On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows

Abstract

SUMMARY We present a new unconditionally positivity‐preserving (PP) implicit time integration method for the DG scheme applied to shallow water flows. This novel time discretization enhances the currently used PP DG schemes, because in the majority of previous work, explicit time stepping is implemented to deal with wetting and drying. However, for explicit time integration, linear stability requires very small time steps. Especially for locally refined grids, the stiff system resulting from space discretization makes implicit or partially implicit time stepping absolutely necessary. As implicit schemes require a lot of computational time solving large systems of nonlinear equations, a much larger time step is necessary to beat explicit time stepping in terms of CPU time. Unfortunately, the current PP implicit schemes are subject to time step restrictions due to a so‐called strong stability preserving constraint. In this work, we hence give a novel approach to positivity preservation including its theoretical background. The 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. In the DG context, we prove consistency of the discretization as well as a truncation error of the third order away from the wet–dry transition. Because of the proposed modification, the implicit scheme can take full advantage of larger time steps and is able to beat explicit time stepping in terms of CPU time. The performance and accuracy of this new method are studied for several classical test cases. Copyright © 2014 John Wiley & Sons, Ltd.