# 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

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topObregon, 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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