An efficient combination of Split-step in time and the Meshless local Petrov-Galerkin methods for solving the Ginzburg-Landau equation in‎‏ ‎two and three dimensions

Document Type : Original Paper


Department of Applied Mathematics, Shiraz University of Technology, Shiraz, Iran


In this paper, an efficient combination of the time-splitting and meshless local Petrov-Galerkin method for the numerical solution of Ginzburg–Landau equation in‎‏ ‎two and three dimensions is presented. ‎The main idea of splitting scheme is separating the original equation in time into two parts, linear and nonlinear‎. ‎Since, solving the nonlinear part based on the weak form is complicated and contains error, the split-step in time will be used. we solve the nonlinear part analytically and linear part numerically by the meshless local Petrov-Galerkin method in space variables and the Crank-Nicolson method in time‎. Hence, the moving Kriging interpolation is used instated of moving least squares. Therefore, the shape functions of the meshless local Petrov-Galerkin method have the Kronecker's delta property and the boundary conditions can be implemented directly and easily‎. ‎ ‎Several examples for two and three dimensions are presented and the results are compared with their analytical solutions to demonstrate the validity and capability of this method‎.


Main Subjects

[1] Weinan, E. (1994). Dynamics of vortices in Ginzburg-Landau theories
with applications to superconductivity
. Physica D: Nonlinear
(4), 383-404.
[2] Chen, Z. (1997). Mixed finite element methods for a dynamical Ginzburg-
Landau model in superconductivity.
Numerische Mathematik
 [3] Xu, Q., and Chang, Q. (2011). Difference methods for computing the
Ginzburg-Landau equation in two dimensions.
Numerical Methods for
Partial Differential Equations
(3), 507-528.
[4] Mu, M., and Huang, Y. (1998). An alternating Crank-Nicolson method for
decoupling the Ginzburg-Landau equations.
SIAM journal on numerical
(5), 1740-1761.
[5] Shokri, A., and Dehghan, M. (2012). A meshless method using radial basis
functions for the numerical solution of two-dimensional complex
Ginzburg-Landau equation. Computer Modeling in Engineering and
(4), 333.
[6] Liu G R.
Mesh free methods: moving beyond the finite element method
CRC press, 2002.
[7] Gingold, R. A., and Monaghan, J. J. (1977). Smoothed particle
hydrodynamics: theory and application to non-spherical stars
. Monthly
notices of the royal astronomical society
(3), 375-389.
[8] Li, Hua., and Shantanu, S. Mulay.
Volume 10, Issue 1
May 2020
Pages 62-87
  • Receive Date: 05 March 2019
  • Revise Date: 21 October 2019
  • Accept Date: 17 January 2020
  • First Publish Date: 21 May 2020