Abstract
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor–Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems.
Highlights
The development of new numerical discretisations based on finite element methods is being driven by the need for more flexibility in mesh geometry
Compatible finite element methods are a form of mixed finite element methods that allow the exact representation of the standard vector calculus identities div-curl=0 and curl-grad=0
In this paper we will use a Taylor-Galerkin discretisation for the potential vorticity equation; this discretisation achieves the same aims as the streamline-upwind Petrov-Galerkin (SUPG) discretisation described above, but arises more naturally in the discrete time setting and we shall postpone our discussion of it until we have described the time-discrete formulation of the full shallow water system
Summary
The development of new numerical discretisations based on finite element methods is being driven by the need for more flexibility in mesh geometry. Compatible finite element methods are a form of mixed finite element methods (meaning that different finite element spaces are used for different fields) that allow the exact representation of the standard vector calculus identities div-curl=0 and curl-grad=0. This necessitates the use of H(div) finite element spaces for velocity, such as Raviart-Thomas and Brezzi-Douglas-Marini, and discontinuous finite element spaces for pressure (stable pairing of velocity and pressure space relies on the existence of bounded commuting projections from continuous to discrete spaces, as detailed in Boffi et al [2013], for example). A survey of the stability and approximation properties of compatible finite element spaces is provided in Natale et al [2016], including a proof of the absence of spurious inertial oscillations
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