Recently, an exponential integrator Fourier pseudo-spectral (EIFP) scheme for the Klein–Gordon–Dirac (KGD) equation in the nonrelativistic limit regime has been proposed (Yi et al., 2019). The scheme is fully explicit and numerical experiments show that it is very efficient due to the fast Fourier transform (FFT). However, the authors did not give a strict convergence analysis and error estimate for the scheme. In addition, the scheme did not satisfy time symmetry which is an important characteristic of the exact solution. In this paper, by setting two-level format for Klein–Gordon part and three-level format for Dirac part, respectively, we proposed a new EIFP scheme for the KGD equation with periodic boundary conditions. The new scheme is time symmetric and fully explicit. By using the standard energy method and the mathematical induction, we make a rigorously convergence analysis and establish error estimates without any CFL condition restrictions on the grid ratio. The convergence rate of the proposed scheme is proved to be at the second-order in time and spectral-order in space, respectively, in a generic $H^m$-norm. The numerical experiments are carried out to confirm our theoretical analysis. Because that our error estimates are given under the general $H^m$-norm, the conclusion can easily be extended to two- and three-dimensional problems without the stability (or CFL) condition under sufficient regular conditions.

最近，提出了非相对论极限状态下 Klein-Gordon-Dirac (KGD) 方程的指数积分器傅立叶伪谱 (EIFP) 方案（Yi 等人，2019）。该方案是完全明确的，数值实验表明由于快速傅立叶变换 (FFT)，它非常有效。但是，作者并没有对该方案进行严格的收敛分析和误差估计。此外，该方案不满足时间对称性，这是精确解的一个重要特征。在本文中，通过分别为 Klein-Gordon 部分设置两级格式和为 Dirac 部分设置三级格式，我们提出了一种新的具有周期性边界条件的 KGD 方程的 EIFP 格式。新方案是时间对称的且完全明确的。通过使用标准能量法和数学归纳法，我们进行了严格的收敛分析并建立了对网格比没有任何 CFL 条件限制的误差估计。证明了所提出方案的收敛速度在时间上分别为二阶和在空间上为谱阶。$H^{米}$-规范。进行数值实验以证实我们的理论分析。因为我们的误差估计是在一般情况下给出的$H^{米}$-范数，结论可以很容易地扩展到二维和三维问题，而没有足够正则条件下的稳定性（或 CFL）条件。