A generalization of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals

Abstract

The Filon–Clenshaw–Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency \(\omega \) it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC\(+\), which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC\(+\) shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages.