GEOMETRIC TWO-SCALE INTEGRATORS FOR HIGHLY OSCILLATORY SYSTEM: UNIFORM ACCURACY AND NEAR CONSERVATIONS

In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter ε ∈ (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 O(1)-amplitude and O(1/ε)-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 ε 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.