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We propose a finite volume scheme for convection-diffusion equations withnonlinear diffusion. Such equations arise in numerous physical contexts. Wewill particularly focus on the drift-diffusion system for semiconductors andthe porous media equation. In these two cases, it is shown that the transientsolution converges to a steady-state solution as t tends to infinity. Theintroduced scheme is an extension of the Scharfetter-Gummel scheme fornonlinear diffusion. It remains valid in the degenerate case and preservessteady-states. We prove the convergence of the scheme in the nondegeneratecase. Finally, we present some numerical simulations applied to the twophysical models introduced and we underline the efficiency of the scheme topreserve long-time behavior of the solutions.

A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme