Petrov-Galerkin flux upwinding for mixed mimetic spectral elements, and its application to vorticity stabilisation in shallow water
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Upwinded mass fluxes are described and analysed for advection operators discretised using mixed mimetic spectral elements. This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics. As for the original operator, the upwinded mass flux advection operator is conservative, however unlike the original operator, which is purely hyperbolic, the upwinded operator adds dissipation which is biased towards high wave numbers. The upwinded operator also removes the spectral gaps present in the dispersion relation for the original operator. As for the original operator, a material form advection operator may be constructed by similarly downwinding the trial functions of the tracer gradients. Both methods allow for the recovery of exact energy conservation for an incompressible flow field via skew-symmetric formulations. However these skew-symmetric formulations are once again purely hyperbolic operators which do not suppress oscillations. Finally, the scheme is implemented within a shallow water code on the sphere in order to diagnose and interpolate the vorticity term. In the absence of other dissipation terms, it is shown to yield more coherent results for a standard test case of barotropic instability.
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