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