Explicit symmetric low-regularity integrators for the semilinear Klein-Gordon equation

TL;DR

This paper introduces symmetric low-regularity integrators for the semilinear Klein-Gordon equation, enabling efficient and accurate long-term simulations under relaxed regularity conditions.

Contribution

It proposes a systematic symmetrization procedure to construct symmetric schemes from explicit integrators and demonstrates their optimal convergence and energy preservation.

Findings

Achieves optimal convergence orders in energy space with relaxed regularity.

Symmetric schemes improve convergence as solution regularity increases.

Second-order scheme nearly preserves energy over long simulations.

Abstract

This paper is concerned with the design and analysis of symmetric low-regularity integrators for the semilinear Klein-Gordon equation. We first propose a general symmetrization procedure that allows for the systematic construction of symmetric schemes from existing explicit (non-symmetric) integrators. Applying this procedure, we derive two novel schemes. Error analyses show that both integrators achieve their optimal convergence orders in the energy space under significantly relaxed regularity assumptions. Furthermore, the symmetry property ensures that the convergence order of a first-order symmetric scheme improves as the regularity of the exact solution increases. A numerical experiment demonstrates that the proposed second-order symmetric scheme nearly preserves the system energy over extended periods.