Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
This work provides a numerical scheme for geophysical fluid dynamics simulations on the sphere, addressing the need for stable and accurate discretizations of shallow water equations.
The paper presents a compatible finite element discretization for the nonlinear rotating shallow water equations on the sphere, integrating consistent upwind stabilization to dissipate enstrophy at the gridscale while maintaining optimal order accuracy. The method is demonstrated on standard test problems.
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.