Multi-revolution composition methods for highly oscillatory
differential equations
Philippe Chartier, Joseba Makazaga, Ander Murua, and Gilles Vilmart
Abstract.
We introduce a new class of multi-revolution composition methods (MRCM) for the
approximation of the Nth-iterate of a given near-identity map. When applied to the
numerical integration of highly oscillatory systems of differential equations, the technique
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 Schršodinger equation, illustrate
the theoretical results, as well as the efficiency and versatility of the methods.
Key Words. near-identity map, highly-oscillatory, averaging, differential equation, composition
method, geometric integration, asymptotic preserving.