A meshless symplectic method for two-dimensional nonlinear Schrödinger equations based on radial basis function approximation

Abstract

For two-dimensional nonlinear Schrödinger equations, we propose a meshless symplectic method based on radial basis function interpolation. With the method of lines, we first discretize the equation in spatial domain by using the radial basis function approximation method and obtain a finite-dimensional Hamiltonian system. Then appropriate time integrator is employed to derive the full-discrete symplectic scheme. Compared with the classical conservative methods that are only valid on uniform grids, our meshless method is conservative for both uniform grids and nonuniform nodes. The accuracy and conservation properties are analyzed in detail. Several numerical experiments are presented to demonstrate the accuracy and the conservation properties of our approach.