Multi-revolution composition methods for highly oscillatory differential equations

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 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.

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

“…In the context of deterministic oscillatory problems, order conditions for MRCMs up to arbitrary high order are derived in [7] using the algebraic framework of labelled rooted trees, and integrators up to order 4 are exhibited. However, it is known [30,31] that a composition method with real coefficients of order strictly larger than 2 necessarily involves negative coefficients, see [4] for an elegant geometric proof.…”

“…It was noted in the deterministic context [7] that in principle, any nonstiff integrator could be used. However, a natural choice for such approximation is to use a splitting method where the oscillatory and non oscillatory parts of the problem are solved separately in an efficient way (sometimes exactly), as proposed and studied recently in [34] in the stochastic context of the Langevin equation.…”

“…In [7], the class of deterministic multi-revolution composition methods was recently introduced for systems of the form (1) with g r = 0, r = 1, . .…”

“…It is shown in [7] that the constant symbolized by O is independent of N ≥ 1 and ε ≤ ε 0 if ϕ ε (y) is a C 3 function of (y, ε). Next, considering for ϕ ε the flow of an highly oscillatory system after one highly oscillatory period permits to design large time step geometric integrators.…”

“…Again, following [7] in the deterministic setting, this problem can be put in the form (1) up to a time transformationt = t/2π, where A ∈ R 12×12 is given by…”