Improved error estimates of the time-splitting methods for the long-time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

Abstract

We provide improved uniform error estimates for the time-splitting Fourier pseudo-spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter $\text{◂+▸}\epsilon \in (0,1]$. We first reformulate the KGDS into a coupled Schrödinger–Dirac system (CSDS) and then apply the second-order Strang splitting method to CSDS with the spatial discretization provided by Fourier pseudo-spectral method. Based on rigorous analysis, we establish improved uniform error bounds for the second-order Strang splitting method at $O\text{◂()▸}(\text{◂◽˙▸}{h}^{m-1}+\epsilon {\tau }^2)$ up to the long time at $\text{◂⋅▸}O(1/\epsilon )$. In addition to the conventional analysis methods, we mainly apply the regularity compensation oscillation technique for the analysis of long time dynamic simulation. The numerical results show that our method and conclusion are not only suitable for one-dimensional problem, but also can be directly extended to higher dimensional problem and highly oscillatory problem. As far as we know there has not been any relevant long time analysis and any improved uniform error bounds for the TSFP method solving the KGDS. Our methods are novel and provides a reference for analyzing the improved error bounds of other coupled systems similar to the KGDS.