A structure-preserving approximation of the discrete split rotating shallow water equations

TL;DR

This paper presents a novel split finite element discretization for a slice model of the rotating shallow water equations that preserves key physical structures and improves computational efficiency.

Contribution

It introduces a structure-preserving split FE scheme for the slice model, enhancing efficiency and maintaining conservation properties.

Findings

Scheme conserves energy, mass, potential vorticity, and enstrophy.

Achieves a twofold increase in computational efficiency.

Provides insights for full 2D rotating shallow water equations.

Abstract

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter, 2019), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.