Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements

TL;DR

This paper introduces a family of mimetic finite element schemes for the rotating shallow-water equations that conserve both energy and enstrophy, applicable on non-orthogonal grids, with numerical verification of these properties.

Contribution

It develops a novel class of mixed finite element discretisations that simultaneously conserve energy and enstrophy for shallow-water equations, avoiding the need for orthogonal grids.

Findings

Conservation of energy and enstrophy demonstrated numerically.

Applicable to non-orthogonal grids, increasing flexibility.

Validated through numerical experiments.

Abstract

This paper presents a family of spatial discretisations of the nonlinear rotating shallow-water equations that conserve both energy and potential enstrophy. These are based on two-dimensional mixed finite element methods and hence, unlike some finite difference methods, do not require an orthogonal grid. Numerical verification of the aforementioned properties is also provided.