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Quadrature Methods for Highly Oscillatory Singular Integrals

Quadrature Methods for Highly Oscillatory Singular Integrals

Year:    2021

Author:    Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey, Jon Trevelyan

Journal of Computational Mathematics, Vol. 39 (2021), Iss. 2 : pp. 227–260

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1911-m2019-0044

Journal of Computational Mathematics, Vol. 39 (2021), Iss. 2 : pp. 227–260

Published online:    2021-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    34

Keywords:    Numerical quadrature Singular highly oscillatory integrals Asymptotic analysis Boundary Element Method Plane wave enrichment Partition of Unity.

Author Details

Jing Gao

Marissa Condon

Arieh Iserles

Benjamin Gilvey

Jon Trevelyan

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