An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations

Abstract

We propose an efficient time-splitting Chebyshev–Tau spectral method for the Ginzburg–Landau–Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg–Landau–Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev–Tau spectral discretization in space and Crank–Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled 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 the method in one dimension is $O(N\log (N))$ compared with that of the non-optimized one, which is $O(N^2)$. By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg–Landau–Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg–Landau–Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg–Landau–Schrödinger equation are also presented.

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

$$(\alpha -i\beta ){\psi }_t=△\psi +(V(\mathbf{x})+\gamma {|\psi |}^2)\psi ,\mathbf{x}\in {\mathbb{R}}^d,$$

along with given initial data

$$\psi (\mathbf{x},0)={\psi }_0(\mathbf{x}),$$

and the following zero/nonzero far-field boundary condition

$$\psi (\mathbf{x},t)\to g(\mathbf{x}),\text{as}|\mathbf{x}|\to \infty .$$

Here $i^2=-1$. α, β and γ are constant ($|\alpha |+|\beta |\ne 0$). $\psi (\mathbf{x},t)$ is a complex function. ${\psi }_0(\mathbf{x})$ and $g(\mathbf{x})$ are some known functions in ${\mathbb{R}}^d$.

The GLSE (1.1) is one of the most well-studied equations in nonlinear science. For example, when $\alpha =1$, $\beta =0$, $\gamma =-\frac{1}{{\epsilon }^2}$ and $V(\mathbf{x})=\frac{1}{{\epsilon }^2}$ (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 $\alpha =0$, $\beta =2$ and $V(\mathbf{x})$ 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 $O(N^2)$ 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 $O(N\log (N))$. 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.