Volume 39, Issue 2
Quadrature Methods for Highly Oscillatory Singular Integrals

Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey & Jon Trevelyan

J. Comp. Math., 39 (2021), pp. 227-260.

Published online: 2020-11

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65D32, 41A55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jgao@xjtu.edu.cn (Jing Gao)

marissa.condon@dcu.ie (Marissa Condon)

ai10@cam.ac.uk (Arieh Iserles)

benjamin.gilvey@durham.ac.uk (Benjamin Gilvey)

jon.trevelyan@durham.ac.uk (Jon Trevelyan)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-227, author = {Gao , Jing and Condon , Marissa and Iserles , Arieh and Gilvey , Benjamin and Trevelyan , Jon }, title = {Quadrature Methods for Highly Oscillatory Singular Integrals}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {39}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1911-m2019-0044}, url = {http://global-sci.org/intro/article_detail/jcm/18373.html} }
TY - JOUR T1 - Quadrature Methods for Highly Oscillatory Singular Integrals AU - Gao , Jing AU - Condon , Marissa AU - Iserles , Arieh AU - Gilvey , Benjamin AU - Trevelyan , Jon JO - Journal of Computational Mathematics VL - 2 SP - 227 EP - 260 PY - 2020 DA - 2020/11 SN - 39 DO - http://doi.org/10.4208/jcm.1911-m2019-0044 UR - https://global-sci.org/intro/article_detail/jcm/18373.html KW - Numerical quadrature, Singular highly oscillatory integrals, Asymptotic analysis, Boundary Element Method, Plane wave enrichment, Partition of Unity. AB -

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.

Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey & Jon Trevelyan. (2020). Quadrature Methods for Highly Oscillatory Singular Integrals. Journal of Computational Mathematics. 39 (2). 227-260. doi:10.4208/jcm.1911-m2019-0044
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