Highly-oscillatory problems with time-dependent vanishing frequency

TL;DR

This paper addresses the challenges of analyzing and numerically solving highly-oscillatory problems with time-dependent frequencies that can vanish, introducing new methods for asymptotic analysis and accurate numerical solutions.

Contribution

It introduces a novel approach for asymptotic analysis and a second order uniformly accurate numerical method for oscillatory problems with vanishing, time-dependent frequencies.

Findings

Asymptotic behavior can be inferred despite vanishing frequencies.

A second order uniformly accurate numerical method is developed.

The approach handles additional complexities from frequency vanishing.

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.