


[Contents]  [Index] 
Cooperation with: S. Ermakov (St. Petersburg University, Russia), A. Garcia (San Jose State University, San Jose, USA), C. Lécot (Université de Savoie, Chambéry, France), O. Muscato (Università di Catania, Italy), S. Rjasanow (Universität des Saarlandes, Saarbrücken)
Supported by: DFG: ``Effektive Steuerung von stochastischen Partikelverfahren für Strömungen in verdünnten Gasen'' (Effective control of stochastic particle methods for rarefied gas flow)
Description:
Rarefied gas flows play an important role in applications like
aerospace design (space shuttle reentry),
vacuum engineering (material processing, pumps), or,
more recently, nanotechnology. Mathematically, such flows are
described (in the simplest case of a monatomic gas)
by the Boltzmann equation
Due to the high dimensionality of the problem (f is a function of 7 independent variables) stochastic algorithms are a commonly used tool for the numerical treatment of equation (1). Stochastic particle methods, like the standard DSMC (direct simulation Monte Carlo) technique, are based on the simulation of an appropriate large system of particles approximating the behavior of the real gas. Such methods provide results that are subject to random fluctuations. Thus, the construction of algorithms with reduced fluctuations is an important issue (variance reduction problem). The main purpose of the project is to contribute to this field of research.
In recent years a new approach to the
variance reduction problem,
called SWPM (stochastic weighted particle method),
has been developed ([1]). It uses a system of weighted particles
The application of equation (1) is restricted to flows, where the mean free path between collisions of molecules is sufficiently large compared to the characteristic length scale of the problem. Some modification of the DSMC method, called Consistent Boltzmann Algorithm, has been developed in the literature, extending the field of application into the direction of moderately dense gases. In [3] the convergence behavior of this algorithm has been studied. The equation was found which is asymptotically solved by the particle system. This equation generalizes the classical Boltzmann equation (1). A further investigation of its relationship to the Enskog equation was the subject of [4]. First an Htheorem for the new equation was established. Then, following the classical derivation by Chapman and Cowling, approximations to the equations of continuity, momentum and energy were found. The firstorder correction terms with respect to the particle diameter turned out to be the same as for the Enskog equation.
A kinetic equation, related to quantum gases, was considered in [5]. In this paper a DSMC algorithm for the UehlingUhlenbeckBoltzmann equation is described in terms of Markov processes. This provides a unifying framework for both the classical Boltzmann case as well as the FermiDirac and BoseEinstein cases. The foundation of the algorithm is established by demonstrating its link to the kinetic equation. By numerical experiments its sensitivity to the number of simulation particles and to the discretization of the velocity space is studied, when approximating the steady state distribution.
References:



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