A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation

TL;DR

This paper introduces a third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation, achieving high accuracy with weak regularity assumptions and outperforming existing methods for non-smooth solutions.

Contribution

A novel third-order low-regularity trigonometric integrator for the Klein-Gordon equation, utilizing Duhamel's formula and twisted functions for improved accuracy with less regularity.

Findings

Achieves third-order accuracy in energy space under weak regularity conditions.

Demonstrates superior accuracy over existing exponential integrators for non-smooth solutions.

Validates effectiveness through numerical experiments.

Abstract

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.