Quadrature Methods for Highly Oscillatory Singular Integrals (Journal of Computational Mathematics, 2020)
We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, theevaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering, and many of these exhibit singularities. We showthat 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 theintegral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructedto approximate the integral. Unlike an asymptotic expansion, the Filon method achieveshigh accuracy for both small and large frequency. Complex-valued quadrature involvesinterpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complex-valued Gaussian quadrature achieves the highestaccuracy when the three methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost ofcomputation of orthogonal polynomials and their zeros.