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This paper is concerned with diffusive approximations of peculiar numericalschemes for several linear (or weakly nonlinear) kinetic models which aremotivated by wide-range applications, including radiative transfer or neutrontransport, run-and-tumble models of chemotaxis dynamics, andVlasov-Fokker-Planck plasma modeling. The well-balanced method applied to suchkinetic equations leads to time-marching schemes involving a '' scatteringS-matrix '' , itself derived from a normal modes decomposition of thestationary solution. One common feature these models share is the type ofdiffusive approximation: their macroscopic densities solve drift-diffusionsystems, for which a distinguished numerical scheme isIl'in/Scharfetter-Gummel's '' exponential fitting '' discretization. We provethat the well-balanced schemes relax, within a parabolic rescaling, towards theIl'in exponential-fitting discretization by means of an appropriatedecomposition of the S-matrix. This is the so-called asymptotic preserving (oruniformly accurate) property.

Some examples of kinetic scheme whose diffusion limit is Il'in's exponential-fitting