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A structure-preserving algorithm for surface water flows with transport processes

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Abstract

We consider a system of coupled equations modeling a shallow water flow with solute transport and introduce an artificial dissipation in order to improve the dissipation properties of the original cell-vertex central-upwind numerical scheme applied to these equations. Namely, a formulation is proposed which involves an artificial dissipation parameter and guarantees a consistency property between the continuity equation and the transport equation at the discrete level and, in addition, ensures the nonlinear stability and positivity of the scheme. A well-balanced positivity-preserving reconstruction is stated in terms of the conservative variable describing the concentration. We establish that constant-concentration states are preserved in space and time for any hydrodynamic flow field in the absence of source terms in the transport equation. Furthermore, we prove the maximum and minimum principles for the concentration. A suitable discretization of the diffusion term is used in combination with the proposed reconstruction procedure and artificial dissipation formulation and this allows us to prove the positivity of the concentration in the presence of diffusion effects. Finally, our numerical experiments confirm the well-balanced and positivity-preserving properties when the artificial dissipation is introduced in the central-upwind scheme, and the accuracy of the scheme for modeling surface water flows with transport processes.

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Acknowledgements

The authors would like to thank the Editor and reviewers for their contribution to improve the quality of this study.

Funding

The funding for this research was provided by UM6P/OCP Group of Morocco.

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Authors and Affiliations

  1. International Water Research Institute, Mohammed VI Polytechnic University, Green City, 43150, Morocco

    Hasan Karjoun & Abdelaziz Beljadid

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Abdelaziz Beljadid

  3. Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75258, Paris, France

    Philippe G. LeFloch

Authors
  1. Hasan Karjoun
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  2. Abdelaziz Beljadid
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  3. Philippe G. LeFloch
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Correspondence to Abdelaziz Beljadid.

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The authors declare no competing interests.

Additional information

Communicated by: Enrique Zuazua.

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Karjoun, H., Beljadid, A. & LeFloch, P.G. A structure-preserving algorithm for surface water flows with transport processes. Adv Comput Math 48, 7 (2022). https://doi.org/10.1007/s10444-021-09918-y

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  • DOI: https://doi.org/10.1007/s10444-021-09918-y

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