Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Levin methods for highly oscillatory integrals with singularities


In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.

This is a preview of subscription content, log in to check access.


  1. 1

    Abramowitz M, Stegun I-A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications, 1972

  2. 2

    Bruno O, Geuzaine C, Monro J, et al. Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: The convex case. Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 2004, 362: 629–645

  3. 3

    Chandler-Wilde S-N, Graham I-G, Langdon S, et al. Numerical-asymptotic boundary integral methods in highfrequency acoustic scattering. Acta Numer, 2012, 21: 89–305

  4. 4

    Chung K-C, Evans G-A, Webster J-R. A method to generate generalized quadrature rules for oscillatory integrals. Appl Numer Math, 2000, 34: 85–93

  5. 5

    Colton D, Kress R. Integral Equation Methods in Scattering Theory. New York: Wiley, 1983

  6. 6

    Deaño A, Huybrechs D, Iserles A. Computing Highly Oscillatory Integrals. Philadelphia: SIAM, 2018

  7. 7

    Domínguez V, Graham I-G, Kim T. Filon-Clenshaw-Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIAM J Numer Anal, 2013, 51: 1542–1566

  8. 8

    Erdelyi A. Asymptotic representations of Fourier integrals and the method of stationary phase. J Soc Ind Appl Math, 1955, 3: 17–27

  9. 9

    Gao J, Condon M, Iserles A. Quadrature methods for highly oscillatory singular integrals., 2016

  10. 10

    Gao J, Iserles A. A generalization of Filon-Clenshaw-Curtis quadrature for hihgly oscillatory integrals. BIT, 2017, 4: 1–19

  11. 11

    He G, Xiang S, Zhu E. Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method. Int J Comput Math, 2014, 93: 1–25

  12. 12

    Huybrechs D, Olver S. Highly oscillatory quadrature. In: Highly Oscillatory Problems. Cambridge: Cambridge University Press, 2009, 25–50

  13. 13

    Huybrechs D, Vandewalle S. On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J Numer Anal, 2006, 44: 1026–1048

  14. 14

    Iserles A. On the numerical quadrature of highly-oscillating integrals i: Fourier transforms. IMA J Numer Anal, 2004, 24: 365–391

  15. 15

    Iserles A, Nørsett S-P. On quadrature methods for highly oscillatory integrals and their implementation. BIT, 2004, 44: 755–772

  16. 16

    Iserles A, Nørsett S-P. Efficient quadrature of highly oscillatory integrals using derivatives. Proc R Soc Lond Ser A Math Phy Eng Sci, 2005, 461: 1383–1399

  17. 17

    Kang H, Xiang S. Efficient integration for a class of highly oscillatory integrals. Appl Math Comput, 2011, 218: 3553–3564

  18. 18

    Kang H, Xiang S, He G. Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. J Comput Appl Math, 2013, 242: 141–156

  19. 19

    Levin D. Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math Comp, 1982, 38: 531–538

  20. 20

    Li J, Wang X, Wang T. A universal solution to one-dimensional oscillatory integrals. Sci China Ser F, 2008 51: 1614–1622

  21. 21

    Lyness J, Lottes J. Asymptotic expansions for oscillatory integrals using inverse functions. BIT, 2009, 49: 397–417

  22. 22

    Ma Y, Xu Y. Computing highly oscillatory integrals. Math Comp, 2017, 87: 309–345

  23. 23

    Olver S. Moment-free numerical integration of highly oscillatory functions. IMA J Numer Anal, 2006, 26: 213–227

  24. 24

    Olver S. Fast, numerically stable computation of oscillatory integrals with stationary points. BIT, 2010, 50: 149–171

  25. 25

    Olver S. Shifted GMRES for oscillatory integrals. Numer Math, 2010, 114: 607–628

  26. 26

    Piessens R, Branders M. On the computation of Fourier transforms of singular functions. J Comput Appl Math, 1992, 43: 159–169

  27. 27

    Shen J, Tang T, Wang L. Spectral Methods: Algorithms, Analysis and Applications. Heidelberg: Springer, 2011

  28. 28

    Spence E-A, Kamotski I-V, Smyshlyaev V-P. Coercivity of combined boundary integral equations in high-frequency scattering. Comm Pure Appl Math, 2014, 68: 1587–1639

  29. 29

    Wang Y, Xiang S. tA Levin method for logarithmically singular oscillatory integrals. ArXiv:1901.05192, 2019

  30. 30

    Xiang S. Efficient Filon-type methods for ∫abf(x) eiwg(x) dx. Numer Math, 2007, 105: 633–658

  31. 31

    Xiang S. Numerical analysis of a fast integration method for highly oscillatory functions. BIT, 2007, 47: 469–482

  32. 32

    Xiang S. On the Filon and Levin methods for highly oscillatory integral. J Comput Appl Math, 2007, 208: 434–439

  33. 33

    Xiang S, Chen X, Wang H. Error bounds for approximation in Chebyshev points. Numer Math, 2010, 116: 463–491

  34. 34

    Xiang, He G, Cho Y. On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals. Adv Comput Math, 2014, 41: 573–597

  35. 35

    Xu Z, Milovanović G-V, Xiang S. Efficient computation of highly oscillatory integrals with Hankel kernel. Appl Math Comput, 2015, 261: 312–322

  36. 36

    Xu Z, Xiang S. Gauss-type quadrature for highly oscillatory integrals with algebraic singularities and applications. Int J Comput Math, 2017, 94: 1123–1137

Download references


This work was supported by National Natural Science Foundation of China (Grant No. 11771454) and Research Fund of National University of Defense Technology (Grant No. ZK19-19). The authors are grateful to the referees’ helpful suggestions and insightful comments, which helped improve the manuscript significantly. The authors thank Dr. Saira and Dr. Suliman at Central South University for their careful checking of numerous details.

Author information

Correspondence to Shuhuang Xiang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Xiang, S. Levin methods for highly oscillatory integrals with singularities. Sci. China Math. (2020).

Download citation


  • Levin method
  • highly oscillatory integral
  • algebraic singularity
  • logarithmic singularity


  • 65D30
  • 65D32
  • 65L99