411 RWTH Publication No: 443043        2014        IGPM411.pdf
TITLE Multiwavelet-Based Grid Adaptation with Discontinuous Galerkin Schemes for Shallow Water Equations
AUTHORS Nils Gerhard , Daniel Caviedes-Voullième, Siegfried Müller, Georges Kesserwani
ABSTRACT We provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography. A challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well-balancing of the approximate solution. A key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process. The underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. Furthermore we use the multiresolution analysis of the underlying data as an additional indicator whether the limiter has to be applied on a cell or not. By this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution. By means of well-known 1D and 2D benchmark problems, we verify that multiwavelet-based grid adaptation can significantly reduce the computational cost by sparsening the computational grids, while retaining accuracy and keeping well-balancing and positivity.
KEYWORDS grid adaptivity, multiresolution analysis, multiwavelets, discontinuous Galerkin, shallow water flows, positivity-preserving, well-balancing, limiters, shock detection
DOI 10.1016/j.jcp.2015.08.030
PUBLICATION Journal of Computational Physics
301, 265-288, 2015