In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when $\eps$ is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be $\mathcal{O}(\varepsilon^{3} \hh^2)$ and $\mathcal{O}(\varepsilon^{4} \hh^4)$ for the time stepsize $\hh$. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency.

翻译：本文研究非线性相对论Klein-Gordon（NRKG）方程的对称积分器，该方程含有一个无量纲参数$0<\varepsilon\ll 1$，该参数与光速成反比。模型的时间高度振荡特性对应于参数$\varepsilon$，且当$\varepsilon$较小时方程具有强非线性。这两个方面给数值方法的设计带来了显著的计算负担。我们提出并分析了一类新型对称积分器，该积分器基于问题的若干形式化方法、傅里叶伪谱方法和指数积分器。通过利用所提出的对称性质和隐式指数积分器的刚性阶条件，我们构造了两个高达四阶的实用积分器。严格研究了所得积分器的收敛性，结果表明时间精度改进为$\mathcal{O}(\varepsilon^{3} \hh^2)$和$\mathcal{O}(\varepsilon^{4} \hh^4)$（其中$\hh$为时间步长）。利用调制傅里叶展开，建立了多级积分器在长时间内的近能量守恒性质。即使采用大步长，这些理论结果仍然成立。针对NRKG方程的数值结果表明，所提出的积分器具有改进的一致误差界、优异的长时间能量守恒性和有竞争力的计算效率。