### CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION

R. Eymard1, C. Guichard2, Xavier Lhébrard3

1. 1. Université Gustave Eiffel, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), UGE, UPEC, CNRS, F-77454, Marne-la-Vallée, France;
2. Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe ANGE, F-75005 Paris;
3. Ecole Normale Supérieure de Rennes, France
• Received:2018-12-23 Revised:2018-12-23 Published:2021-04-12
• Contact: R. Eymard,Email:robert.eymard@u-pem.fr
• Supported by:
The authors thank Professor Thierry Gallouët for fruitful discussions. This work was supported by the French Agence Nationale de la Recherche (CHARMS project, ANR-16-CE06-0009).

R. Eymard, C. Guichard, Xavier Lhébrard. CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION[J]. Journal of Computational Mathematics, 2021, 39(3): 428-452.

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of θ for stabilising the scheme.

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