Direct Statistical Simulation Method and Master Kinetic Equation

In this work the direct statistical simulation method for spatially uniform relaxation in rarefied gas is developed on the basis of a probabilistic interpretation for the integral representation of the master kinetic equation (Kats equation). It is shown that under certain conditions the conventional schemes for the relaxation simulation follow directly from this equation. A new accurate simulation scheme is proposed, the majorant frequency scheme, which requires a computing capacity that scales directly with the number of sampling particles. The relation between the solutions of the master kinetic equation of rarefied gas and the Boltzmann equation is studied for the uniform case. It is shown that the correlations occurring in the finite-number particle system affect significantly the statistical simulation results. The criterion for estimating the influence of such correlations during the computation process is suggested. Introduction At present, the direct statistical simulation method, based on splitting up the evolution of a gas system in two stages, is widely used in the dynamics of rarefied gas. The method is realized in the following way. The flowfield computed is divided into cells of a finite size Ax and, according to the initial distribution function, N sampling particles are placed into each cell. Then the spatially-uniform relaxation stage and the stage of a free-molecular transition are successively carried out in all the cells. The free-molecular transition simulation can be performed without difficulties. In this case, the realization of the spatially-uniform relaxation stage is of primary importance. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. * Institute of Theoretical and Applied Mechanics. t Computing Center, 171 172 M. S. IVANOV ET AL For the simulation of collisional relaxation, the numerical schemes suggested in Refs. 1-4 are used. All these schemes are derived from heuristic considerations on the basis of physical understanding of the relaxation process in an actual gas; as a result there is no direct relation with the Boltzmann kinetic equation. The heuristic character of these schemes allow comparative analysis only qualitatively, with the use of the "Boltzmannian" collision frequency as the main criterion. " The stochastic process for an approximate solution of the Boltzmann equation is constructed in Ref. 8 using the Euler scheme for the Boltzmann spatiallyuniform equation with its further randomization. In such an approach, for colliding particles the conservation laws are not valid, and this is the basic difference from the schemes given in Refs. 1-4. It seems reasonable to consider the known numerical schemes for the statistical simulation of rarefied gas flows in light of a general theory of Monte Carlo techniques. Such a unified consideration enables one to carry out a comparative analysis to show the inner relation between these schemes and also justifies the use of various Monte Carlo weight techniques. ^ Derivation of Simulation Technique from Master Kinetic Equation In the construction of the Monte Carlo numerical technique we shall directly proceed from the master kinetic equation for the N-particle distribution function, which in the spatially-uniform case has the following r°° {fN(t,cjj) fN(t,c)} | vi-vj | by dbij dey l C) K2(t'->t I c) cp (t,c) dc dt + cp0(t,c) /2) Jo J where 9 (t,c) = t) (c) f^ (t,c) is the collision density. DIRECT SIMULATION AND MASTER KINETIC EQUATION 173 9o(t,c)=fN(o,c)\)(c)exp{-\)(c)t} _ . _ • _ _ N _ _• '-* C) = -^Y D'^WCv^V;' I Vi,Vj) II 5(vm-Vm)