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

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