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

Telescopic projective integration for kinetic equations with multiple relaxation times