Numerical schemes for kinetic equations in diffusive regimes

The diffusive scaling of many finite-velocity kinetic models leads to a small-relaxation time behavior governed by reduced systems which are parabolic in nature. Here we demonstrate that standard numerical methods far hyperbolic conservation laws with stiff relaxation fail to capture the right asymptotic behavior. We show how to design numerical schemes for the study of the diffusive limit that possess the discrete analogue of the continuous asymptotic limit. Numerical results for a model of relaxing heat flow and for a model of nonlinear diffusion are presented.