### QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

Jing Gao1, Marissa Condon2, Arieh Iserles3, Benjamin Gilvey4, Jon Trevelyan4

1. 1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China;
2. School of Electronic Engineering, Dublin City University, Ireland;
3. DAMTP, Centre for Mathematical Sciences, University of Cambridge, UK;
4. Department of Engineering, Durham University, UK
• Received:2019-02-19 Revised:2019-10-21 Published:2021-03-15
• Contact: Benjamin Gilvey,Email:benjamin.gilvey@durham.ac.uk;Jon Trevelyan,Email:jon.trevelyan@durham.ac.uk
• Supported by:
The work is supported by Royal Society International Exchanges (grant IE141214), the Projects of International Cooperation and Exchanges NSFC-RS (Grant No. 11511130052), the Key Science and Technology Program of Shaanxi Province of China (Grant No. 2016GY-080) and the Fundamental Research Funds for the Central Universities.

Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey, Jon Trevelyan. QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS[J]. Journal of Computational Mathematics, 2021, 39(2): 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.

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