HOC–ADI schemes for two-dimensional Ginzburg–Landau equation in superconductivity

Abstract

In this paper, we apply the high-order compact scheme coupled with alternating direction implicit (HOC–ADI) method to solve the two-dimensional Ginzburg–Landau (2D GL) equation. Five HOC–ADI schemes with second order accuracy in time and fourth order in space are proposed for 2D GL equation. Scheme I and Scheme II are both nonlinear, which need nonlinear iteration.To overcome this shortcoming, Scheme III and Scheme IV are proposed in order to avoid nonlinear iteration. With the three-level ADI scheme and the method of extrapolation, we obtain a linearized scheme V. Some numerical experiments are shown to testify and compare the superiority of the new numerical schemes.

Introduction

Ginzburg–Landau (GL) equation was proposed in the 1950s by V.L. Ginzburg and L. Landau on the basis of second order Landau phase transformation theory. This model is used to describe superconductivity phenomena. In this paper, we are interested in the initial–boundary value problem of the following type [5]

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}-c\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+d{\left|u\right|}^{2}u-\gamma u=0,(x,y,t)\in \Omega \times (0,T],\hfill \\ u(x,y,0)=u_0\left(x,y\right),(x,y)\in \bar{\Omega },\hfill \\ u(x,y,t)=0,(x,y)\in \partial \Omega ,t\in [0,T],\hfill \end{array}\right.$$

where $c=\nu +i\alpha$, $d=\lambda +i\beta$ are complex parameters, $i=\sqrt{-1}$, $\Omega =\left(x_L,x_R\right)\times \left(y_L,y_R\right)$, and $\overline{\Omega }$ is the closure of $\Omega$, $u(x,y,t)$ is an unknown complex function, $\nu >0$, $\lambda >0$, $\alpha$, $\beta$ are given real parameters. $u_0(x,y)$ is a given complex function, $\gamma$ is the coefficient of the linear evolution term. Eq. (1) is called 2D GL equation.

This model is used to describe the dynamics of electromagnetic behavior of a superconductor in an external magnetic field. It is also employed to study the phase transitions in superconductors near their critical temperature. Moreover, the GL theory is irreplaceable in investigating the dynamics of Abrikosov vortices in type II superconductors.

After some direct computing, we can derive the following estimate about the solution of the 2D GL equation (1):

Proposition 1

The solution $u\left(x,y,t\right)$ for Eq. (1) is bounded, that is, there exists a constant C, such that the mass ${\Vert u(\cdot ,t)\Vert }^2$ of the system satisfies

$${\Vert u(\cdot ,t)\Vert }^2={\int }_{\Omega }{\left|u(x,y,t)\right|}^{2}dxdy\le C.$$

In more detail, we have

• When $\gamma <0$, it has

  $${\Vert u(\cdot ,t)\Vert }^2\le e^{2\gamma t}{\Vert u_0\Vert }^2.$$

  This indicates that the mass exponentially decays to zero with the exponent rate $2\gamma$ as $\gamma <0$.

• When $\gamma >0$, we have

  $${\Vert u(\cdot ,t)\Vert }^2\le e^{\gamma T}{\Vert u_0\Vert }^2.$$

  This suggests that the mass is bounded in a finite time interval $[0,T]$.

This proposition demonstrates that the initial–boundary valued problem (1) is $L^2$-stable. For more details about this estimate, we refer to [8] and references therein.

In view of the importance of GL equation in mathematics and physics, many physicists and mathematicians have studied this model, and obtained a lot of theoretical results [1], [12], [15]. In terms of numerical methods, Du et al. studied the model from pure mathematics and numerical aspect by finite elements method [4]. Lord studied its attractors and finite difference method [14]. López considered the problem of computation and deformation of group orbits of solutions of the complex GL equation with cubic nonlinearity in $1+1$ space–time dimension invariant under the action of the three-dimensional Lie group of symmetries [13]. Shokri proposed a meshless method by radial basis functions [19] and a predictor-correction HOC–ADI scheme [18]. Wang & Huang constructed an implicit midpoint scheme for fractional GL equation [21]. Wang & Guo proposed some finite difference schemes for 2D GL equations [20], and proved that these schemes are of second order convergence rate in the $l^2$ norm. A linearized difference scheme has been proposed by Xu & Chang which can be used to solve the GL equation under periodic boundary conditions [23]. Zhang & Yan proposed a Lattice Boltzmann method for 3D GL equation [24]. Due to the limitations of computer performance and other aspects, the research on numerical methods of 2D problems is relatively rare. Kong et al. constructed some efficient numerical schemes [8], [9], and also analyzed the stability and convergence of the schemes. However, when such a scheme is used to solve a multi-dimensional problem, it requires solving a nonlinear system of equations with huge degree. When the considered spatial domain is large or the mesh is densely divided, the computing efficiency is not ideal.

In this paper, we are devoting to constructing some high-order compact and alternative direction implicit (HOC–ADI) schemes for the periodic initial–boundary value problem of the 2D GL equation (1). The HOC–ADI schemes are obtained by combining the HOC scheme [10], [11], [22] and ADI technique [2], [3], [6], [7], [16], [17]. It is discretized with the HOC method in space and the ADI method in time. This method takes full advantages of the ADI method and HOC scheme. By reducing the order of the algebraic equations, it saves storage space and calculation time. At the same time, it is efficient and unconditionally stable for solving multidimensional problems.

The rest of this paper is organized as follows. Section 2 reviews the basic idea about HOC method. Five difference schemes are presented in Section 3 which share the advantages of HOC method and ADI technique. Section 4 provides some numerical results to validate the efficiency and superiority by comparisons. Some conclusions are given in the final section to end the paper.