A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations in space and time
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Methods for upwinding the potential vorticity in a finite element discretisation of the rotating shallow water equations are studied. These include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a recent approach where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent, resulting in a turbulent state where the potential enstrophy is more strongly damped than for the other schemes. While the SUPG scheme is widely known to be consistent, since it modifies the test functions only, the downwinded trial function formulation results in the advection of downwind corrections. Results of the SUPG and downwinded trial function schemes are very similar in terms of both potential enstrophy conservation and turbulent spectra. The main difference between these schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are greatly improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme. We also present a formulation by which potential enstrophy is exactly conserved in time. The application of the APVM with exact temporal conservation of potential enstrophy yields conservation errors that are significantly smaller than those of the SUPG or downwinded trial function formulations using a potential enstrophy dissipating time discretisation. Exact conservation of the potential enstrophy in time allows for stable simulation in the absence of dissipation, despite the uncontrolled aliasing of grid scale turbulence.
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