Geodesic
Analyse numérique, Equations aux dérivée partielles
On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations
Cancès, Clément1  ; Venel, Juliette2
1 Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
2 Univ. Polytechnique Hauts-de-France, INSA Hauts-de-France, CERAMATHS – Laboratoire de Matériaux Céramiques et de Mathématiques, F-59313 Valenciennes, France.
Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 535-558

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We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.

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DOI : 10.5802/crmath.421
Classification : 65M08, 65M12, 35K51, 35Q92, 92D25

Cancès, Clément 1 ; Venel, Juliette 2

1 Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
2 Univ. Polytechnique Hauts-de-France, INSA Hauts-de-France, CERAMATHS – Laboratoire de Matériaux Céramiques et de Mathématiques, F-59313 Valenciennes, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {535--558},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
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     year = {2023},
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Cancès, Clément; Venel, Juliette. On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations. Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 535-558. doi: 10.5802/crmath.421

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