Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates
arXiv:2012.04410 · doi:10.1016/j.cnsns.2023.107119
Abstract
The two-dimensional shallow water equations in Eulerian and Lagrangain coordinates are considered. Lagrangian and Hamiltonian formalism of the equations is given. The transformations mapping the two-dimensional shallow water equations with a circular or plane bottom into the gas dynamics equations of a polytropic gas with polytropic exponent is represented. Group properties of the equations are considered, and the group classification for the case of the elliptic paraboloid bottom topography is performed. The properties of the two-dimensional shallow water equations in Lagrangian coordinates are discussed from the discretization point of view. New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.
24 pages, 2 figures, 1 table
References in corpus (4)
- Conservation Laws of the Two-Dimensional Gas Dynamics Equations
- Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates
- Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations
- Conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Lagrange's variables