On Numerical Solution of the Gardner–Ostrovsky Equation

M. A. Obregon; Y. A. Stepanyants

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 113-130
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Obregon, M. A., and Stepanyants, Y. A.. "On Numerical Solution of the Gardner–Ostrovsky Equation." Mathematical Modelling of Natural Phenomena 7.2 (2012): 113-130. <http://eudml.org/doc/222297>.

@article{Obregon2012,
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.},
author = {Obregon, M. A., Stepanyants, Y. A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {KdV equation; Ostrovsky equation; soliton; terminal decay; Petviashvili method; numerical scheme},
language = {eng},
month = {2},
number = {2},
pages = {113-130},
publisher = {EDP Sciences},
title = {On Numerical Solution of the Gardner–Ostrovsky Equation},
url = {http://eudml.org/doc/222297},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Obregon, M. A.
AU - Stepanyants, Y. A.
TI - On Numerical Solution of the Gardner–Ostrovsky Equation
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 113
EP - 130
AB - 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.
LA - eng
KW - KdV equation; Ostrovsky equation; soliton; terminal decay; Petviashvili method; numerical scheme
UR - http://eudml.org/doc/222297
ER -

References

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