Highly-oscillatory problems with time-dependent vanishing frequency
P. Chartier, M. Lemou, F. Méhats, and G. Vilmart
Abstract. 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.
Key Words. highly-oscillatory problems, time-dependent vanishing frequency, asymptotic expansion, uniform accuracy.