In this paper, we formulate and analyse a symmetric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and the proposed integrator has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures the good long-time energy conservation which is rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods.

翻译：本文针对d维空间（d=1,2,3）中的非线性Klein-Gordon方程，构建并分析了一种对称低正则性积分器。该积分器基于两步三角法构造，形式简洁。严格误差估计表明，在$H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$正则性要求下，积分器可在能量空间实现二阶时间精度。此外，通过调制傅里叶展开技术严格证明了该格式的时间对称性可确保良好的长时间能量守恒。数值实验展示了新积分器相较于现有方法的优越性。