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A convergent finite volume scheme for dissipation driven models with volume filling constraint

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Abstract

In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications.

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Acknowledgements

We thank the two referees for their very careful reading of the manuscript and their constructive comments. The authors have been supported by the multilateral project of the Austrian Agency for International Co-operation in Education and Research (OeAD), Grant FR 01/2021 and by the Campus France PHC AMADEUS program (Grant 46397PA). C. Cancès acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01) and from the COMODO project (ANR-19-CE46-0002). A. Zurek has been partially supported by the Austrian Science Fund (FWF), Grants P30000, P33010, F65, and W1245.

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  1. Inria, Univ. Lille, CNRS, UMR 8524–Laboratoire Paul Painlevé, 59000, Lille, France

    Clément Cancès

  2. Université de Technologie de Compiègne, LMAC, 60200, Compiègne, France

    Antoine Zurek

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  1. Clément Cancès
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Cancès, C., Zurek, A. A convergent finite volume scheme for dissipation driven models with volume filling constraint. Numer. Math. 151, 279–328 (2022). https://doi.org/10.1007/s00211-022-01270-7

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