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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), pp. 345–374.
T. B. Benjamin, Internal Waves of permanent form in fluids of great depth, J. Fluid Mechanics, 29 (1967), pp. 559–592.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, 1988.
H. Kalisch and J. L. Bona, Models for internal waves in deep water, Disc. Cont. Dyn. Sys., 6 (2000), pp. 1–19.
H.-O. Kreiss and J. Oliger, Stability of the Fourier method, SIAM J. Numer. Anal., 16 (1979), pp. 421–433.
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.
H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), pp. 1082–1091.
J. E. Pasciak, Spectral and pseudospectral methods for advection equations, Math. Comput., 35 (1980), pp. 1081–1092.
J. E. Pasciak, Spectral methods for a nonlinear initial value problem involving pseudodifferential operators, SIAM J. Numer. Anal., 19 (1982), pp. 142–153.
B. Pelloni and V. A. Dougalis, Numerical solution of some nonlocal, nonlinear dispersive wave equations, J. Nonlinear Sci., 10 (2000), pp. 1–22.
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.
V. Thomeé and A. S. Vasudeva Murthy, A numerical method for the Benjamin–Ono equation, BIT, 38 (1998), pp. 597–611.
Author information
Authors and Affiliations
Corresponding author
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
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
Issue Date:
DOI: https://doi.org/10.1007/s10543-005-2636-x