  This paper is concerned with diffusive approximations of peculiar numerical
schemes for several linear (or weakly nonlinear) kinetic models which are
motivated by wide-range applications, including radiative transfer or neutron
transport, run-and-tumble models of chemotaxis dynamics, and
Vlasov-Fokker-Planck plasma modeling. The well-balanced method applied to such
kinetic equations leads to time-marching schemes involving a '' scattering
S-matrix '' , itself derived from a normal modes decomposition of the
stationary solution. One common feature these models share is the type of
diffusive approximation: their macroscopic densities solve drift-diffusion
systems, for which a distinguished numerical scheme is
Il'in/Scharfetter-Gummel's '' exponential fitting '' discretization. We prove
that the well-balanced schemes relax, within a parabolic rescaling, towards the
Il'in exponential-fitting discretization by means of an appropriate
decomposition of the S-matrix. This is the so-called asymptotic preserving (or
uniformly accurate) property.
