On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

Clément Cancès (1, 2) , Juliette Venel (3, 4)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 CERAMATHS - Laboratoire de Matériaux Céramiques et de Mathématiques
4 UPHF - Université Polytechnique Hauts-de-France

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.

Mots clés: Nonlinear convection diffusion; finite volume method; energy dissipation; convergence

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Clément Cancès, Juliette Venel. On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations. 2022. ⟨hal-03693887v1⟩