Highly Oscillatory Quadrature: The Story so Far

Abstract

The last few years have witnessed substantive developments in the com- putation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin-type methods. All these methods share the surprising property that their accuracy increases with growing oscillation. These developments are described in a unified fashion, taking the multivariate integral ?Ω f(x)eiωg(x)dV as our point of departure