Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities

TL;DR

This paper introduces and analyzes a numerical quadrature method for highly oscillatory integrals with logarithmic singularities, demonstrating its robustness, stability, and efficiency through theoretical error estimates and numerical experiments.

Contribution

The paper develops a novel Filon-Clenshaw-Curtis quadrature rule specifically designed for integrals with oscillations and logarithmic singularities, including detailed error analysis and implementation guidance.

Findings

Error decreases as oscillation rate increases for fixed nodes

Method is numerically stable and efficient

Numerical experiments confirm theoretical error estimates

Abstract

In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of $N$, the number of nodes, $k$ the rate of oscillations and a Sobolev-like regularity of the function. We prove that that the method is not only robust but the error even decreases, for fixed $N$, as $k$ increases. Practical issues about the implementation of the rule are also covered in this paper by: (a) writing down ready-to-implement algorithms; (b) analysing the numerical stability of the computations and (c) estimating the overall computational cost. We finish by showing some numerical experiments which illustrate the theoretical results presented in this paper.