Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations

Abstract

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, absolute vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.