A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations

Philippe Chartier (1, 2, 3) , Mohammed Lemou (2, 4, 3) , Florian Méhats (2, 5, 3) , Gilles Vilmart (6, 7)

1 Inria - Institut National de Recherche en Informatique et en Automatique
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 MINGUS - Multi-scale numerical geometric schemes
4 CNRS - Centre National de la Recherche Scientifique
5 UR - Université de Rennes
6 Section de mathématiques [Genève]
7 UNIGE - Université de Genève = University of Geneva

We introduce a new methodology to design uniformly accurate methods for oscil-latory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.

Mots clés: stroboscopic averaging; standard averaging; uniform accuracy; micro-macro method; highly-oscillatory problems; pullback method

Domaines: Analyse numérique [math.NA]

Philippe Chartier, Mohammed Lemou, Florian Méhats, Gilles Vilmart. A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations. Foundations of Computational Mathematics, 2018. ⟨hal-01666472v1⟩