Construction and analysis of an explicit fourth-order symmetric low-regularity integrator for the semilinear Klein-Gordon equation

In this paper, we propose and analyse a novel explicit fourth-order symmetric 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, the employment of a twisted function tailored for trigonometric integrals and the symmetrization procedure of trigonometric integrators. Robust error analysis is conducted, demonstrating that the proposed scheme achieves fourth-order accuracy in the energy space under a weak regularity requirement in $H^{2+\max (\mu ,1)}({\mathbb{T}}^d)\times H^{1+\max (\mu ,1)}({\mathbb{T}}^d)$ with $\mu >\frac{d}{2}$. A numerical experiment shows that the proposed low-regularity integrator is much more accurate than some well-known exponential integrators for approximating the Klein-Gordon equation with non-smooth solutions.