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Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts

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Abstract

We study the flux globalization based central-upwind scheme from Cheng et al. (J Sci Comput 80:538–554, 2019) for the Saint-Venant system of shallow water equations. We first show that while the scheme is capable of preserving moving-water equilibria, it fails to preserve much simpler “lake-at-rest” steady states. We then modify the computation of the global flux variable and develop a well-balanced scheme, which can accurately handle both still- and moving-water equilibria. In addition, we extend the flux globalization based central-upwind scheme to the case when “dry” and/or “almost dry” areas are present. To this end, we introduce a hybrid approach: we use the flux globalization based scheme inside the “wet” areas only, while elsewhere we apply the central-upwind scheme from Bollermann et al. (J Sci Comput 56:267–290, 2013), which is designed to accurately capture wet/dry fronts. We illustrate the performance of the proposed schemes on a number of numerical examples.

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Data Availability

The data that support the findings of this study and FORTRAN codes developed by the authors and used to obtain all of the presented numerical results are available from the corresponding author upon reasonable request.

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Funding

The work of A. Chertock was supported in part by NSF Grant DMS-1818684. The work of A. Kurganov was supported in part by NSFC Grants 11771201 and 1201101343, and by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001).

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Alina Chertock

  2. Department of Mathematics, SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, 518055, China

    Alexander Kurganov

  3. Numerical Environmental Prediction Section, Environment and Climate Change Canada, Dorval, QC, H9P1J3, Canada

    Xin Liu

  4. Department of Civil Engineering, University of Ottawa, Ottawa, ON, K1N6N5, Canada

    Xin Liu

  5. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Yongle Liu

  6. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Yongle Liu

  7. Department of Mathematics, University of Texas, San Antonio, TX, 78249, USA

    Tong Wu

Authors
  1. Alina Chertock
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  2. Alexander Kurganov
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  3. Xin Liu
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  4. Yongle Liu
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  5. Tong Wu
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Correspondence to Yongle Liu.

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Chertock, A., Kurganov, A., Liu, X. et al. Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts. J Sci Comput 90, 9 (2022). https://doi.org/10.1007/s10915-021-01680-z

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  • DOI: https://doi.org/10.1007/s10915-021-01680-z

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