Uniform error bounds of an exponential wave integrator for the Klein–Gordon–Schrödinger system in the nonrelativistic and massless limit regime

Abstract

We propose an exponential wave integrator Fourier pseudo-spectral (EWI-FP) method and establish the uniform error bounds for the Klein–Gordon–Schrödinger system (KGSS) with $\varepsilon \in (0,1]$. In the nonrelativistic and massless limit regime ($0<\varepsilon \ll 1$), the solution of KGSS propagates waves with wavelength $O\left(\varepsilon \right)$ in time and amplitude at $O\left({\varepsilon }^{{\alpha }^{†}}\right)$ where ${\alpha }^{†}=min\{\alpha ,\beta +1,2\}$ with two parameters $\alpha$ and $\beta$. The parameters satisfy $\alpha \ge 0$ and $\beta \ge -1$. In this regime, due to the oscillation in time, it is very difficult to develop efficient schemes and make the corresponding error analysis for KGSS. In this paper, firstly, in order to overcome the difficulty of controlling the nonlinear terms, we transform the KGSS into a system with higher derivative. Then we construct an EWI-FP method and provide the error estimates with two bounds at $O(h^{\sigma +2}+min\{\tau /{\varepsilon }^{1-{\alpha }^{\ast }},{\tau }^2/{\varepsilon }^{2-{\alpha }^{†}}\})$ and $O(h^{\sigma +2}+{\tau }^2+{\varepsilon }^{{\alpha }^{†}})$, respectively, where ${\alpha }^{\ast }=min\{1,\alpha ,1+\beta \}$, $\sigma$ has to do with the smoothness of the solution in space, $h$ is mesh size and $\tau$ is time step. From the two error bounds, we obtain the error estimates $O(h^{\sigma +2}+{\tau }^{{\alpha }^{†}})$ for $\alpha \ge 0$ and $\beta \ge -1$. Hence, we get uniform second-order error bounds at $O(h^{\sigma +2}+{\tau }^2)$ in time when $\alpha \ge 2$ and $\beta \ge 1$, and uniformly accurate first-order error estimates for any $\alpha \ge 1$ and $\beta \ge 0$. We also get uniformly accurate spatial spectral accuracy for any $\alpha \ge 0$ and $\beta \ge -1$. Our numerical results support our conclusions.