265
Views
12
CrossRef citations to date
0
Altmetric
Original Articles

Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method

, &
Pages 83-107 | Received 03 Mar 2014, Accepted 04 Nov 2014, Published online: 16 Dec 2014

References

  • M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.
  • S. Arden, S.N. Chandler-Wilde, and S. Langdon, A collocation method for high-frequency scattering by convex polygons, J. Comput. Appl. Math. 204 (2007), pp. 334–343. doi: 10.1016/j.cam.2006.03.028
  • A. Asheim and D. Huybrechs, Complex Gaussian quadrature for oscillatory integral transforms, IMA J. Numer. Anal. 112 (2013), pp. 197–219.
  • G. Bao and W. Sun, A fast algorithm for the electromagnetic scattering from a large cavity, SIAM J. Sci. Comput. 27 (2005), pp. 553–574. doi: 10.1137/S1064827503428539
  • J.S. Ball and N.H.F. Beebe, Efficient Gauss-related quadrature for two classes of logarithmic weight functions, ACM Trans. Math. Softw. 33(3) (2007), 21 p. Article 19. doi: 10.1145/1268769.1268773
  • N. Bleistein and R. Handelsman, A generalization of the method of steepest descent, IMA J. Numer. Anal. 10 (1972), pp. 211–230.
  • S.N. Chandler-Wilde, I.G. Graham, S. Langdon, and E.A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numerica 21 (2012), pp. 89–305. doi: 10.1017/S0962492912000037
  • S.N. Chandler-Wilde and S. Langdon, A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J. Numer. Anal. 45 (2007), pp. 610–640. doi: 10.1137/06065595X
  • R. Chen and C. An, On evaluation of Bessel transforms with oscillatory and algebraic singular integrands, J. Comput. Appl. Math. 264 (2014), pp. 71–81. doi: 10.1016/j.cam.2014.01.009
  • A. Deaño and D. Huybrechs, Complex Gaussian quadrature of oscillatory integrals, Numer. Math. 112 (2009), pp. 197–219. doi: 10.1007/s00211-008-0209-z
  • V. Domínguez, Filon–Clenshaw–Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities, J. Comput. Appl. Math. 261 (2014), pp. 299–319. doi: 10.1016/j.cam.2013.11.012
  • V. Domínguez, I.G. Graham, and V.P. Smyshlyaev, A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering, Numer. Math. 106 (2007), pp. 471–510. doi: 10.1007/s00211-007-0071-4
  • V. Domínguez, I.G. Graham, and V.P. Smyshlyaev, Stability and error estimates for Filon–Clenshaw–Curtis rules for highly-oscillatory integrals, IMA J. Numer. Anal. 31 (2011), pp. 1253–1280. doi: 10.1093/imanum/drq036
  • V. Domínguez, I.G. Graham, and T. Kim, Filon–Clenshaw–Curtis rules for highly-oscillatory integrals with algebraic singularities and stationary points, SIAM J. Numer. Anal. 51 (2013), pp. 1542–1566. doi: 10.1137/120884146
  • I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007.
  • A. Glaser, X. Liu, and V. Rokhlin, A fast algorithm for the calculation of the roots of special functions, SIAM J. Sci. Comput. 29 (2007), pp. 1420–1438. doi: 10.1137/06067016X
  • G.H. Golub and J.A. Welsch, Calculation of Gauss quadrature rules, Math. Comput. 23 (1969), pp. 221–230. doi: 10.1090/S0025-5718-69-99647-1
  • W. Gautschi, Orthogonal polynomials: Computation and Approximation, Oxford University Press, New York, 2004.
  • W. Gautschi, Gauss quadrature routines for two classes of logarithmic weight functions, Numer. Algor. 55 (2010), pp. 265–277. doi: 10.1007/s11075-010-9366-0
  • D. Huybrechs and S. Vandewalle, On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation, SIAM J. Numer. Anal. 44 (2006), pp. 1026–1048. doi: 10.1137/050636814
  • D. Huybrechs and S. Vandewalle, A sparse discretisation for integral equation formulations of high frequency scattering problems, SIAM J. Sci. Comput. 29 (2007), pp. 2305–2328. doi: 10.1137/060651525
  • A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A 461 (2005), pp. 1383–1399. doi: 10.1098/rspa.2004.1401
  • A. Iserles and S.P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comput. 75 (2006), pp. 1233–1258. doi: 10.1090/S0025-5718-06-01854-0
  • S.U. Islam and S. Zaman, New quadrature rules for highly oscillatory integrals with stationary points, J. Comput. Appl. Math. 278 (2015), pp. 75–89.
  • S.U. Islam, A.S. Al-Fahid, and S. Zaman, Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points, Eng. Anal. Bound. Elem. 37 (2013), pp. 1136–1144.
  • S.U. Islam, I. Aziz, and W. Khan, Numerical integration of multidimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions, Eng. Anal. Bound. Elem. 36 (2012), pp. 1684–1695.
  • H. Kang, S. Xiang, and G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions, J. Comput. Appl. Math. 242 (2013), pp. 141–156. doi: 10.1016/j.cam.2012.10.016
  • T.L. Li, J.H. Lee, and Y.F. Gao, An approximate formulation of the effective indentation modulus of elastically anisotropic film-on-substrate systems, Int. J. Appl. Mech. 1(03) (2009), pp. 515–525. doi: 10.1142/S1758825109000241
  • D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comput. 38 (1982), pp. 531–538. doi: 10.1090/S0025-5718-1982-0645668-7
  • D. Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math. 67 (1996), pp. 95–101. doi: 10.1016/0377-0427(94)00118-9
  • D. Levin, Analysis of a collocation method for integrating rapidly oscillatory functions, J. Comput. Appl. Math. 78 (1997), pp. 131–138. doi: 10.1016/S0377-0427(96)00137-9
  • R. Piessens and M. Branders, On the computation of Fourier transforms of singular functions, J. Comput. Appl. Math. 43 (1992), pp. 159–169. doi: 10.1016/0377-0427(92)90264-X
  • G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.
  • H. Wang and S. Xiang, On the evaluation of Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 234 (2010), pp. 95–100. doi: 10.1016/j.cam.2009.12.007
  • G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1966.
  • R. Wong, Quadrature formulas for oscillatory integral transforms, Numer. Math. 39 (1982), pp. 351–360. doi: 10.1007/BF01407868
  • S. Xiang, G. He, and Y. Cho, On error bounds of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals, Adv. Comput. Math. (2014). DOI 10.1007/s10444-014-9377-9.
  • S. Xiang, Y. Cho, H. Wang, and H. Brunner, Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications, IMA J. Numer. Anal. 31 (2011), pp. 1281–1314. doi: 10.1093/imanum/drq035
  • S. Xiang, On the Filon and Levin methods for highly oscillatory integral ∫abf(x)eiωg(x)dx, J. Comput. Appl. Math. 208 (2007), pp. 434–439. doi: 10.1016/j.cam.2006.10.006
  • S. Xiang and H. Wang, On the Levin iterative method for oscillatory integrals, J. Comput. Appl. Math. 217 (2008), pp. 38–45. doi: 10.1016/j.cam.2007.06.012

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.