Quadrature methods for highly oscillatory singular integrals

Gao, J.; Condon, M.; Iserles, A.; Gilvey, B.D.; Trevelyan, J.

Abstract

We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering, and many of these exhibit singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructed to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves high accuracy for both small and large frequency. Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost of computation of orthogonal polynomials and their zeros.

Citation

Gao, J., Condon, M., Iserles, A., Gilvey, B., & Trevelyan, J. (2021). Quadrature methods for highly oscillatory singular integrals. Journal of computational mathematics, 39(2), 227-260. https://doi.org/10.4208/jcm.1911-m2019-0044

• Journal Article Type: Article
• Acceptance Date: Nov 11, 2019
• Publication Date: 2021
• Deposit Date: Nov 11, 2019
• Publicly Available Date: Nov 11, 2020
• Journal: Journal of Computational Mathematics.
• Print ISSN: 0254-9409
• Electronic ISSN: 1991-7139
• Publisher: 科学出版社 = Science Press
• Peer Reviewed: Peer Reviewed
• Volume: 39
• Issue: 2
• Pages: 227-260
• DOI: https://doi.org/10.4208/jcm.1911-m2019-0044
• Public URL: https://durham-repository.worktribe.com/output/1277772