Improved uniform error bounds of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Schrödinger equation with the small coupling constant

Abstract

Recently, the long time numerical simulation of PDEs with weak nonlinearity (or small potentials) becomes an interesting topic. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant ɛ∈(0,1], we proposed a time-splitting Fourier pseudo-spectral (TSFP) scheme by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through rigorous error analysis, we establish improved error bounds for the scheme at O(hm+ɛτ2) up to the long time at O(1/ɛ) where h is the mesh size and τ is the time step, respectively, and m depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGSE with the small coupling constant. The tools for error analysis mainly include the mathematical induction and the standard energy method as well as the regularity compensation oscillation (RCO) technique which has been developed recently. The numerical experiments support our theoretical analysis. Our scheme is novel because that to the best of our knowledge there has not been any TSFP scheme and any relevant long time analysis for the KGSE.