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

• Published in: Journal of nonlinear mathematical physics Vol. 28; no. 1; pp. 90 - 107
• Main Authors: Dorodnitsyn, V. A., Kaptsov, E. I., Meleshko, S. V.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2021; Springer Nature B.V
• Subjects: Classification; Conservation laws; Fluid flow; Lagrange coordinates; Lie groups; Research Article; Topography; Traveling waves; Lie group; conservation laws; group classification; invariant solutions; invariant difference schemes
• ISSN: 1402-9251; 1776-0852; 1776-0852
• DOI: 10.2991/jnmp.k.200922.007

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.