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Quadrature methods for highly oscillatory singular integrals.

Gao, J. and Condon, M. and Iserles, A. and Gilvey, B.D. and Trevelyan, J. (2021) 'Quadrature methods for highly oscillatory singular integrals.', Journal of computational mathematics., 39 (2). pp. 227-260.


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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Publisher statement:First published in Gao, J., Condon, M., Iserles, A., Gilvey, B.D. & Trevelyan, J. (2021). Quadrature methods for highly oscillatory singular integrals. Journal of Computational Mathematics 39(2): 227-260 published by Global Science Press. © Copyright Global Science Press, All right reserved.
Date accepted:11 November 2019
Date deposited:11 November 2019
Date of first online publication:November 2020
Date first made open access:11 November 2020

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