Geometric two-scale integrators for highly oscillatory system: uniform accuracy and near conservations
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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, a nonlinear Schrödinger equation and a charged-particle system illustrate the performance of the proposed methods over the existing ones.
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