Long-time oscillatory energy conservation of total energy-preserving methods for highly oscillatory Hamiltonian systems

TL;DR

This paper analyzes the long-term near conservation of oscillatory energy by the AAVF method in highly oscillatory Hamiltonian systems, demonstrating its effectiveness through modulated Fourier expansion techniques.

Contribution

It provides a rigorous analysis of the long-time oscillatory energy behavior of the AAVF method, extending understanding of energy conservation in highly oscillatory Hamiltonian systems.

Findings

AAVF method preserves total energy exactly.

Long-time near conservation of oscillatory energy established.

Extension to multi-frequency systems included.

Abstract

For an integrator when applied to a highly oscillatory system, the near conservation of the oscillatory energy over long times is an important aspect. In this paper, we study the long-time near conservation of oscillatory energy for the adopted average vector field (AAVF) method when applied to highly oscillatory Hamiltonian systems. This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly. This paper is devoted to analysinganother important property of AAVF method, i.e., the near conservation of its oscillatory energy in a long term. The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion. A similar result of the method in the multi-frequency case is also presented in this paper.