Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant

Abstract

Recently, the long-time numerical simulation and error analysis of PDEs with weak nonlinearity (or small potentials) become an interesting topic. However, the existing results of long-time error analysis mostly focus on the single equations. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant $\varepsilon \in (0,1]$, we propose an exponential wave integrator Fourier pseudo-spectral (EWIFP) method by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through careful and rigorous analysis, we establish improved error bounds for the numerical solution at $O(h^m + \varepsilon \tau^2)$ in the long-time domain up to $O(1/\varepsilon)$ where $m$ is determined by the regularity conditions, h is the mesh size and τ is the time step, respectively. Compared with the existing results, our analysis shows the long-time errors of numerical solution for the KGSE. In error analysis, in addition to the classical tools such as energy method and cut-off technique, we also adopt the regularity compensation oscillation (RCO) technique which has been developed recently to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the long-term stability of discrete mass and energy. To the best of our knowledge, there has not been any relevant long-time error analysis for the KGSE and any improved uniform error bounds for an exponential wave integrator. Our work is novel and provides a reference for analyzing the improved error bounds of the numerical methods for other coupled equations.