Geometric Two-Scale Integrators for Highly Oscillatory System: Uniform Accuracy and Near Conservations

Abstract

In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter . 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 -amplitude and -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 at finite times and some near-conservation laws in long times. Numerical experiments on a Hénon–Heiles model, two nonlinear Schrödinger equations, and a charged-particle system illustrate the performance of the proposed methods over the existing ones.