Skip to main content
SpringerLink
Log in

Error Analysis of a Spectral Projection of the Regularized Benjamin–Ono Equation

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The regularized Benjamin–Ono equation appears in the modeling of long-crested interfacial waves in two-fluid systems. For this equation, Fourier–Galerkin and collocation semi-discretizations are proved to be spectrally convergent. A new exact solution is found and used for the experimental validation of the numerical algorithm. The scheme is then used to study the interaction of two solitary waves.

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

Similar content being viewed by others

References

  1. J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), pp. 345–374.

    Article  Google Scholar 

  2. T. B. Benjamin, Internal Waves of permanent form in fluids of great depth, J. Fluid Mechanics, 29 (1967), pp. 559–592.

    Google Scholar 

  3. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, 1988.

  4. H. Kalisch and J. L. Bona, Models for internal waves in deep water, Disc. Cont. Dyn. Sys., 6 (2000), pp. 1–19.

    Google Scholar 

  5. H.-O. Kreiss and J. Oliger, Stability of the Fourier method, SIAM J. Numer. Anal., 16 (1979), pp. 421–433.

    Article  Google Scholar 

  6. Y. Maday and A. Quarteroni, Error analysis for spectral approximation of the Korteweg–de Vries equation, RAIRO Modl. Math. Anal. Numr., 22 (1988), pp. 499–529.

    Google Scholar 

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

    Article  Google Scholar 

  8. J. E. Pasciak, Spectral and pseudospectral methods for advection equations, Math. Comput., 35 (1980), pp. 1081–1092.

    Google Scholar 

  9. J. E. Pasciak, Spectral methods for a nonlinear initial value problem involving pseudodifferential operators, SIAM J. Numer. Anal., 19 (1982), pp. 142–153.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  12. V. Thomeé and A. S. Vasudeva Murthy, A numerical method for the Benjamin–Ono equation, BIT, 38 (1998), pp. 597–611.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, NTNU, Trondheim, 7491, Norway

    Henrik Kalisch

Authors
  1. Henrik Kalisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henrik Kalisch.

Additional information

AMS subject classification (2000)

35Q53, 65M12, 65M70.

Received September 2004. Revised January 2005. Communicated by Uri Ascher.

Henrik Kalisch: This work was supported in part by the BeMatA program of the Research Council of Norway.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kalisch, H. Error Analysis of a Spectral Projection of the Regularized Benjamin–Ono Equation. Bit Numer Math 45, 69–89 (2005). https://doi.org/10.1007/s10543-005-2636-x

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-005-2636-x

Keywords

Search

Navigation