Invariant discretization schemes for the shallow-water equations
This work provides a novel method for constructing invariant numerical schemes for shallow-water equations, which is important for computational fluid dynamics and geophysical modeling.
The authors derive invariant discretization schemes for the shallow-water equations, extending difference invariants to finite volume methods. They propose invariant Eulerian schemes using an invariant moving mesh generator and evaluate conservation properties, finding that invariant schemes better preserve energy, mass, and momentum compared to non-invariant ones.
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.