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A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes. (English) Zbl 1298.76120
Summary: A well-balanced, spatially arbitrary high order accurate semi-implicit discontinuous Galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non-orthogonal grids. The semi-implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime. For \(p=0\) the resulting scheme becomes a generalization of the classical semi-implicit finite-volume/finite difference scheme of V. Casulli and R. A. Walters [Int. J. Numer. Methods Fluids 32, No. 3, 331–348 (2000; Zbl 0965.76061)], but with less conditions on the grid geometry. The method proposed in this paper allows large time steps with respect to the surface wave speed \(\sqrt{gH}\) and is thus particularly suitable for low Froude number flows. The approach is validated on some typical academic benchmark problems using polynomial degrees up to \(p=6\).

MSC:
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
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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