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__Mimetic Finite Element methods applied to the Shallow Water Equations__

A. McRae and C. Cotter

**Abstract**

In the context of weather forecasting, the evolution of the ocean and atmosphere is governed by the Navier-Stokes equations. The shallow water equations are a simplification of the full equations that are relevant for atmosphere/ocean modelling. The analytic shallow water equations conserve several physically meaningful quantities, such as the total energy and entropy within the system. In this talk, I will present a discretisation of the shallow water equations, based on mimetic finite elements, which reproduces many of the same conservation properties. This is achieved through using distinguished choices of function spaces. The function spaces are linked by differential operators, which allows some operations to be represented exactly. This leads to discrete equivalents of identities such as `div-curl = 0' and `curl-grad = 0', which turn out to be fundamental in proving the discrete conservation properties.