A discontinuous Galerkin method for two-dimensional flow and transport in shallow water |
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| Institution: | 1. Advanced Engineering Centre, School of Computing, Engineering and Mathematics, University of Brighton, Brighton, BN24GJ, UK;2. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, F-76801, Saint-Étienne-du-Rouvray, France;1. Friedrich–Alexander University of Erlangen–Nürnberg, Department of Mathematics, Cauerstrasse 11, 91058 Erlangen, Germany;2. Rice University, Department of Computational and Applied Mathematics, 6100 Main Street–MS 134, Houston, TX 77005-1892, USA;1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA;2. The Institute for Computational Engineering And Sciences, The University of Texas at Austin, Austin, TX 78712, USA;3. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA;1. Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France;2. Centre of Mathematics, Minho University, Campus de Gualtar - 4710-057 Braga, Portugal;3. École Centrale de Nantes, 1 rue de La Noë, BP 92101 44321 Nantes Cedex 3, France |
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| Abstract: | A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species. |
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