A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes

TL;DR

This paper introduces a novel primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, combining finite element and finite volume techniques to improve accuracy and preserve physical properties.

Contribution

The paper presents a new finite element method using compound and dual finite element spaces with discrete Hodge star operators for better mimetic properties on polygonal meshes.

Findings

Method achieves improved accuracy over previous schemes.

Preserves key physical properties like conservation and balance.

Effective on standard test cases.

Abstract

A new numerical method is presented for solving the rotating shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.