# Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

## TL;DR

This paper introduces an energy conserving finite element scheme for the rotating shallow water equations that incorporates upwinding for enhanced stability, extending previous Hamiltonian formulations.

## Contribution

It develops a novel compatible finite element discretisation that includes upwinding in both velocity and depth fields within an energy conserving Hamiltonian framework.

## Key findings

- Energy conservation validated through coupled time discretisation.
- Upwinding improves stability and field development.
- New method extends previous energy conserving schemes.

## Abstract

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.