In this paper, we formulate and analyse a geometric 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 thus it 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 its 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：本文提出并分析了一种用于求解d维空间（d=1,2,3）中非线性Klein-Gordon方程的几何低正则积分子。该积分子基于两步三角法构造，因此形式简洁。本文严格给出了误差估计，表明在能量空间中，当正则性要求满足$H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$时，该积分子可实现二阶时间精度。此外，该格式的时间对称性保证了其良好的长期能量守恒特性，这一结论通过调制傅里叶展开技术得到严格证明。文中给出了数值实验，数值结果表明新积分子相较于现有方法具有显著优势。