Filon–Clenshaw–Curtis formulas for highly oscillatory integrals in the presence of stationary points

Abstract

Numerical approximation of a general class of one-dimensional highly oscillatory integrals over bounded intervals with exponential oscillators is considered. A Filon-type method based on modified Clenshaw–Curtis quadrature rules is developed and its stability is established when the stationary points of the oscillator function are all of order two. Also, an error estimate for the method is provided, which shows that the method is convergent as the number of Clenshaw–Curtis points increases, and the rate of convergence depends only on the Sobolev regularity of the integrand. Using some numerical experiments, the theoretical results are illustrated.