
QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS
Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey, Jon Trevelyan
Journal of Computational Mathematics
2021, 39 (2):
227260.
DOI: 10.4208/jcm.1911m20190044
We address the evaluation of highly oscillatory integrals, with powerlaw and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the evaluation of oscillatory integrals dominates the runtime for waveenriched 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 Filontype method is constructed to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves high accuracy for both small and large frequency. Complexvalued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complexvalued 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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