Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates

TL;DR

This paper analyzes the symmetries and conservation laws of the (1+2)-dimensional shallow water equations in Lagrangian coordinates, introduces invariant finite-difference schemes, and explores their properties and discretizations.

Contribution

It provides a group classification for these equations with specific bottom topographies and constructs new invariant conservative finite-difference schemes.

Findings

Group classification for elliptic paraboloid bottom topography

Construction of invariant conservative finite-difference schemes

Schemes preserve mass and energy conservation laws

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 $\gamma=2$ 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.