Telescopic Projective Integration for Multiscale Kinetic Equations with a Specified Relaxation Profile

We study the design of a general, fully explicit numerical method for simulating kinetic equations with an extended BGK collision model allowing for multiple relaxation times. In that case, the problem is stiff and we show that 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. They 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 multiple relaxation time BGK models. We show that the number of projective levels only depends on the number of clusters and the size of the outer level time step only depends on the slowest time scale present in the model. Both do not depend on the small-scale parameter. We analyze stability and illustrate with numerical results.