A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations
Kopera, Michal A.
Constantinescu, Emil M.
Giraldo, Francis X.
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The high-order numerical solution of the non-linear shallow water equations is susceptible to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by presenting a shock capturing model based on the numerical residual of the solution. Via numerical tests, we demonstrate that the model removes the spurious oscillations in the proximity of strong wave fronts while preserving their strength. Furthermore, for coarse grids, it prevents energy from building up at small wave-numbers. The model has no tunable parameter and, if applied to the continuity equation to stabilize the water surface, the addition of the shock capturing scheme does not affect mass conservation. We found that our model improves the continuous and discontinuous Galerkin solutions alike in the proximity of sharp fronts propagating on wet surfaces. In the presence of wet/dry interfaces, however, the model needs to be enhanced with the addition of an inundation scheme. In this paper, we simply rely on the presence of a relatively small layer of water in the regions that should be dry.
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