Abstract
The shallow water equations with non-flat bottom topography may describe flows in rivers, lakes or coastal areas. It is well known that this system of balance laws admits discontinuous solutions and numerical schemes have to account for this difficulty. In this contribution, we use the discontinuous Galerkin method to solve these equations. In order to introduce a small but sufficient amount of numerical dissipation to the scheme, we apply a spectral viscosity based damping strategy developed in [10, 11]. This strategy consists of efficient adaptive modal filtering which is directly applied to the coefficients of the numerical solution. In the context of non-flat bottom topography, an extra challenge is posed by steady states with non-zero flux gradients that are exactly balanced by the non-zero source term, hence well-balancedness is required. In addition, non-negativity of the water height has to be preserved. In this contribution, we extend the work of Xing, Zhang and Shu [18] regarding positivity preservation and well-balancedness to triangulations but stay with filtering procedures as our shock capturing strategy.
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Meister, A., Ortleb, S. (2013). The DG Scheme on Triangular Grids with Adaptive Modal and Variational Filtering Routines Applied to Shallow Water Flows. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33221-0_15
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DOI: https://doi.org/10.1007/978-3-642-33221-0_15
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