An improved meshless method for solving two- and three-dimensional coupled Klein–Gordon–Schrödinger equations on scattered data of general-shaped domains

Abstract

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to obtain the solution of two- and three-dimensional coupled Klein–Gordon–Schrodinger (KGS) equations. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the polyharmonic splines are used as the basis functions and to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The aim of this paper is to show that the SMRPI procedure is suitable for the treatment of the KGS equations. The results of numerical experiments are compared with analytical solution and stability analysis is checked to confirm the accuracy and efficiency of the presented scheme.