A Positivity-preserving Limiter for the Runge-Kutta Discontinuous Galerkin Approximation to the Shallow Water Equations

Maximilian Bremer, University of Texas

Discontinuous Galerkin discretizations of the shallow water equations have been very successful in approximating hurricane storm surge. This project is part of a larger effort to design a high-order scheme to run on next-generation HPC architectures. A crucial part of capturing hurricane-induced flooding is accurate treatment of inundation. The algorithm must be robust at high order and phenomenologically consistent and must preserve the computational efficiency of the discontinuous Galerkin scheme. Existing algorithms fail to satisfy the three criteria simultaneously to the extent required for our simulations. The fundamental notion of these algorithms as well as our proposed algorithm is to examine the solution at each stage of the time-stepping scheme and apply a positivity limiter in partially wetted regions, where negative water column height is observed. We propose a novel method of reconstructing these problematic regions using Bernstein polynomials and locally available data. In addition, we will present a proof that, under a modest CFL-type restraint, the mass within each element will be guaranteed to remain positive throughout the simulation. One-dimensional results will be presented as far as they are available.

Abstract Author(s): M. Bremer, C. Michoski, C. Dawson