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

TL;DR

This paper introduces modified Filon-Clenshaw-Curtis quadrature rules for oscillatory integrals with nonlinear oscillators, improving accuracy and efficiency through interpolation-based approaches and error analysis.

Contribution

It develops new modifications of Filon-Clenshaw-Curtis rules for nonlinear oscillators, including error estimates and numerical comparisons.

Findings

Modified rules improve accuracy for nonlinear oscillators

Error estimates are validated by numerical experiments

Composite rules perform well near stationary points

Abstract

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.