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Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects pp 303–311Cite as

Convergence of Finite Volume Scheme for Degenerate Parabolic Problem with Zero Flux Boundary Condition

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 77)

Abstract

This note is devoted to the study of the finite volume methods used in the discretization of degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The notion of an entropy-process solution, successfully used for the Dirichlet problem, is insufficient to obtain a uniqueness and convergence result because of a lack of regularity of solutions on the boundary. We infer the uniqueness of an entropy-process solution using the tool of the nonlinear semigroup theory by passing to the new abstract notion of integral-process solution. Then, we prove that numerical solution converges to the unique entropy solution as the mesh size tends to 0.

Keywords

  • Dirichlet Problem
  • Integral Solution
  • Entropy Solution
  • Young Measure
  • Accretive Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgments

This work has been supported by the French ANR project CoToCoLa.

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Authors and Affiliations

  1. Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comte, 16 route de Gray, 25030, Besançon, France

    Boris Andreianov & Mohamed Karimou Gazibo

  2. Institut Für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Boris Andreianov

Authors
  1. Boris Andreianov
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  2. Mohamed Karimou Gazibo
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Corresponding author

Correspondence to Boris Andreianov .

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Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

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Andreianov, B., Gazibo, M.K. (2014). Convergence of Finite Volume Scheme for Degenerate Parabolic Problem with Zero Flux Boundary Condition. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_29

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