  We study a general, high-order, fully explicit numerical method for
simulating kinetic equations with a BGK-type collision model with multiple
relaxation times. In that case, the problem is stiff and its spectrum consists
of multiple separated eigenvalue clusters. Projective integration methods are
explicit integration schemes that first take a few small (inner) steps with a
simple, explicit method, after which the solution is extrapolated forward in
time over a large (outer) time step. These are very efficient schemes, provided
there are only two clusters of eigenvalues. Telescopic projective integration
methods generalize the idea of projective integration methods by constructing a
hierarchy of projective levels. Here, we show how telescopic projective
integration methods can be used to efficiently integrate kinetic equations with
multiple relaxation times. We show that the required number of projective
levels depends on the number of clusters, which in turn depends on the
stiffness of the BGK source term. The size of the outer level time step only
depends on the slowest time scale present in the model and is independent of
the stiffness of the problem. We discuss stability and illustrate the approach
with simulations in one and two spatial dimensions.
