In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the employment of a twisted function tailored for trigonometric integrals. Robust error analysis is conducted, demonstrating that the proposed scheme achieves third-order accuracy in the energy space under a weak regularity requirement in $H^{1+\max(\mu,1)}(\mathbb{T}^d)\times H^{\max(\mu,1)}(\mathbb{T}^d)$ with $\mu> \frac{d}{2}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than some well-known exponential integrators of order three for approximating the Klein-Gordon equation with non-smooth solutions.

翻译：本文针对$d$维空间（$d=1,2,3$）中具有非光滑解的半线性Klein-Gordon方程，提出并分析了一种新颖的三阶低正则性三角积分器。该积分器的构造基于对Duhamel公式的充分利用，以及为三角积分量身定制的扭曲函数的运用。我们进行了稳健的误差分析，证明在$H^{1+\max(\mu,1)}(\mathbb{T}^d)\times H^{\max(\mu,1)}(\mathbb{T}^d)$（其中$\mu> \frac{d}{2}$）的弱正则性要求下，所提格式在能量空间中可实现三阶精度。数值实验表明，对于具有非光滑解的Klein-Gordon方程，所提出的三阶低正则性积分器比一些著名的三阶指数积分器具有更高的精度。