  We present Chapman--Enskog and Hilbert expansions applied to the $\BigO(v/c)$
Boltzmann equation for the radiative transfer of neutrinos in core-collapse
supernovae. Based on the Legendre expansion of the scattering kernel for the
collision integral truncated after the second term, we derive the diffusion
limit for the Boltzmann equation by truncation of Chapman--Enskog or Hilbert
expansions with reaction and collision scaling. We also give asymptotically
sharp results obtained by the use of an additional time scaling. The diffusion
limit determines the diffusion source in the \emph{Isotropic Diffusion Source
Approximation (IDSA)} of Boltzmann's equation for which the free streaming
limit and the reaction limit serve as limiters. Here, we derive the reaction
limit as well as the free streaming limit by truncation of Chapman--Enskog or
Hilbert expansions using reaction and collision scaling as well as time
scaling, respectively. Finally, we motivate why limiters are a good choice for
the definition of the source term in the IDSA.
