On energy conservation by trigonometric integrators in the linear case with application to wave equations

TL;DR

This paper introduces a modified energy concept for trigonometric integrators applied to oscillatory linear Hamiltonian systems, ensuring exact energy preservation and extending to linear wave equations.

Contribution

It derives a modified energy that is exactly conserved by trigonometric integrators under certain conditions, extending energy conservation results to linear wave equations.

Findings

Modified energy is exactly preserved by the integrators.

Extension of energy conservation to linear wave equations.

Applicable under specific filter function conditions.

Abstract

Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered. Under a condition of Hairer & Lubich on the filter functions in the method, a modified energy is derived that is exactly preserved by trigonometric integrators. This implies and extends a known result on all-time near-conservation of energy. The extension can be applied to linear wave equations.