A combined Filon/asymptotic quadrature method for highly oscillatory problems

A crossing between the asymptotic expansion of an oscillatory integral and Filon-type methods is obtained by applying a Filon-type method to the error term in the asymptotic expansion, which is in itself an oscillatory integral. The efficiency of the approach is investigated through analysis and numerical experiments, revealing a method which in many cases performs better than the Filon-type method. It is shown that considerable savings in terms of the required number of potentially expensive moments can be expected. The case of multivariate oscillatory integrals is discussed briefly.