Geometric two-scale integrators for highly oscillatory system: uniform accuracy and near conservations

TL;DR

This paper introduces two new geometric integrators for highly oscillatory Hamiltonian systems that maintain uniform accuracy and near conservation properties across different regimes, demonstrated through multiple physical models.

Contribution

The paper develops two novel time-symmetric integrators using a two-scale approach, achieving uniform second-order accuracy and near conservation laws for all oscillation scales.

Findings

Methods achieve uniform second-order accuracy for all 

Numerical experiments show improved performance over existing methods

Near conservation laws are observed in long-term simulations

Abstract

In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter $\varepsilon\in(0,1]$. The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations with $\mathcal{O}(1)$-amplitude and $\mathcal{O}(1/\varepsilon)$-frequency, which makes classical numerical methods inefficient. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy for all $\varepsilon$ at finite times and some near-conservation laws in long times. Numerical experiments on a H\'{e}non-Heiles model, a nonlinear Schr\"{o}dinger equation and a charged-particle system illustrate the performance of the proposed methods over the existing ones.