Abstract

This work is devoted to the construction of efficient numerical schemes for a wide class of highly oscillatory models including kinetic Vlasov models, nonlinear Schr ödinger and Klein-Gordon equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency. This is a stronger future than the usual so called “Asymptotic preserving” property, the last being therefore satisfied by our scheme in the highly oscillatory limit. Our strategy enables to simulate the oscillatory problem without using any mesh or time step refinement, and the order of the scheme is preserved uniformly in all regimes. The method is based on a “double-scale” reformulation of the original equation, with the introduction of an additional variable. Then a link is made with well-known strategies based on Chapman-Enskog expansions in kinetic theory, which we extend to the dispersive context of Schrödinger-type equations.