  We introduce a second-order time discretization method for stiff kinetic
equations. The method is asymptotic-preserving (AP) -- can capture the Euler
limit without numerically resolving the small Knudsen number; and
positivity-preserving -- can preserve the non-negativity of the solution which
is a probability density function for arbitrary Knudsen numbers. The method is
based on a new formulation of the exponential Runge-Kutta method and can be
applied to a large class of stiff kinetic equations including the BGK equation
(relaxation type), the Fokker-Planck equation (diffusion type), and even the
full Boltzmann equation (nonlinear integral type). Furthermore, we show that
when coupled with suitable spatial discretizations the fully discrete scheme
satisfies an entropy-decay property. Various numerical results are provided to
demonstrate the theoretical properties of the method.
