Premium
On unconditionally positivity preserving DG schemes for shallow water flows with shock capturing by adaptive filtering procedures
Author(s) -
Ortleb Sigrun,
Meister Andreas
Publication year - 2013
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201310245
Subject(s) - discretization , time stepping , ode , spacetime , space time , shallow water equations , computer science , nonlinear system , mathematics , context (archaeology) , scheme (mathematics) , mathematical optimization , control theory (sociology) , mathematical analysis , geology , artificial intelligence , physics , quantum mechanics , chemical engineering , engineering , control (management) , paleontology
In this work, we present an unconditionally positivity preserving implicit time integration scheme for the DG method applied to shallow water flows. For locally refined grids with very small elements, the ODE resulting from space discretization is stiff and requires implicit or partially implicit time stepping. However, for simulations including wetting and drying or regions with small water height, severe time step restrictions have to be imposed due to positivity preservation. Nevertheless, as implicit time stepping demands a significant amount of computational time in order to solve a large system of nonlinear equations for each time step we need large time steps to obtain an efficient scheme. In this context, we propose a novel approach to the strategy of positivity preservation. This new technique is based on the so‐called Patankar trick and guarantees non‐negativity of the water height for any time step size while still preserving conservativity. Due to this modification, the implicit scheme can take full advantage of larger time steps and is therefore able to beat explicit time stepping in terms of CPU time. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)