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

• Published in: Journal of computational physics 2024-09, Vol.513, p.113183, Article 113183
• Main Author: Le Roux, Daniel Y.
• Format: Article
• Language: English
• Subjects: Discontinuous Galerkin methods; Engineering Sciences; Fourier/stability analysis; Geophysical fluid dynamics; Inertia-gravity waves; Mathematics; Rossby waves; Shallow water equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.113183

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré 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 P1DG and non conforming P1NC 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 P1NC discrete frequencies compared to the P1DG 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 P1DG and P1NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P1NC approach. •A Fourier analysis of the 2D shallow-water model is performed for the linear DG and NC Galerkin discretizations.•A long time stability result is proven for all schemes and fluxes examined in this paper.•A super-convergent result is demonstrated for the NC frequencies compared to the DG ones.•We show that the Roe flux yields spurious frequencies in O(h) for the Rossby frequency for both the DG and NC methods.•The CFL number of the NC method is 5/3 as large as that of the DG method.