Skip to main content
SpringerLink
Log in

Convergence of finite difference schemes for the Benjamin–Ono equation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we analyze finite difference schemes for Benjamin–Ono equation, \(u_t= u u_x + H u_{xx}\), where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data are sufficiently regular, fully discrete finite difference schemes shown to converge to a classical solution. Finally, the convergence is illustrated by several examples.

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

Similar content being viewed by others

References

  1. Ablowitz, M.J., Fokas, A.S.: The inverse scattering transform for the Benjamin–Ono equation, a pivot for multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benjamin, T.B.: Internal waves of permanent form in fluid of great depth. J. Fluid Mech. 29, 559–592 (1967)

    Article  MATH  Google Scholar 

  3. Boyd, J.P., Xu, Z.: Comparison of three spectral methods for the Benjamin–Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions. Wave Motion 48, 702–706 (2011)

    Article  MathSciNet  Google Scholar 

  4. Burq, N., Planchon, F.: On well-posedness for the Benjamin–Ono equation. Math. Ann. 340, 497–542 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deng, Z., Ma, H.: Optimal error estimates of the Fourier spectral method for a class of nonlocal, nonlinear dispersive wave equations. Appl. Numer. Math. 59, 988–1010 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dutta, R., Holden, H., Koley, U., Risebro, N.H.: Operator splitting for the Benjamin–Ono equation. J. Differ. Equ. 259, 6694–6717 (2015). doi:10.1016/j.jde.2015.08.002

  7. Holden, H., Koley, U., Risebro, N.H.: Convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation. IMA J. Numer. Anal. 35, 1047–1077 (2015). doi:10.1093/imanum/dru040

  8. Ionescu, A., Kenig, C.E.: Global well posedness of the Benjamin–Ono equation in low regularity spaces. J. Am. Math. Soc. 20, 753–798 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iorio, R.: On the Cauchy problem for the Benjamin–Ono equation. Commun. Part. Differ. Equ. 11, 1031–1081 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. King, F.W.: Hilbert Transforms. Vol. 1. Cambridge UP, Cambridge (2009)

  11. Molinet, L.: Global well-posedness in \(L^2\) for the periodic Benjamin–Ono equation. Am. J. Math. 130, 635–683 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ono, H.: Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39(4), 1082–1091 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pelloni, B., Dougalis, V.A.: Numerical solution of some nonlocal, nonlinear dispersive wave equations. J. Nonlinear Sci. 10, 1–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pelloni, B., Dougalis, V.A.: Error estimate for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations. Appl. Numer. Math. 37, 95–107 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ponce, G.: On the global well posedness of the Benjamin–Ono equation. Differ. Int. Equ. 4, 527–542 (1991)

    MathSciNet  MATH  Google Scholar 

  16. Satsuma, J., Ishimori, Y.: Periodic wave and rational soliton solutions of the Benjamin–Ono equation. J. Phys. Soc. Japan 46, 681–687 (1979)

    Article  Google Scholar 

  17. Sjöberg, A.: On the Korteweg–de Vries equation: existence and uniqueness. J. Math. Anal. Appl. 29, 569–579 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tao, T.: Global well-posedness of the Benjamin–Ono equation in \(H^1(\mathbb{R})\). J. Hyperbolic Differ. Equ. 1(1), 27–49 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thomee, V., Murthy, A.S.V.: A numerical method for the Benjamin–Ono equation. BIT 38(3), 597–611 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Rajib Dutta, Helge Holden & Nils Henrik Risebro

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  3. Tata Institute of Fundamental Research Centre, Centre For Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Sharada Nagar, Chikkabommasandra, Bangalore, 560065, India

    Ujjwal Koley

Authors
  1. Rajib Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helge Holden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ujjwal Koley
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nils Henrik Risebro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helge Holden.

Additional information

Supported in part by the Research Council of Norway and the Alexander von Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dutta, R., Holden, H., Koley, U. et al. Convergence of finite difference schemes for the Benjamin–Ono equation. Numer. Math. 134, 249–274 (2016). https://doi.org/10.1007/s00211-015-0778-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0778-6

Mathematics Subject Classification

Search

Navigation