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

Filon-Clenshaw-Curtis rules are among rapid and accurate quadrature rules for computing highly oscillatory integrals. In the implementation of the Filon-Clenshaw-Curtis rules in the case when the oscillator function is not linear, its inverse should be evaluated at some points. In this paper, we solve this problem by introducing an approach based on the interpolation, which leads to a class of modifications of the original Filon-Clenshaw-Curtis rules. In the absence of stationary points, two kinds of modified Filon-Clenshaw-Curtis rules are introduced. For each kind, an error estimate is given theoretically, and then illustrated by some numerical experiments. Also, some numerical experiments are carried out for a comparison of the accuracy and the efficiency of the two rules. In the presence of stationary points, the idea is applied to the composite Filon-Clenshaw-Curtis rules on graded meshes. An error estimate is given theoretically, and then illustrated by some numerical experiments

Modified Filon-Clenshaw-Curtis rules for oscillatory integrals with a nonlinear oscillator. ETNA - Electronic Transactions on Numerical Analysis

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