A monotone numerical flux for quasilinear convection diffusion equation
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- by C. Chainais-Hillairet, R. Eymard and J. Fuhrmann
- Math. Comp. 93 (2024), 203-231
- DOI: https://doi.org/10.1090/mcom/3870
- Published electronically: June 30, 2023
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Abstract:
We propose a new numerical 2-point flux for a quasilinear convection–diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results.References
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Bibliographic Information
- C. Chainais-Hillairet
- Affiliation: Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe, Université Lille 1 Sciences et Technologies, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
- MR Author ID: 647104
- ORCID: 0000-0002-6281-4603
- Email: claire.chainais@univ-lille.fr
- R. Eymard
- Affiliation: LAMA - UMR 8050, Université Gustave Eiffel, 5 boulevard Descartes, 77454 Marne-la-Vallée cedex 2, France
- MR Author ID: 262329
- Email: robert.eymard@univ-eiffel.fr
- J. Fuhrmann
- Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
- ORCID: 0000-0003-4432-2434
- Email: juergen.fuhrmann@wias-berlin.de
- Received by editor(s): October 3, 2022
- Received by editor(s) in revised form: April 17, 2023
- Published electronically: June 30, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 203-231
- MSC (2020): Primary 65N08, 65N12, 35A23, 35B40
- DOI: https://doi.org/10.1090/mcom/3870
- MathSciNet review: 4654620