  In this paper, we study the large--time behavior of a numerical scheme
discretizing drift-- diffusion systems for semiconductors. The numerical method
is finite volume in space, implicit in time, and the numerical fluxes are a
generalization of the classical Scharfetter-- Gummel scheme which allows to
consider both linear or nonlinear pressure laws. We study the convergence of
approximate solutions towards an approximation of the thermal equilibrium state
as time tends to infinity, and obtain a decay rate by controlling the discrete
relative entropy with the entropy production. This result is proved under
assumptions of existence and uniform-in-time $L $\infty$$ estimates for
numerical solutions, which are then discussed. We conclude by presenting some
numerical illustrations of the stated results.
