Shallow water equations in Lagrangian coordinates: symmetries, conservation laws and its preservation in difference models

TL;DR

This paper explores the symmetries and conservation laws of shallow water equations in Lagrangian coordinates, constructs invariant difference schemes preserving these laws, and demonstrates their effectiveness through numerical tests.

Contribution

It introduces invariant difference schemes for shallow water equations in Lagrangian coordinates that preserve key conservation laws, including mass, momentum, and energy.

Findings

Invariant schemes preserve conservation laws in numerical models.

Exact invariant solutions are constructed for the schemes.

Numerical tests show the schemes' effectiveness compared to existing methods.

Abstract

The one-dimensional shallow water equations in Eulerian and Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian (potential) coordinates and symmetries and conservation laws in mass Lagrangian variables. For equations in Lagrangian coordinates with a flat bottom an invariant difference scheme is constructed which possesses all the difference analogues of the conservation laws: mass, momentum, energy, the law of center of mass motion. Some exact invariant solutions are constructed for the invariant scheme, while the scheme admits reduction on subgroups as well as the original system of equations. For an arbitrary shape of bottom it is possible to construct an invariant scheme with conservation of mass and momentum or energy. Invariant conservative difference scheme for the case of a flat bottom tested numerically in comparison with other known schemes.