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

• Published in: International journal for numerical methods in fluids Vol. 76; no. 2; pp. 69 - 94
• Main Authors: Meister, A., Ortleb, S.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 20.09.2014; Wiley Subscription Services, Inc
• Subjects: Central processing units; DG method; Discretization; implicit time integration; Patankar trick; Polypropylenes; positivity preservation; Preserving; Shallow water; shallow water equations; Stability; Time integration; Truncation errors; Water flow; well-balancedness
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.3921

SUMMARYWe 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. In this work, we present a new unconditionally positivity‐preserving implicit time integration method for the DG scheme applied to shallow water flows. 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. Because of the proposed modification, implicit schemes can now take full advantage of larger time steps and are able to beat explicit time stepping in terms of CPU time.