Modified Filon-Clenshaw-Curtis rules for oscillatory integrals with a nonlinear oscillator

Abstract

Filon-Clenshaw-Curtis (FCC) rules rank among the rapid and accurate quadrature rules for computing oscillatory integrals. In the implementation of the FCC rules, when the oscillator of the integral is nonlinear, its inverse has to be evaluated at several points. In this paper we suggest an approach based on interpolation, which leads to a class of modifications of the original FCC rules in such a way that the modified rules do not involve the inverse of the oscillator function. In the absence of stationary points, two reliable and efficient algorithms based on the modified FCC (MFCC) rules are introduced. For each algorithm, an error estimate is verified theoretically and then illustrated by some numerical experiments. Also, some numerical experiments are carried out in order to compare the convergence speed of the two algorithms. In the presence of stationary points, an algorithm based on composite MFCC rules on graded meshes is developed. An error estimate is derived and illustrated by some numerical experiments.