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

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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.

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Acknowledgments

Research of the first author is sponsored by the National Science Foundation grant DMS-1216454, ORNL’s Laboratory Directed Research and Development funds, and the U. S. Department of Energy, Office of Advanced Scientific Computing Research. The work was partially performed at ORNL, which is managed by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725.

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Authors and Affiliations

  1. Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Yulong Xing

  2. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Yulong Xing

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Xiangxiong Zhang

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  1. Yulong Xing
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  2. Xiangxiong Zhang
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Correspondence to Yulong Xing.

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Xing, Y., Zhang, X. Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes. J Sci Comput 57, 19–41 (2013). https://doi.org/10.1007/s10915-013-9695-y

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  • DOI: https://doi.org/10.1007/s10915-013-9695-y

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