Long-time energy analysis of extended RKN integrators for muti-frequency highly oscillatory Hamiltonian systems

TL;DR

This paper investigates the long-term energy conservation properties of extended RKN integrators when applied to multi-frequency highly oscillatory Hamiltonian systems, combining theoretical analysis and numerical experiments.

Contribution

It provides a theoretical framework for understanding the long-time energy behavior of ERKN integrators in complex oscillatory systems, supported by numerical validation.

Findings

ERKN integrators exhibit remarkable long-time energy conservation.

Theoretical analysis using modulated multi-frequency Fourier expansions supports numerical results.

Numerical experiments confirm the near conservation of energies over long simulation times.

Abstract

In this paper, we study the long-time near conservation of the total and oscillatory energies for extended RKN (ERKN) integrators when applied to muti-frequency highly oscillatory Hamiltonian systems. We consider one-stage explicit symmetric integrators and show their long-time behaviour of numerical energy conservations by using modulated multi-frequency Fourier expansions. Numerical experiments are carried out and the numerical results demonstrate the remarkable long-time near conservation of the energies for the ERKN integrators and support our theoretical analysis presented in this paper.