Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations

Many kinetic models of the Boltzmann equation have a diiusive scaling that leads to the Navier-Stokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter, from the rareeed kinetic regimes to the hydrodynamic diiusive regimes. An earlier approach in 11] reformulates such systems into the common hyperbolic relaxation system by Jin and Xin for hyperbolic conservation laws used to construct the relaxation schemes, and then use a multistep time splitting method to solve the relaxation system. Here we observe that the combination of the two time-split steps may yield a hyperbolic-parabolic systems that are more advantageous, in both stability and eeciency, for numerical computations. We show that such an approach yields a class of asymptotic-preserving (AP) schemes which are suitable for the computation of multiscale kinetic problems. We use the Goldstein-Taylor and the Carleman models to illustrate this approach.