  We develop a family of second-order implicit-explicit (IMEX) schemes for the
stiff BGK kinetic equation. The method is asymptotic-preserving (can capture
the Euler limit without numerically resolving the small Knudsen number) as well
as positivity-preserving --- a feature that is not possessed by any of the
existing second or high order IMEX schemes. The method is based on the usual
IMEX Runge-Kutta framework plus a key correction step utilizing the special
structure of the BGK operator. Formal analysis is presented to demonstrate the
property of the method and is supported by various numerical results. Moreover,
we show that the method satisfies an entropy-decay property when coupled with
suitable spatial discretizations. Additionally, we discuss the generalization
of the method to some hyperbolic relaxation system and provide a strategy to
extend the method to third order.
