Highly-oscillatory problems with time-dependent vanishing frequency
For researchers in numerical analysis and asymptotic analysis, this is a first step toward handling a previously intractable class of oscillatory problems.
This work addresses highly-oscillatory evolution problems where the frequency depends on time and vanishes at some instants, a scenario previously avoided. It derives the asymptotic behavior of the solution and presents a second-order uniformly accurate numerical method.
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.