An Adaptive Filon Algorithm for Highly Oscillatory Integrals

Abstract

Based on the error analysis of Extended Filon Method (EFM), we present an adaptive Filon method to calculate highly oscillatory integrals. The main idea is to allow interpolation points depend upon underlying frequency in order to minimize the error. Typically, quadrature error need be examined in two regimes. Once frequency is large, asymptotic behaviour dominates and we need to choose interpolation points accordingly, while for small frequencies good choice of interpolation points is similar to classical, non-oscillatory quadrature. In this paper we choose frequency-dependent interpolation points according to a smooth homotopy function and the accuracy is superior to other EFMs. The basic algorithm is presented in the absence of stationary points but we extend it to cater for highly oscillatory integrals with stationary points. The presentation is accompanied by numerical experiments which demonstrate the power of our approach.