  In this paper, we propose a general framework to design asymptotic preserving
schemes for the Boltzmann kinetic kinetic and related equations. Numerically
solving these equations are challenging due to the nonlinear stiff collision
(source) terms induced by small mean free or relaxation time. We propose to
penalize the nonlinear collision term by a BGK-type relaxation term, which can
be solved explicitly even if discretized implicitly in time. Moreover, the
BGK-type relaxation operator helps to drive the density distribution toward the
local Maxwellian, thus natually imposes an asymptotic-preserving scheme in the
Euler limit. The scheme so designed does not need any nonlinear iterative
solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly
small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler)
limit even if the small scale determined by the Knudsen number is not
numerically resolved. It is also consistent to the compressible Navier-Stokes
equations if the viscosity and heat conductivity are numerically resolved. The
method is applicable to many other related problems, such as hyperbolic systems
with stiff relaxation, and high order parabilic equations.
