  We present high-order, fully explicit projective integration schemes for
nonlinear collisional kinetic equations such as the BGK and Boltzmann equation.
The methods first take a few small (inner) steps with a simple, explicit method
(such as direct forward Euler) to damp out the stiff components of the
solution. Then, the time derivative is estimated and used in an (outer)
Runge-Kutta method of arbitrary order. The procedure can be recursively
repeated on a hierarchy of projective levels to construct telescopic projective
integration methods. Based on the spectrum of the linearized collision
operator, we deduce that the computational cost of the method is essentially
independent of the stiffness of the problem: with an appropriate choice of
inner step size, the time step restriction on the outer time step, as well as
the number of inner time steps, is independent of the stiffness of the
(collisional) source term. In some cases, the number of levels in the
telescopic hierarchy depends logarithmically on the stiffness. We illustrate
the method with numerical results in one and two spatial dimensions.
