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

Abstract

We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative entropy functionals. For this kind of models including the 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 the resolution of the stationary equation in order to preserve steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After describing the numerical scheme, we present several numerical results which confirm the accuracy and underline the efficiency to preserve the large-time asymptotic.