Invariant discretization schemes for the shallow-water equations

TL;DR

This paper develops invariant discretization schemes for shallow-water equations, extending difference invariants to finite volume methods, and compares their conservation properties and convergence behavior.

Contribution

It introduces invariant finite volume schemes for shallow-water equations using difference invariants and invariant moving meshes, expanding the applicability of invariant discretization methods.

Findings

Classical invariant schemes converge to the Lagrangian formulation.

Invariant schemes require grid redistribution based on fluid velocity.

Energy, mass, and momentum conservation are evaluated for both schemes.

Abstract

Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e., the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations in computational coordinates. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy, mass and momentum is evaluated for both the invariant and non-invariant schemes.