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Artificial Viscosity to Get Both Robustness and Discrete Entropy Inequalities

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Abstract

In the present work, we consider the numerical approximation of the weak solutions of first-order system of evolution laws supplemented with entropy inequalities. The systems under consideration are hyperbolic as soon as a conservation form is satisfied, but such stability property may be lost for non-conservative systems. Here, we show that the robustness and the entropy stability of any finite volume numerical scheme can be restored by introducing a suitable artificial numerical viscosity.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

Manuel J. Castro acknowledges financial support from the Spanish Government and FEDER through the coordinated Research project RTI2018-096064-B-C21 and the Andalusian Government Research projects UMA18-FEDERJA-161 and P18-RT-3163. Arnaud Duran acknowledges financial support from the French National Research Agency project NABUCO, Grant ANR-17-CE40-0025 and from the French National program INSU-CNRS (Institut National des Sciences de l’ Univers - Centre National de la Recherche Scientifique) program LEFE-MANU (Les Enveloppes Fluides et Environnement - Méthodes Mathématiques et Numériques), project DWAVE. Tomás Morales acknowledges financial support from the Spanish Government and FEDER through the coordinated Research project RTI2018-096064-B-C22.

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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. Departamento de Análisis Matemático, Estadística e Investigación Operativa, y Matemática Aplicada, Universidad de Málaga, Campus de Teatinos s/n, 29080, Málaga, Spain

    Manuel J. Castro Díaz

  3. 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

  4. Departamento de Matemáticas, Universidad de Córdoba, Campus de Rabanales, 14071, Córdoba, Spain

    Tomás Morales de Luna

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  1. Christophe Berthon
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Berthon, C., Castro Díaz, M.J., Duran, A. et al. Artificial Viscosity to Get Both Robustness and Discrete Entropy Inequalities. J Sci Comput 97, 65 (2023). https://doi.org/10.1007/s10915-023-02385-1

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