Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

TL;DR

This paper develops invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates, ensuring conservation laws are preserved numerically across various bottom topographies.

Contribution

It introduces a novel invariant conservative finite-difference scheme for the modified shallow water equations in Lagrangian coordinates, maintaining key conservation laws.

Findings

The scheme preserves mass, momentum, and energy conservation laws.

Numerical tests show improved accuracy over naive invariant schemes.

The approach applies to different bottom topographies, including inclined and parabolic.

Abstract

The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian coordinates. For equations in Lagrangian coordinates an invariant finite-difference scheme is constructed for all cases for which conservation laws exist in the differential model. Such schemes possess the difference analogues of the conservation laws of mass, momentum, energy, the law of center of mass motion for horizontal, inclined and parabolic bottom topographies. Invariant conservative difference scheme is tested numerically in comparison with naive approximation invariant scheme.