Error analysis of trigonometric integrators for semilinear wave equations

TL;DR

This paper provides an error analysis for trigonometric integrators applied to semilinear wave equations, demonstrating optimal second-order convergence under finite energy conditions and covering various methods including impulse and mollified impulse methods.

Contribution

It offers a uniform error analysis for multiple trigonometric integrators, explaining their convergence behavior and applicability to wave equations with periodic boundary conditions.

Findings

Achieves optimal second-order convergence for the methods analyzed.

The analysis is uniform in the spatial discretization parameter.

Explains convergence behavior of Störmer-Verlet/leapfrog in time.

Abstract

An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semi-discretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal second-order convergence is shown requiring only that the exact solution is of finite energy. The analysis is uniform in the spatial discretization parameter. It covers the impulse method which coincides with the method of Deuflhard and the mollified impulse method of Garc\'ia-Archilla, Sanz-Serna & Skeel as well as the trigonometric methods proposed by Hairer & Lubich and by Grimm & Hochbruck. The analysis can also be used to explain the convergence behaviour of the St\"ormer-Verlet/leapfrog discretization in time.