  We are interested in the large-time behavior of solutions to finite volume
discretizations of convection-diffusion equations or systems endowed with
non-homogeneous Dirichlet and Neumann type boundary conditions. Our results
concern various linear and nonlinear models such as Fokker-Planck equations,
porous media equations or drift-diffusion systems for semiconductors. For all
of these models, some relative entropy principle is satisfied and implies
exponential decay to the stationary state. In this paper we show that in the
framework of finite volume schemes on orthogonal meshes, a large class of
two-point monotone fluxes preserve this exponential decay of the discrete
solution to the discrete steady state of the scheme. This includes for instance
upwind and centered convections or Scharfetter-Gummel discretizations. We
illustrate our theoretical results on several numerical test cases.
