Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

Abstract

ABSTRACTIn this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete norm. The proposed scheme is shown to be convergent with the convergence order of in the discrete norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experiments are conducted to verify the theoretical analysis.