Previous Articles    

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.

CLC Number: 

[1] S. Amari, J. Bornemann, Efficient numerical computation of singular integrals with applications to electromagnetics. IEEE Trans. Antennas Propag., 43(1995), 1343-1348.
[2] A. Asheim, A. Deaño, D. Huybrechs, H. Wang, A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discret. Contin. Dyn. Syst. A, 34(2014), 883-901.
[3] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. Auckland:McGraw-Hill; 1978.
[4] S.N. Chandler-Wilde, I.G. Graham, S. Langdon, E.A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer., 21(2012), 89-305.
[5] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration[2nd ed]. New York:Academic Press; 1984.
[6] A. Deaño, D. Huybrechs, A. Iserles, The kissing polynomials and their Hankel derminants. 2015. http://www.damtp.cam.ac.uk/user/na/NApapers/NA201501.pdf.
[7] A. Deaño, D. Huybrechs, A. Iserles, Computing Highly Oscillatory Integrals. Philadelphia:SIAM; 2018.
[8] V. Dominguez, Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities. J. Comp. Appl. Math., 261(2014), 299-319.
[9] G.A. Evans, K.C. Chung, Some theoretical aspects of generalised quadrature methods. J. Complexity, 19(2003), 272-285.
[10] L.O. Fichte, S. Lange, M. Clemens, Numerical quadrature for the approximation of singular oscillating integrals. Adv. Radio Sci., 4(2006), 11-15.
[11] M.E. Honnor, J. Trevelyan, D. Huybrechs, Numerical evaluation of 2D partition of unity boundary integrals for Helmholtz problems. J. Comp. Appl. Math., 234(2010), 1656-1662.
[12] D. Huybrechs, S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal., 44(2006), 1026-1048.
[13] A. Iserles, S.P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math., 44(2004), 755-772.
[14] A. Iserles, S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives. Proc. Royal Soc. A, 461(2005), 1383-1399.
[15] H. Kang, S. Xiang, G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. J. Comp. Appl. Math., 242(2013), 141-156.
[16] D. Levin, Procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations. Math. Comput., 38(1982), 531-538.
[17] A. Maher, N.B. Pleshchinskii, Plane electromagnetic wave scattering and diffraction in a stratified medium. Paper presented at:International Conference on Mathematical Methods in Electromagnetic Theory, 2000, Kharkov, Ukraine.
[18] K. Nesvit, Scattering and propagation of the TE/TM waves on pre-fractal impedance grating in numerical results. Paper presented at:8th European Conference on Antennas and Propagation; (2014), 2773-2777. Hague, Netherlands.
[19] J. Niegemann, Efficient cubature rules for the numerical integration of logarithmic singularities. Paper presented at:2014 International Conference on Electromagnetics in Advanced Applications; (2014), 601-604. Palm Beach, Aruba.
[20] S. Olver, Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Anal., 26(2006), 213-227.
[21] E. Perrey-Debain, J. Trevelyan, P. Bettess, Wave boundary elements:a theoretical overview presenting applications in scattering of short waves. Eng. Anal. Bound. Elem., 28(2004), 131- 141.
[22] P. Sjölin, Some remarks on singular oscillatory integrals and convolution operators. Proc. Am. Math. Soc., 145(2017), 3843-3848.
[23] E.M. Stein, Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ:Princeton University Press; 1993.
[24] R. Wong, Asymptotic Approximations of Integrals. New York:Academic Press; 1989.
[1] Heiko Gimperlein, Ceyhun Özdemir, Ernst P. Stephan. TIME DOMAIN BOUNDARY ELEMENT METHODS FOR THE NEUMANN PROBLEM: ERROR ESTIMATES AND ACOUSTIC PROBLEMS [J]. Journal of Computational Mathematics, 2018, 36(1): 70-89.
[2] H. Harbrecht, M. Utzinger. ON ADAPTIVE WAVELET BOUNDARY ELEMENT METHODS [J]. Journal of Computational Mathematics, 2018, 36(1): 90-109.
[3] Ernst P. Stephan. The hp-Version of BEM - Fast Convergence, Adaptivity and Efficient Preconditioning [J]. Journal of Computational Mathematics, 2009, 27(2-3): 348-359.
[4] Sheng Zhang,Dehao Yu. MULTIGRID ALGORITHM FOR THE COUPLING SYSTEM OF NATURAL BOUNDARYELEMENT METHOD AND FINITE ELEMENT METHOD FOR UNBOUNDED DOMAINPROBLEMS [J]. Journal of Computational Mathematics, 2007, 25(1): 13-026.
[5] Yun-qing Huang,Wei Li,Fang Su. OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCALPOLYNOMIAL APPROXIMATION SPACES [J]. Journal of Computational Mathematics, 2006, 24(3): 365-372.
[6] Ping Bing MING,Zhong Ci SHI. MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q1 ELEMENT Ⅲ: THE EFFECT OF NUMERICAL INTEGRATION [J]. Journal of Computational Mathematics, 2003, 21(3): 287-294.
Viewed
Full text


Abstract