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A Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Method for the Nonlinear Shallow Water Equations

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Abstract

In this paper, we consider the development of central discontinuous Galerkin methods for solving the nonlinear shallow water equations over variable bottom topography in one and two dimensions. A reliable numerical scheme for these equations should preserve still-water stationary solutions and maintain the non-negativity of the water depth. We propose a high-order technique which exactly balances the flux gradients and source terms in the still-water stationary case by adding correction terms to the base scheme, meanwhile ensures the non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.

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Acknowledgements

ML is partially supported by the Fundamental Research Funds for the Central Universities in China (Project No. 106112015CDJXY100008) and the NSFC (Grant No. 11501062). PG is partially supported by the NSF (Grant No. DMS-1615480) and the Simons Foundation (Grant No. 246170). FL is partially supported by the NSF (Grant No. DMS-1318409). LX is partially supported by the NSFC (Grant No. 11371385), the start-up fund of Youth 1000 plan of China and that of Youth 100 plan of Chongqing University.

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  1. Liwei Xu

    Present address: School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, 611731, People’s Republic of China

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Maojun Li & Liwei Xu

  2. Institute of Computing and Data Sciences, Chongqing University, Chongqing, 400044, People’s Republic of China

    Maojun Li & Liwei Xu

  3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716-2553, USA

    Philippe Guyenne

  4. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180-3590, USA

    Fengyan Li

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Correspondence to Liwei Xu.

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Li, M., Guyenne, P., Li, F. et al. A Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Method for the Nonlinear Shallow Water Equations. J Sci Comput 71, 994–1034 (2017). https://doi.org/10.1007/s10915-016-0329-z

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  • DOI: https://doi.org/10.1007/s10915-016-0329-z

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