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A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates. (English) Zbl 1218.76028
Summary: A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in \(\mathbb R^3\), are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations.
The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order \(k\) on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadrature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step.
The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, P. N. Swarztrauber, J. Comput. Phys. 102, No. 1, 211–224 (1992; Zbl 0756.76060)], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of \(O(\Delta x^{k+1})\) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step \(\Delta t\) is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order \(k\).

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76B07 Free-surface potential flows for incompressible inviscid fluids
chammp; AMATOS
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