Meshless symplectic and multi-symplectic local RBF collocation methods for nonlinear Schrödinger equation

Abstract

This study formulates a novel meshless symplectic and multi-symplectic local radial basis function (RBF) collocation method (LRBFCM) for the nonlinear Schrödiner equation. The LRBFCM is employed for spatial discretization and then symplectic integrator is conducted for time discretization. The properties of the space differentiation matrix of LRBFCM are discussed in detail. The conservativeness of the proposed method is discussed and the accuracy is assessed. The LRBFCM is simple and efficient, since it can avoid the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. Numerical tests with uniform knots and random knots are designed to show the effectiveness of the method.