Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein–Gordon–Schrödinger (KGS) equations

Abstract

In the present study, three numerical meshless methods are being considered to solve coupled Klein–Gordon–Schrödinger equations in one, two and three dimensions. First, the time derivative of the mentioned equation will be approximated using an implicit method based on Crank–Nicolson scheme then Kansa’s approach, RBFs-Pseudo-spectral (PS) method and generalized moving least squares (GMLS) method will be used to approximate the spatial derivatives. The proposed methods do not require any background mesh or cell structures, so they are based on a meshless approach. Applying three techniques reduces the solution of the one, two and three dimensional partial differential equations to the solution of linear system of algebraic equations. As is well-known, the use of Kansa’s approach makes the coefficients matrix in the above linear system of algebraic equations to be ill-conditioned and we applied LU decomposition technique. But when we employ PS method (Fasshauer, 2007), the matrix of coefficients in the obtained linear system of algebraic equations is well-conditioned. Also the GMLS technique yields a well-conditioned linear system, because a shifted and scaled polynomial basis will be used. At the end of this paper, we provide some examples on one, two and three-dimensions for obtaining numerical simulations. Also the obtained numerical results show the applicability of the proposed three methods to find the numerical solution of the KGS equations.