Long-time simulations of highly-oscillatory problems

Highly-oscillatory phenomena combine the well-known numerical challenges of stiff equations and geometric problems, such as order reduction and energy preservation. Using asymptotic expansions to derive new non-stiff models is common, but this fails when the characteristic period is larger that the time-step. High-order averaging allows to asymptotically separate the "drift" dynamics from the oscillations using formal calculations, while keeping track of the error of approximation. This method is used to generates new, modified problems, namely a micro-macro problem or a pulled-back problem, which can be solved with better numerical accuracy.

In this talk, I will introduce high-order averaging using a (somewhat recent) closed form approach which simplifies the framework of the method, and exhibit the improvement in numerical accuracy it allows. I will then present the geometric properties of the method, notably the preservation of a Hamiltonian structure, and demonstrate how this may be used for long-time simulations.