Structure-preserving algorithms with uniform error bound and long-time energy conservation for highly oscillatory Hamiltonian systems

TL;DR

This paper introduces blended structure-preserving algorithms for highly oscillatory Hamiltonian systems that maintain symplecticity and energy with uniform error bounds over long time periods.

Contribution

It develops new algorithms combining structure-preserving and uniform error bound techniques for nonlinear Hamiltonian systems with high oscillations.

Findings

Algorithms preserve symplecticity and energy accurately.

Uniform error bounds are established for highly oscillatory solutions.

Numerical experiments confirm theoretical advantages.

Abstract

Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear Hamiltonian systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Moreover, the proposed algorithms are shown to have uniform error bound for the highly oscillatory structure. A numerical experiment is carried out to support the theoretical results established in this paper by showing the performance of the blended algorithms.