Since the early 1970's the dominant method for computation of RGD has been the Direct Simulation Monte Carlo (DSMC) method, which moves particles according to their velocities and performs collisions between randomly chosen particles. This method has been very successful in a wide range of applications. There is one important flow regime, however, in which the DSMC method loses its effectiveness: flow for which the Knudsen number is small enough that the collision rate is large, but not small enough that the flow is well described by fluid mechanics. In this regime, the appropriate length and time scales are nearly those for fluid mechanics, but the collisional length and time scales are quite small. Since accuracy of DSMC depends on resolution of the collisional length and time scales, it becomes slower and less accurate in this regime. The construction of numerical schemes that are able to describe correctly the asymptotic behavior of the Boltzmann equation in the fluid limit, by avoiding severe restriction on the time stepping, has been the subject of many recent papers and inspired the development of asymptotic-preserving (AP) methods which are nowadays very popular in different fields. In this talks we survey some recent advancements in this direction, including several results obtained in collaboration with Russ.
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