A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure

Francis Filbet (1) , Maxime Herda (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MMCS - Modélisation mathématique, calcul scientifique

We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative en-tropy functionals. For this kind of models including porous media equations, Fokker-Planck equations for plasma physics or dumbbell models for polymer flows, it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme is built from a discretization of the steady equation and preserves steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After proving well-posedness, stability, exponential return to equilibrium and convergence, we present several numerical results which confirm the accuracy and underline the efficiency to preserve large-time asymptotic.

Mots clés: 65M12; MSC2010 subject classifications 65M08; porous media; 76X05; 76S05; 82D60; polymers; magnetized plasma; conditions; Finite volume methods; relative entropy; non-homogeneous Dirichlet boundary

Domaines: Analyse numérique [math.NA]

Francis Filbet, Maxime Herda. A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure. Numerische Mathematik, 2017. ⟨hal-01326029v4⟩