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A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations

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Abstract

In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet–dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun. Math. Sci., 5(1):133–160, 2007). The positivity of the computed water height is ensured following (Bollermann et al., Commun. Comput. Phys., 10:371–404, 2011): The outgoing fluxes are limited in case of draining cells.

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Acknowledgments

The first ideas for this work were discussed by the authors at a meeting at the “Mathematisches Forschungsinstitut Oberwolfach”. The authors are grateful for the support and inspiring atmosphere there. The research of A. Kurganov was supported in part by the NSF Grant DMS-1115718 and the ONR Grant N000141210833. The research of A. Bollermann, G. Chen and S. Noelle was supported by DFG Grant NO361/3-1 and No361/3-2. G. Chen is partially supported by the National Natural Science Foundation of China (No. 11001211, 51178359).

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Authors and Affiliations

  1. IGPM, RWTH Aachen University of Technology, Templergraben 55, 52062 , Aachen, Germany

    Andreas Bollermann, Guoxian Chen & Sebastian Noelle

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Guoxian Chen

  3. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Alexander Kurganov

Authors
  1. Andreas Bollermann
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  2. Guoxian Chen
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  3. Alexander Kurganov
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  4. Sebastian Noelle
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Correspondence to Sebastian Noelle.

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Bollermann, A., Chen, G., Kurganov, A. et al. A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations. J Sci Comput 56, 267–290 (2013). https://doi.org/10.1007/s10915-012-9677-5

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  • DOI: https://doi.org/10.1007/s10915-012-9677-5

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