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Discontinuous Galerkin solver for the shallow-water equations in covariant form on the sphere and the ellipsoid. (English) Zbl 1436.65136

Summary: A Discontinuous Galerkin (DG) method for the solution of the shallow-water equations (SWE) on arbitrary 2-dimensional (2D) manifolds is presented. To this purpose the SWE are formulated in covariant form using tensor notation. This allows to correctly transform the numerical fluxes between the local coordinate systems of any two neighboring grid cells. In particular, the covariant form of the numerical diffusion term in the Lax-Friedrichs numerical flux has been derived, too. This general approach has the advantage that it avoids any coordinate singularity. It is tested for the SWE on the sphere with several standard test setups. Beyond this, a recently published test case with an analytic solution for linear inertial-gravity wave expansion has been performed. The derived formalism on arbitrary 2D manifolds allows an easy extension from the sphere to the ellipsoid. The comparison of a barotropic instability test case for the earth shows a non-negligible difference between the solution on these two bodies. The presented approach may be a starting point for the development of a dynamical core for numerical weather and climate prediction models based on the DG method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35R01 PDEs on manifolds
86A05 Hydrology, hydrography, oceanography

Software:

chammp; ICON
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Full Text: DOI

References:

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