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Pré-publication, Document de travail Année : 2024

A super-convergent quasi-discontinuous Galerkin method for long time behavior of transport-dominated problems in geophysical flows

Résumé

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 (DG) and non conforming (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 demonstrated for all schemes and fluxes examined here. Further, a super-convergent result is proven for the discrete frequencies of the NC method compared to the DG one, except for the slow mode in the Roe flux case. Indeed, the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode for both the DG and NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the NC approach.
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Dates et versions

hal-04447547, version 1 (08-02-2024)

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  • HAL Id : hal-04447547 , version 1

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Daniel Le Roux. A super-convergent quasi-discontinuous Galerkin method for long time behavior of transport-dominated problems in geophysical flows. 2024. ⟨hal-04447547⟩
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Dernière date de mise à jour le 26/06/2024
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