Derivative-free high-order uniformly accurate schemes for highly oscillatory systems

In this paper we address the computational aspects of uniformly accurate numerical methods for solving highly oscillatory evolution equations. In particular, we introduce an approximation strategy that allows the construction of arbitrary high-order methods using solely the right-hand side of the differential equation. No derivative of the vector field is required, while uniform accuracy is retained. The strategy is then applied to two different formulations of the problem, namely the two-scale and the micro–macro formulations. Numerical experiments on the Hénon–Heiles system, as well as on the Klein–Gordon equation and a Vlasov-type problem, confirm the validity of the new strategy.