Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates

TL;DR

This paper develops invariant conservative difference schemes for 1D shallow water equations in Eulerian and Lagrangian coordinates, preserving conservation laws and handling complex bottom topographies with numerical verification.

Contribution

It introduces new invariant difference schemes that maintain conservation laws for shallow water equations in both coordinate systems, including cases requiring moving meshes.

Findings

Schemes preserve all key conservation laws.

Numerical tests confirm accuracy across various bottom topographies.

Moving meshes are effectively used in Eulerian coordinates.

Abstract

The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations, and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogues of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.