Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes

Authors:
Yulong Xing;Xiangxiong Zhang
Affiliations:
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, USA 37831 and Department of Mathematics, University of Tennessee, Knoxville, USA 37996;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA 02139
Venue:
Journal of Scientific Computing
Year:
2013

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Abstract

The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In "Xing et al. Adv. Water Resourc. 33: 1476---1493, 2010)", the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.