Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations
arXiv:2008.12852 · doi:10.2991/jnmp.k.200922.007
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.
22 pages, 8 figures
References in corpus (1)
Cited by in corpus (9)
- Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates
- Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates
- Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates
- Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations
- Lagrangian formalism and Noether-type theorems for second-order delay ODEs
- Group classification of the two-dimensional Green-Naghdi equations with a time dependent bottom topography
- Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation
- Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates
- On the correspondence between variational principles in Eulerian and Lagrangian descriptions