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On Numerical Solution of the Gardner–Ostrovsky Equation

Published online by Cambridge University Press:  29 February 2012

M. A. Obregon  and
Y. A. Stepanyants
M. A. Obregon
Affiliation:
E.T.S. Ingeniería Industrial, University of Malaga, Dr Ortiz Ramos s/n, 29071, Malaga, Spain
Y. A. Stepanyants*
Affiliation:
Department of Mathematics and Computing, Faculty of Sciences, University of Southern Queensland, Toowoomba, Australia
*
⋆ Corresponding author. E-mail: Yury.Stepanyants@usq.edu.au
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Abstract

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovsky equation (ut + cux + α uux + α1u2ux + βuxxx)x = γu which is also known as the extended rotation-modified Korteweg–de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth’ rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.

Keywords

KdV equationOstrovsky equationsolitonterminal decayPetviashvili methodnumerical scheme
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 113 - 130
Copyright
© EDP Sciences, 2012

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