Error analysis of an explicit symmetric fourth-order exponential wave integrator Fourier pseudo-spectral method for the Klein-Gordon-Schrödinger equation

Abstract

In this paper, we propose a fourth-order exponential wave integrator Fourier pseudo-spectral method for the Klein-Gordon-Schr\”{o}dinger (KGS) equation. The proposed method is time symmetric and explicit so it is easy to apply by the fast Fourier transform (FFT). By using the standard energy method and the mathematical induction, we make a rigorously convergence analysis and establish error estimates without any CFL condition restrictions on the grid ratio. The convergence rates of the proposed scheme are proved to be at the fourth-order in time and spectral-order in space, respectively, in a generic $H^m$-norm. Extensive numerical results are reported to confirm our error bounds. Because that our error estimates are given under the general $H^m$-norm, the conclusion can easily be extended to two- and three-dimensional problems without the stability (or CFL) condition under sufficient regular conditions. The proposed fourth-order method could also find applications to solve other coupled system such as the Klein-Gordon-Dirac system.