A super-convergent quasi-discontinuous Galerkin method for approximating inertia-gravity and Rossby waves in geophysical flows

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincare and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P-1(DG) and non conforming P-1(Nc) Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the PNc 1 discrete frequencies compared to the P-1(DG) ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P-1(DG) and P-1(Nc) methods. Finally, numerical solutions of linear and nonlinear test problems confirm the theoretical results and reveal the computational cost efficiency of the P-1(Nc) approach.