  In this work we numerically study the diffusive limit of run & tumble kinetic
models for cell motion due to chemotaxis by means of asymptotic preserving
schemes. It is well-known that the diffusive limit of these models leads to the
classical Patlak-Keller-Segel macroscopic model for chemotaxis. We will show
that the proposed scheme is able to accurately approximate the solutions before
blow-up time for small parameter. Moreover, the numerical results indicate that
the global solutions of the kinetic models stabilize for long times to steady
states for all the analyzed parameter range. We also generalize these
asymptotic preserving schemes to two dimensional kinetic models in the radial
case. The blow-up of solutions is numerically investigated in all these cases.
