SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Multi-revolution composition methods for highly oscillatory differential equations

P. Chartier, J. Makazaga, A. Murua and G. Vilmart

Abstract
We introduce a new class of multi-revolution composition methods (MRCM) for the approximation of the $N^{th}$-iterate of a given near-identity map. When applied to the numerical integration of highly-oscillatory systems of differential equations, the technique we propose benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.

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