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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Clément Cancès 1 ; Juliette Venel 2
CC-BY 4.0
@article{CRMATH_2023__361_G2_535_0,
author = {Cl\'ement Canc\`es and Juliette Venel},
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},
year = {2023},
doi = {10.5802/crmath.421},
language = {en},
}
TY - JOUR AU - Clément Cancès AU - Juliette Venel TI - On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations JO - Comptes Rendus. Mathématique PY - 2023 SP - 535 EP - 558 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.421 LA - en ID - CRMATH_2023__361_G2_535_0 ER -
Clément Cancès; Juliette Venel. On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 535-558. doi : 10.5802/crmath.421. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.421/
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