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积分器。该积分器基于两步三角方法构造，形式简洁。我们严格给出了误差估计，证明在$H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$正则性要求下，该积分器在能量空间内能达到二阶时间精度。此外，格式的时间对称性保证了良好的长时间能量守恒性质，并通过调制傅里叶展开技术得到严格证明。数值算例表明，新积分器相较于现有方法具有明显优势。