In this paper, we propose and analyze a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the technique of twisted function to the trigonometric integrals. Rigorous error estimates are presented and the proposed method is shown to have third-order accuracy in the energy space under a weak regularity requirement in $H^{2}\times H^{1}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than the well-known exponential integrators of order three for approximating the Klein-Gordon equation with nonsmooth solutions.

翻译：本文针对$d$维空间（$d=1,2,3$）中的半线性Klein-Gordon方程，提出并分析了一种新型三阶低正则性三角积分方法。该积分方法基于Duhamel公式的充分应用以及三角积分中的扭函数技术构建。我们给出了严格的误差估计，证明该方法在$H^{2}\times H^{1}$弱正则性条件下，在能量空间中具有三阶精度。数值实验表明，所提出的三阶低正则性积分方法在逼近具有非光滑解的Klein-Gordon方程时，计算精度显著优于经典的三阶指数积分方法。