We consider Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic and
kinetic equations in the diffusion limit. In such regime the system relaxes
towards a parabolic convection-diffusion equation and it is desirable to have a
method that is able to capture the asymptotic behavior with an implicit
treatment of the limiting diffusive terms. To this goal we reformulate the
problem by properly combining the limiting diffusion flux with the convective
flux. This, however, introduces new difficulties due to the dependence of the
stiff source term on the gradient. Thus, by an accurate analysis of the
different type of IMEX schemes, we proposed several schemes that under some
assumptions show good behavior with respect to the small scaling parameter in
the zero relaxation limit. In particular, at variance with the classical
fluid-limit, our approach originates in the zero relaxation limit an IMEX
method for the corresponding convection-diffusion system. Several numerical
examples including neutron transport equations confirm the theoretical
analysis.