Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 90 - 107

Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-dimensional Green-Naghdi Equations

Authors
V.A. Dorodnitsyn1, E.I. Kaptsov1, 2, S.V. Meleshko2, *
1Keldysh Institute of Applied Mathematics, Russian Academy of Science, Miusskaya Pl. 4, Moscow, 125047, Russia
2School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand
*Corresponding author. Email: sergey@math.sut.ac.th
Corresponding Author
S.V. Meleshko
Received 29 June 2020, Accepted 20 July 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.200922.007How to use a DOI?
Keywords
Lie group, group classification, invariant solutions, conservation laws, invariant difference schemes
Abstract

The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated form of the same order. The study is performed in Lagrangian coordinates which allows one to find Lagrangians for the analyzed equations. Complete group classification of both cases of the Green-Naghdi equations with respect to the bottom topography is presented. Applying Noether’s theorem, the obtained Lagrangians and the group classification, conservation laws of the one-dimensional Green-Naghdi equations with uneven bottom topography are obtained. Difference schemes which preserve the symmetries of the original equations and the conservation laws are constructed. Analysis of the developed schemes is given. The schemes are tested numerically on the example of an exact traveling-wave solution.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 1
Pages
90 - 107
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.200922.007How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - V.A. Dorodnitsyn
AU  - E.I. Kaptsov
AU  - S.V. Meleshko
PY  - 2020
DA  - 2020/12
TI  - Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-dimensional Green-Naghdi Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 90
EP  - 107
VL  - 28
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.200922.007
DO  - https://doi.org/10.2991/jnmp.k.200922.007
ID  - Dorodnitsyn2020
ER  -