In this paper, we propose a time fourth-order exponential wave integrator (EWI) Fourier pseudo-spectral method for solving the nonlinear Dirac equation with periodic boundary value conditions. The new numerical method is designed by using Fourier pseudo-spectral method for spatial derivatives combined with a new fourth-order EWI for temporal derivatives. We give rigourously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. In more details, the proposed method has the fourth-order temporal accuracy and spectral spatial accuracy, respectively, in general $H^m$-norm. Extensive numerical experiments are reported to confirm the theoretical analysis and show the superiority of the new method.

在本文中，我们提出了一种时间四阶指数波积分器（EWI）傅里叶伪谱方法，用于求解具有周期性边值条件的非线性狄拉克方程。新的数值方法是利用傅里叶伪谱法对空间导数结合新的四阶EWI对时间导数进行设计。我们对没有任何 CFL 类型条件约束的数值解进行严格的误差分析并建立误差界限。更详细地说，所提出的方法通常分别具有四阶时间精度和光谱空间精度$H^{米}$-规范。大量的数值实验证实了理论分析并显示了新方法的优越性。