A finite element exterior calculus framework for the rotating shallow-water equations

TL;DR

This paper develops finite element exterior calculus discretizations for the rotating shallow-water equations on the sphere, ensuring conservation, stability, and oscillation-free potential vorticity advection, extending previous mimetic frameworks.

Contribution

It introduces a unified exterior calculus framework for discretizing shallow water equations, including primal and primal-dual formulations with conservation and stability properties.

Findings

Both formulations conserve mass and potential vorticity.

Discretizations are stable and free of spurious oscillations.

The framework generalizes mimetic finite difference methods.

Abstract

We describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler, Thuburn, Klemp, and Skamarock (Journal of Computational Physics, 2010). The exterior calculus notation provides a guide to which finite element spaces should be used for which physical variables, and unifies a number of desirable properties. We present two formulations: a ``primal'' formulation in which the finite element spaces are defined on a single mesh, and a ``primal-dual'' formulation in which finite element spaces on a dual mesh are also used. Both formulations have velocity and layer depth as prognostic variables, but the exterior calculus framework leads to a conserved diagnostic potential vorticity. In both formulations we show how to construct discretisations that have mass-consistent (constant potential vorticity stays constant), stable and oscillation-free potential vorticity advection.