear equation and a linear equation. We solve the nonlinear equation exactly; while we discretize the linear equation in 1D with Chebyshev–Tau spectral discretization in space and Crank–Nicolson method in time. To solve efficiently the associated discretized system, we decouple them into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of our discretization in 1D is thus . By applying the ADI technique, we extend the newly proposed method for the GLSE in 1D and construct an efficient solver for the GLSE in higher dimensions.
The paper is organized as follows. In Section 2 we present the time-splitting Chebyshev–Tau spectral method to solve the GLSE in 1D, 2D and 3D with the zero/nonzero far-field conditions. Detailed numerical algorithms are provided. In Section 3 we test the numerical accuracy of the method for the GLSE in 1D, 2D and 3D and apply the method to study the semiclassical limits of GLSE in 1D and investigate the quantized vortex dynamics of GLSE in 2D. In Section 4, some conclusions are drawn.
Section snippets
The time-splitting Chebyshev–Tau spectral method for GLSE
In this section, we present an efficient and unconditionally stable numerical method-time-splitting Chebyshev–Tau spectral(TSCS) method for solving the GLSE (1.1), (1.2), (1.3). In the practical implementation, we truncate the problem (1.1), (1.2), (1.3) to a bounded computational domain with an inhomogeneous Dirichlet boundary condition: where we choose in 1D (or in 2D, or in
Numerical results
In this section, we first test space accuracy of the proposed TSCS method, next apply the method to study the semiclassical limit of a one-dimensional NLSE with the nonzero far-field boundary conditions, and finally investigate quantized vortex dynamics governed by the GLSE with nonzero far-field boundary conditions.
Conclusions
We have presented an efficient time-splitting Chebyshev–Tau spectral method for the Ginzburg–Landau–Schrödinger equation with zero/nonzero far-field boundary conditions, which is difficult to solve numerically. We have employed the fast Sine transformation in the implementation, which make the computation cost of our newly proposed method in 1D being . By applying the ADI technique, we have extended the method to the Ginzburg–Landau–Schrödinger equation in higher dimensions and get a
Acknowledgements
The author thanks the support from the Ministry of Education of Singapore grant No. R-158-000-002-112, the Yunnan University of Finance and Economics grant No. 80025092012 and the National Natural Science Foundation of China grant No. 10901134. The author also thanks very stimulating discussions with Professor Weizhu Bao in the subject and the referees for their valuable comments to improve the manuscript. This work was partially done while the author was visiting Department of Mathematics,
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2017, Applied Mathematics and ComputationCitation Excerpt :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].