Eur. Phys. J. Plus (2020) 135: 873
https://doi.org/10.1140/epjp/s13360-020-00890-y

Regular Article

A meshless local moving Kriging method for solving Ginzburg–Landau equation on irregular domains

Department of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

^* e-mail: ilati@sut.ac.ir

Received: 3 July 2020
Accepted: 25 October 2020
Published online: 1 November 2020

Abstract

In this paper, a meshless local moving Kriging method is applied for numerical solving of nonlinear Kuramoto–Tsuzuki and Ginzburg–Landau equations on regular and irregular computational domains. These models are considered with different types of boundary conditions such as Dirichlet, Neumann and periodic boundary conditions. At first, a finite difference formula is employed for obtaining a time-discrete scheme. Then, the spatial derivatives are approximated using the moving Kriging technique. The moving Kriging method is a truly meshless method in which the unknown function can be approximated locally, and this leads to the sparsity of the coefficient matrix. Also, this method can be easily applied on irregular computational domains. All these capabilities make it an efficient method for solving the nonlinear partial differential equations especially Kuramoto–Tsuzuki and Ginzburg–Landau equations. For numerical experiments, various cases of these models are solved on regular and irregular computational domains and the efficiency of the proposed method is shown. Finally, it is concluded that the local moving Kriging method can be considered as an attractive alternative to the existing mesh-based methods in solving nonlinear Kuramoto–Tsuzuki and Ginzburg–Landau equations.