A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

TL;DR

This paper provides a long-term energy-preserving analysis of symmetric and symplectic extended Runge--Kutta--Nyström integrators for highly oscillatory Hamiltonian systems, demonstrating near conservation of energy over extended periods.

Contribution

It introduces a novel long-term energy analysis for both symmetric and symplectic ERKN integrators without assuming symplecticity or symmetry, using new techniques for non-symmetric methods.

Findings

Both symmetric and symplectic ERKN integrators exhibit near energy conservation over long times.

A relationship between symmetric ERKN and trigonometric integrators is established for analysis.

The use of modulated Fourier expansion demonstrates almost-invariants for symplectic ERKN integrators.

Abstract

The primary objective of this paper is to present a long-term numerical energy-preserving analysis of one-stage explicit symmetric and/or symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both kinds of ERKN integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for symmetric integrators, a relationship between symmetric ERKN integrators and trigonometric integrators is established by using Strang splitting and based on this connection, the long-time conservation is derived. For the long-term analysis of symplectic ERKN integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion developed in SIAM J. Numer. Anal. 38 (2000) by Hairer and Lubich. By taking some novel adaptations of this essential technology for non-symmetric methods, we derive the modulated Fourier expansion for symplectic ERKN integrators. Moreover, it is shown that the symplectic ERKN integrators have two almost-invariants and then the near energy conservation over a long term is obtained.