INVARIANT DISCRETIZATION SCHEMES FOR THE SHALLOW-WATER EQUATIONS

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 use of difference invariants to allow construction 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 redistribution of 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 noninvariant schemes.