Long-time analysis of extended RKN integrators for Hamiltonian systems with a solution-dependent high frequency

TL;DR

This paper analyzes the long-term performance of extended RKN integrators in solving highly oscillatory Hamiltonian systems with variable high frequency, demonstrating their near conservation of energy and action over extended periods.

Contribution

It provides the first rigorous analysis of the long-time behavior of ERKN integrators for systems with solution-dependent high frequency, using modulated Fourier expansion techniques.

Findings

ERKN integrators approximately conserve a modified action and energy over long times

Numerical experiments confirm theoretical predictions

Similar long-time behavior observed for RKN methods

Abstract

In this paper, we analyse the long-time behaviour of the extended RKN (ERKN) integrators for solving highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. We prove that a symmetric ERKN integrator approximately conserves a modified action and a modified total energy over long time intervals based on the technique of varying-frequency modulated Fourier expansion. An illustrative numerical experiment is carried out and the numerical results strongly support the theoretical analysis presented in this paper. As a byproduct of this work, similar long-time behaviour is also investigated for an RKN method.