On the numerical quadrature of weakly singular oscillatory integral and its fast implementation

TL;DR

This paper introduces a fast, accurate Clenshaw-Curtis-Filon-type method for numerically integrating weakly singular oscillatory integrals with Fourier and Hankel kernels, leveraging recurrence relations and efficient computation of modified moments.

Contribution

It develops a new recurrence relation-based approach for computing modified moments, enabling efficient and accurate quadrature of weakly singular oscillatory integrals with theoretical error bounds.

Findings

Method achieves $O(N \, \log N)$ computational complexity.

Numerical results confirm high accuracy and efficiency.

Error bounds demonstrate effectiveness for high-frequency integrals.

Abstract

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N\log N)$ operations. The method requires the accurate computation of modified moments. We first give a method for the derivation of the recurrence relation for the modified moments, which can be applied to the derivation of the recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.