Construction of Particlesets to Simulate Rareeed Gases

In this paper a new method is introduced to construct asymptotically f {distributed sequences of points in the IR d. The algorithm is based on a transformation proposed by E.Hlawka and R.M uck.By exploitation of the special structure the method proves to be very eecient. For the numerical tests a new procedure to evaluate the f {discrepancy in two dimensions is proposed. 1 The problem 1.1 The reentry of a space shuttle For the reentry of a space shuttle into the atmosphere, one is interested in the ow eld around the shuttle to be able to adapt the surface in shape and material such that the produced heat can be controlled. The rst phase of the reentry, which takes place in an already rareeed gas is not that crucial in heat production but it determines the initial conditions for the later phases. In this state we can not use normal continuum ow mechanics, i.e. we can not use Navier{Stokes or Euler equations to describe the ow properly. An attempt to govern this region was done by Bird 5] (starting 1968) by direct simulation methods. From a mathematical point of view it is not quite satisfactory, because there is no underlying equation to be solved and hence no possibility of testing a convergence of the quality of the simulation is given. On the other hand the equation governing this region, the Boltzmann equation, is quite complicated. Nanbu 16] proposed in 1980 a method directly connected to the Boltzmann equation, starting from that a group in Kaiserslautern is developing numerical schemes for it. These methods are called \Finite Pointset Methods" FPM. Convergence could be proven by Babovsky et al. ((3] and 4]). 1.2 The mathematical formulation We will follow the formulation for gases, consisting of only one kind of molecules, given in 9]. The Boltzmann equation describes the evolution of the density in the position{density space. We start with an initial density