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.


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