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Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities

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Abstract

This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted Finite-Volume scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.

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Acknowledgements

We would like to thank the “Fédération de Recherche Mathématiques des Pays de Loire” (FMPL) as well as the SHARK-FV Conference for helping to initiate this project.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322, Nantes, France

    Christophe Berthon

  2. Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Arnaud Duran & Khaled Saleh

  3. Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Ecole Centrale de Nantes, 1 rue de la Noë, 44300, Nantes, France

    Françoise Foucher

  4. LAMI, Institut Universitaire de Technologie, Université Nazi BONI, O1 BP 1091, Bobo-Dioulasso 01, Burkina Faso

    Jean De Dieu Zabsonré

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  1. Christophe Berthon
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Berthon, C., Duran, A., Foucher, F. et al. Improvement of the Hydrostatic Reconstruction Scheme to Get Fully Discrete Entropy Inequalities. J Sci Comput 80, 924–956 (2019). https://doi.org/10.1007/s10915-019-00961-y

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