Skip to main content
SpringerLink
Log in

Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The original solutions of highly oscillatory integral equations usually have rapid oscillation, which means that conventional numerical approaches used to solve these equations have poor convergence. In order to overcome this difficulty, in this paper, we propose and analyze an oscillation-preserving Legendre-Galerkin method for second kind integral equations with highly oscillatory kernels. Concretely, we first incorporate the standard approximation subspace of Legendre polynomial basis with a finite number of oscillatory functions which capture the oscillation of the exact solutions. Then, we construct an efficient numerical integration scheme, yielding a fully discrete linear system. Making use of best approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces and the compactness operator theory, we establish that the fully discrete approximate equation has a unique solution and the approximate solution reaches an optimal convergence order without the influence of the wave number. In addition, we prove that for sufficiently large wave number, the spectral condition number of the corresponding linear system is uniformly bounded. At last, we use numerical examples to demonstrate the effectiveness of the proposed method.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Atkinson, K.E.: The Numerical Solution of Integral Equations of Second Kind. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  2. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  3. Brunner, H.: On Volterra integral operators with highly oscillatory kernels. Discr. Cont. Dynam. Syst. 34, 915–929 (2014)

    Article  MathSciNet  Google Scholar 

  4. Brunner, H., Iserles, A., Norsett, S.P.: The computation of the spectra of highly oscillatory Fredholm integral operators. J. Int. Equat. Appl. 23, 467–519 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Brunner, H., Iserles, A., Norsett, S.P.: The spectral problem for a class of highly oscillatory Fredholm integral operators. IMA J. Numer. Anal. 30, 108–130 (2010)

    Article  MathSciNet  Google Scholar 

  6. Brunner, H., Ma, Y., Xu, Y.: The oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernels. J. Integ. Equ. Appl. 27, 455–487 (2015)

    Article  MathSciNet  Google Scholar 

  7. Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations. Math. Comput. 79, 147–167 (2010)

    Article  Google Scholar 

  8. Domspnguez, V., Graham, I.G., Kim, T.: Filon-Clenshaw-Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIAM J. Numer. Anal. 51, 1542–566 (2013)

    Article  MathSciNet  Google Scholar 

  9. Domspnguez, V., Graham, I.G., Smyshlyaev, V.P.: Stability and error estimates for Filon-Clenshaw- Curtis rules for highly oscillatory integrals. IMA J. Numer Anal. 31, 1253–1280 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Huybrechs, D., Vandewalle, S.: The construction of cubature rules for multivariate highly oscilla- tory integrals. Math. Comp. 76, 1955–1980 (2007)

    Article  MathSciNet  Google Scholar 

  12. Iserles, A., Norsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. Amer. Math. Soc. 461, 1383–1399 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Iserles, A., Norsett, S.P.: Quadrature methods for multivariate highly oscillatory integrals using derivatives. Math. Comp. 75, 1233–1258 (2006)

    Article  MathSciNet  Google Scholar 

  14. Kress, R.: Linear Integral Equations. Springer, Berlin (2001)

    MATH  Google Scholar 

  15. Levin, D.: Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comp. Appl. Math. 78, 131–138 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Ma, Y., Xu, Y.: Computing integrals involved the Gaussian function with a small standard deviation. J. Sci. Comput. 78, 1744–1767 (2019)

    Article  MathSciNet  Google Scholar 

  18. Ma, Y., Xu, Y.: Computing highly oscillatory integrals. Math. Comput. 87, 309–345 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, New York (2011)

    Book  Google Scholar 

  22. Stein, E., Harmonic analysis: Real-variable method, orthogonality, and oscillatory integral (1993)

  23. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra type integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numerica 21, 89–305 (2012)

    Article  MathSciNet  Google Scholar 

  25. Chandler-Wilde, S.N., Langdon, S., Ritter, L.: A high¨Cwavenumber boundary¨Celement method for an acoustic scattering problem. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362, 647–671 (2004)

    Article  MathSciNet  Google Scholar 

  26. Wang, H., Xiang, S.: Asymptotic expansion and Filon-type methods for Volterra integral equation with a highly oscillatory kernel. IMA J. Numer Anal. 31, 469–490 (2011)

    Article  MathSciNet  Google Scholar 

  27. Wang, Y., Xu, Y.: Oscillation preserving Galerkin Methods for Fredholm integral equations of the second kind with oscillatory kernels. arXiv:1507.01156 (2015)

  28. Wang, Y., Xiang, S.: Levin methods for highly oscillatory integrals with singularities. Sci. China-Math. 63. https://doi.org/10.1007/s11425-018-1626-x (2020)

  29. Pastore, P.: The numerical treatment of Love’s integral equation having very small parameter. J. Comput. Appl. Math. 236, 1167–1281 (2011)

    Article  MathSciNet  Google Scholar 

  30. Fermo, L., Russo, M.G., Serafini, G.: Numerical treatment of the generalized Love integral equation. Numer. Algor. 86, 179–1789 (2021)

    Article  MathSciNet  Google Scholar 

  31. Fermo, L., Van Der Mee, C.: Volterra integral equations with highly oscillatory kernels: a new numerical method with applications. ETNA 54, 333–354 (2021)

    Article  MathSciNet  Google Scholar 

  32. Xiang, S.: Effcient Filon-type methods for \({\int \limits }_{a}^{b}f(x)e^{i{\omega } g(x)}dx\). Numer. Math. 105, 633–658 (2007)

    Article  MathSciNet  Google Scholar 

  33. Xiang, S., Brunner, H.: Effcient methods for Volterra integral equations with highly oscillatory Bessel kernels. BIT Numer. Math. 53, 241–263 (2013)

    Article  Google Scholar 

  34. Xiang, S., Cho, Y.J., Wang, H., Brunner, H.: Clenshaw-Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications. IMA J. Numer. Anal. 31, 1281–1314 (2011)

    Article  MathSciNet  Google Scholar 

  35. Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53, 414–434 (2012)

    Article  MathSciNet  Google Scholar 

  36. Yi, L., Guo, B.: An h-p Version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53, 2677–2704 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The research of this author is supported by the National Natural Science Foundation of China (12171278, 11971259) and by National Science Foundation of Shandong Province (ZR2020MA047).

Author information

Authors and Affiliations

  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Haotao Cai

Authors
  1. Haotao Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haotao Cai.

Ethics declarations

Conflict of interest

The author declares no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cai, H. Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels. Numer Algor 90, 1091–1115 (2022). https://doi.org/10.1007/s11075-021-01223-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01223-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation