Highly-oscillatory problems with time-dependent vanishing frequency

Philippe Chartier (1) , Mohammed Lemou (1) , Florian Méhats (2) , Gilles Vilmart (3)

1 MINGUS - Multi-scale numerical geometric schemes
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Section de mathématiques [Genève]

In the analysis of highly-oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. This work is a first step towards the resolution of these difficulties. In particular, we show that it is still possible in this situation to infer the asymptotic behaviour of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.

Mots clés: highly-oscillatory problems; time-dependent vanishing frequency; asymptotic expansion; uniform accuracy

Domaines: Analyse numérique [math.NA]

Philippe Chartier, Mohammed Lemou, Florian Méhats, Gilles Vilmart. Highly-oscillatory problems with time-dependent vanishing frequency. 2018. ⟨hal-01845614v1⟩