Optimal error estimate of a linear Fourier pseudo-spectral scheme for two dimensional Klein–Gordon–Schrödinger equations

Abstract

The focus of this paper is on the optimal error bounds of a Fourier pseudo-spectral conservative scheme for solving the 2-dimensional nonlinear Klein–Gordon–Schrödinger equations. The proposed Fourier pseudo-spectral scheme not only conserves the mass and energy in the discrete level but also is efficient in practical computation because only two linear systems need to be solved at each time step. Based on the equivalence between the semi-norm derived by the Fourier pseudo-spectral method and that by the finite difference method, the pseudo-spectral solution of the proposed scheme is proved strictly to be bounded and convergent with the order of O(N−r+τ2) in the discrete L2 norm, where N is the number of nodes and τ is the time step size. Some numerical experiments are carried out to validate the theoretical analysis.