# ABSTRACTS

### Monte Carlo simulation of rarefied gas flows between two coaxial circular cylinders

Kazuo Aoki*

We consider a rarefied gas between two coaxial circular cylinders and investigate the flows caused by the rotation of the cylinders by means of the DSMC method as well as other numerical and asymptotic methods. Here, we focus on the following two problems.

(a) Inverted velocity profile in cylindrical Couette flow
(collaboration with A. L. Garcia, H. Yoshida, and T. Nakanishi)

We first consider cylindrical Couette flow under the diffuse-specular reflection condition in the case where the inner cylinder is rotating whereas the outer cylinder is at rest. When the accommodation coefficients of the cylinders are small (i.e., when the major part of the molecules undergo specular reflection), the flow speed of the gas increases with the distance from the inner cylinder, which is contrary to the ordinary velocity profile of the Couette flow when only the inner cylinder is rotating. We show some results, obtained by DSMC method, that demonstrate such inversion of the velocity profiles. We also investigate the parameter range in which the inversion appears.

(b) Flow bifurcation in Taylor-Couette problem}
(collaboration with H. Yoshida)

Next, we consider axisymmetric Taylor-Couette problem, where the constraint of axial uniformity is released, under the diffuse reflection condition. We investigate gas flows in an annular domain bounded by the top and bottom boundaries where the specular reflection is assumed. We show some examples of the flow field, obtained by the DSMC method, that demonstrate bifurcation of steady flows, that is, coexistence of a single-vortex and a double-vortex flow. The parameter range in which the bifurcation appears is investigated in detail by means of the DSMC computation. Stability of the Couette flow is also discussed on the basis of the BGK model.

### General DSMC Programs for Engineering and Scientific Studies

G.A. Bird*

Early applications of the DSMC method to particular problems employed dedicated programs that were written by the investigator especially for that problem. These programs were typically comprised of several hundred lines of executable code and an experienced worker could produce one in about a month. Most contemporary DSMC applications are by workers who neither have nor desire to have any knowledge of the details of the method. They instead employ general-purpose DSMC programs that involve tens of thousands of lines of code and require years of development time. There are many issues concerning the utility, reliability and availability of these programs. The history of the general-purpose DSMC codes will be outlined and discussed. Many scientific and most engineering applications require three-dimensional programs and the new DS3V program will be presented. This program features an interactive visual interface and integrated three-dimensional displays of the results that remove the need for any post-processing of the output. The program incorporates automatic cell adaption to a specified number of molecules per cell, transient sub-cells, variable time steps, and the sampling may be for continuing unsteady or eventually steady flows. The options offer surface and gas phase chemical reactions and a number of special boundary conditions that include non-uniform flows, periodic boundaries, constant pressure boundaries, and moving surfaces. A parallel version with automatic domain adaption is under development.

### The Information Preservation Method: A Variant of DSMC With Reduced Statistical Fluctuations

Iain D. Boyd*, Quanhua Sun, and Wen-Lan Wang

The Information Preservation (IP) method is a variant of the DSMC method that successfully reduces statistical fluctuations particularly with application to low-speed rarefied flows. In our work, we have further developed the IP method for micro-scale low speed flows by incorporating new physical models and by conducting a series of validation studies. We have identified an additional benefit of the reduced statistical fluctuations provided by the IP method: namely, that this characteristic provides an effective pathway to create continuum-particle hybrid methods. We discuss some of our experiences with the IP method and its use in hybrid methods. Results are shown from our work in micro-scale and hypersonic rarefied flows.

### Transversal inhomogeneities in dilute fluidized granular fluids

J.J. Brey* and M. J. Ruiz-Montero+

One of the main features of granular fluids is their tendency to present spontaneous symmetry breaking phenomena. They occur in isolated as well as in driven systems, with and without gravitational field acting on the particles. Here, we will discuss the continuous spontaneous symmetry breaking taking place in the direction perpendicular to the energy flux in a vibrofluidized granular gas. This effect is investigated for low density granular gases by means of DSMC, paying attention to the nature of the transition and also to the shape of the hydrodynamic profiles beyond the bifurcation. The results are compared with molecular dynamics simulations and with a marginal stability analysis based on the hydrodynamic description.

### Chemical Kinetics behind Shock Waves

D. Bruno*

In this work we show the implementation of a DSMC code to simulate chemical kinetics behind shock waves. In this case, besides a description of viscosity and diffusion, a careful modelling of the kinetics of internal degrees of freedom is needed. The particle method allows to treat the kinetic problem on a state-to-state basis by simulating on a microscopic scale the rotational, vibrational and dissociation kinetics behind the shock: the input data for describing energy exchange processes are in the form of cross sections. The method thus can give account of the interplay between the different energy exchange channels in the first moments of the relaxation when the hierarchy of relaxation times is not established yet. In particular, in this work we assess the role of multi-quantum vibrational energy exchange and dissociation processes in atom-molecule encounters. The physical input data used to model these processes are derived from a complete set of cross sections for the atom-diatom system obtained from QCT calculations. Next, we show how the method is suited to the simulation of fast chemical reactions whose time scale compete with the characteristic time of the flow, i.e. a detonation.

### The attractors of the Rayleigh-Benard Flow of a Rarefied Gas

S.Stefanov+, V. Roussinov and C. Cercignani*

We have studied the Rayleigh-Benard flow of a rarefied gas for Knudsen numbers between 0.001 and 0.04, Froude numbers between 0.1 and 1,000 and a fixed temperature ratio of 0.1. The calculations are performed by both the DSMC and the numerical solution of the Navier-Stokes equations, with a remarkable agreement between the methods. We exhibit chaotic behavior and also a hysteresis cycle.

### The one-dimensional dissipative Boltzmann equation for point particles up to the clustering regime

P. Cordero * and J.M. Pasini

A new way of integrating Boltzman's equation of a one-dimensional gas of point-like particles not subject to gravity between two walls at granular-temperatures T- and T+, with T- < T+ gives a different physical picture of the mechanism that inhibit/create a cluster. It is known that depending on the normalized temperature difference D = (T+ - T-)/(T+ + T-) the system may be completely fluidized, or in a mixed state in which a cluster coexists with the fluidized gas. We devise and explain in detail a method for integrating the one-dimensional dissipative Boltzmann equation in the test-particle limit for the stationary case. The behavior of the system in its fluid phase is dominated by characteristic lines which are trajectories of particles subjected to a force which attracts them to a fixed point. If this point is between the two walls a cluster forms, if not then the system remains fluid.

### Models for non-equilibrium structure of the vapor-liquid interface

A. Frezzotti*

For a long time kinetic theory studies of evaporation or condensation phenomena have been mainly devoted to investigate the structure of the Knudsen layer in the vapor phase, whereas the condensed phase has been described by simple phenomenological models.

Recently, the validity of such models has been questioned in a number of studies in which the details of molecular absorption and reemission at the vapor-liquid interface has been investigated. The talk aims at reviewing and discussing the implications of theese results for evaporation/condensation flows. Moreover, the application of Enskog-Vlasov kinetic equation, as an approximate mathematical model of non-equilibrium flows accross the vapor-liquid interface , will be discussed through some model problems, like the formation of inverted gradient temperature profiles.

### The Direct Simulation Monte Carlo Method and Chapman-Enskog Gas Flows

M. A. Gallis*, J. R. Torczynski, and D. J. Rader

As interest in MEMS and microsystems grows, it is important to determine whether the near-continuum gas flows in these devices can be simulated with the Direct Simulation Monte Carlo (DSMC) method of Bird, which was developed originally for strongly noncontinuum conditions. Gas flows in the near-continuum regime are described by Chapman-Enskog (CE) theory. To assess the behavior of DSMC in this regime, a test problem is developed and simulated in which a monatomic gas is confined between parallel walls held at slightly different temperatures. Simulations are performed using molecules with the mass and reference viscosity of argon (at 273.15 K), fully accommodating walls that are 1 mm apart, gas pressures of 1-2 torr (133.3-266.6 Pa), and wall temperature differences of , +/- 10, +/- 20, , and +/- 50 K from the reference temperature. The Variable Soft Sphere (VSS) model is used to describe collisions of molecules that interact via an inverse-power-law repulsive force, including hard-sphere (HS) and Maxwell (MX) interactions. The DSMC code Icarus is used on ASCI Red to simulate the above, with typical simulations requiring 1000 2-processor nodes for 24 hours. Care is taken to ensure that the computational mesh, the time step, the number of molecules per cell, the transient startup time, and the averaging time are appropriately chosen to minimize discretization and statistical errors. The resulting temperature profiles exhibit the expected nearly linear behavior in the central region of the domain, with Knudsen layers (including temperature jumps) adjacent to the walls. Appropriate moments of the molecular velocity distribution are accumulated and compared to the corresponding quantities of the CE distribution. In the central portion of the domain (i.e., outside the Knudsen layers) and at the lower temperature differences, the DSMC results are in excellent agreement with the corresponding theoretical values for both HS and MX interactions. However, at the highest temperature difference, systematic differences in the DSMC results from the theoretical CE results are observed in the central portion of the domain. These differences appear to indicate the departure of gas behavior from CE theory at large heat fluxes. Thus, DSMC quantitatively reproduces CE behavior in the near-continuum regime and provides insight into the breakdown of CE theory beyond this regime.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. 10 ± 20 ± 50 ±

### Fluctuations in DSMC

A. Garcia*

Fluctuations in DSMC are a double-edged sword. If one is only interested in mean values, spontaneous fluctuations are a nuisance, introducing statistical error to measured quantities. However, as DSMC produces the correct hydrodynamic fluctuations, phenomena such as non-equilibrium mode coupling and Brownian motion may studied.

The talk will review a variety of topics related to fluctuations in DSMC, including a discussion of how to estimate confidence intervals and how fluctuations can bias the measurement of hydrodynamic variables.

### DSMC: A link among physical regimes and used in novel applications

David Goldstein*

DSMC, as a simulation of physics, is an intuitively appealing approach whose longevity and future prospects come largely from its flexibility. DSMC provides a special link between the stochastic micro-world view and macro-scopic effects such as drag or observed radiation signatures. Until the mid '90s, two of the major goals of DSMC development research were to improve efficiency to enable computations into virtually continuum flow and to improve physical modeling. While physical modeling can still be improved, the efficiency goal has largely been met both by coding improvements (vector and parallel processing, sub-cells, adaptive griding and time stepping, etc) and by continued computer improvements via Moore's law. But DSMC can do more by providing a better link between different modeling points of view. We, as well as others, have developed hybrid DSMC/CFD codes to handle increasingly difficult near-continuum problems. And we have developed a discrete velocity/energy/position version of DSMC (IDSMC) that can smoothly bridge the gap between the most simple discrete velocity and cellular automata models of flow and the full physics of DSMC. Meanwhile, on the third side of the bridge, work is underway on linking DSMC to deterministic molecular dynamics.

The other area where DSMC will continue to be important will be in applications in which the fluid physics to be examined is novel. Simulations of unsteady near-continuum 2D and 3D phenomena are now becoming practical and fluid instability growth appears a promising target of opportunity. While several groups are concentrating on small physical scales (e.g., MEMS), we have looked in the other direction toward applying DSMC to astronomical scales. At those large space and time scales, somewhat different physical considerations (such as radiation coupling) become important.

### DSMC in micro/nanoscale science and technology

Recent interest in small scale science and technology has resulted in renewed interest in high Knudsen number, non-continuum gaseous flows. In this regime, DSMC remains the simulation tool of choice by combining simplicity with accuracy and numerical efficiency. In this talk we report on our work on methods development, that is, algorithmic developments which extend the applicability of DSMC to a wider range of practical applications.

We first discuss hybrid DSMC-continuum simulation methods. We present results for hybrid methods for both compressible and incompressible problems and discuss how the widely different characteristics of the numerical methods used to solve the compressible and incompressible formulations affect the resulting hybrid algorithm.

We also discuss the use of DSMC to model chemical vapor deposition with simple surface chemistry. We have developed a profile simulator based on DSMC which captures the deposition dynamics over a wide range of Knudsen numbers from slip flow to free molecular flow.

This work was supported, in part, by the Lawrence Livermore National Laboratory. Collaborators: H. Al-Mohssen, A. L. Garcia, R. Hornung, S. Wijesinghe

### Statistical simulation of reentry capsules in hypersonic near-continuum flows

M.S. Ivanov*

Accurate prediction of force and heat aerodynamic characteristics of future and present reentry vehicles and spacecraft requires comprehensive investigation of hypersonic flows along the entire flight trajectory. Recently, the part of CFD that studies rarefied flows in the transitional regime has drawn much attention, because a large portion of flight trajectory of modern space vehicles lies at high altitudes. Therefore, a new independent branch of CFD, namely, Rarefied Hypersonic Computational Fluid Dynamics, has been formed. For modeling strongly nonequilibrium high-altitude hypersonic flows, the Boltzmann equation should be used. The DSMC method is a simulation technique most frequently used to obtain solutions of the Boltzmann equation. This method has become, de facto, the main tool for the study of complex multidimensional flows of rarefied hypersonic aerothermodynamics. The application of the DSMC method has been limited mostly to altitudes between 300 and 80 km from the free-molecular regime to the near-continuum regime through the transitional regime. The main problem of the DSMC method it that it is extremely computationally expensive for near-continuum, three-dimensional flows.

The talk will include the results of investigation of high-altitude aerothermodynamics for two types of reentry capsules (Progress and Expert) and comparison of these results with available experiments. Three-dimensional parallel computations were performed for Knudsen numbers below 0.01 with real gas effects.

### Molecular Dynamics in Mesoscopic Environments

R. Kapral*

The talk will focus on several aspects of simulating the dynamics of molecules in solvents modeled at the mesoscopic level. The mesoscopic solvent dynamics consists of free streaming interrupted by multi-particle collisions. The multi-particle collisions are carried out by performing random rotations of particle velocities in predetermined cells in a manner that conserves mass, momentum and energy. A hybrid molecular dynamics-mesoscopic solvent model, where full molecular dynamics of molecules is combined with the mesoscopic dynamics of the surrounding fluid, will be described. Applications of this hybrid algorithm will be discussed. Finally, extensions of such schemes to treat molecular solvents, reactive flows and phase segregating fluids will be outlined.

### Sensitivity of thermal explosion to departure from partial equilibrium

A. Lemarchand*

We study an exothermal reactive system in the vicinity of the bifurcation leading to bistability. Close to a bifurcation, a dynamical system is highly sensitive to weak perturbations. In particular, its macroscopic behavior may be strongly modified by small departures from partial equilibrium, induced for example by a chemical reaction. We determine, in the case of the Semenov model, analytical corrections to the macroscopic dynamics induced by perturbations of particle velocity distribution. Our theoretical predictions agree well with the results of microscopic simulations using DSMC method. We show that a departure from Maxwellian velocity distribution induces qualitative and quantitative changes in the system properties. These nonequilibrium effects not only influence the deterministic evolution but also the stochastic behavior. Whereas the system was bistable according to the standard deterministic approach, it can become monostable in the presence of nonequilibrium effects. It means that explosion occurs in a parameter domain that was considered as safe in the standard description without nonequilibrium corrections. Even when the system remains in the same regime, the quantitative effects can be enormous. In the bistable domain, the mean first passage time can be diminished by several orders of magnitude. In the explosive regime, the ignition time can be decreased by a factor of 10. Contrary to the corrections introduced in the deterministic dynamics by mesoscopic fluctuations, the nonequilibrium effects studied here induce macroscopic perturbations that do not vanish in the limit of large systems. Prevention of hazards implies taking into account small departures from partial equilibrium that one are used to neglect.

### Monte Carlo Simulation of gas-phase coherent Processes

S.Longo*, D.Bruno+, M.Capitelli+, and P.Minelli

Effects due to quantum coherence become relevant for very fast chemical reactions, when the phase relations between the quantum states of the transition under examination become manifest, for example in the form of Rabi oscillations, in the time domain, or wiggling of the absorption spectra, in the frequency domain. The kinetics of such reactions is usually based on closed transport equations for the density matrix of the internal levels of the ensemble under examination,i.e. the optical Rabi equation or the Redfield one. We have proposed recently a more detailed description, based on distribution functions over the space of possible (quantum) internal states of the atoms. For example, for spin 1/2 atoms in a magnetic field or, generally, 'two-level' systems one can consider the internal density matrix of any atom, leading to a distribution in the space of Bloch vectors, each equivalent to the density matrix. This helps in keeping the quantum and classical fluctuations distinct, whereas they are mixed in the density matrix formalism. The motivation for this is that the classical fluctuations can have a more detailed spectrum in non-equilibrium cases. Such a representation is very appropriate for gases undergoing translational relaxation, when collision processes are described by sudden, random changes of the atomic state. Keeping in mind the distinction of proper and improper mixture introduced by d'Espagnat long ago, we arrive at an equivalent approach by setting up a kinetic model with one-particle distribution functions over quantum and classical degrees of freedom that keep track of existing correlations between internal state and gas dynamic variables. To this end we associate to any atom a mixed internal (quantum)/translational state. The resulting equations for the kinetics of the ensemble are solved by a version of the DSMC method. In particular, we have studied the decay of correlations between the translational and internal state in a model gas, which can result in a quantum revival phenomenon. This is a peculiar kind of‘dissipative ordering’ cast in a global increase of enthropy. The research in progress is now aimed at developing new methods to include the kinetics of quantum correlations into the scheme, correlations which are produced by collision events. To this aim, approximate forms of the relevant kinetics equations are proposed, which can be conveniently solved by dedicated versions of the DSMC method.

### From dilute to very dense (granular, dissipative) gases

S. Luding *

Dilute, medium dense and very dense granular gases with particles of different size are examined and simulation results are compared with theoretical predictions concerning the transport coefficients pressure, shear stress, viscosity and heat-conductivity.

When dissipation is added, typically clusters form in the system, so that very dilute and very dense systems co-exist. Event Driven simulations are compared with DSMC simulations of this system. The long range structure is reproduced, but the short range correlations are not caught by DSMC.

The pressure, i.e. the equation of state, shows a transition from disordered fluid state to the ordered, dense, solid like state at a transition density of about 0.7 in two dimensions. Due to the excluded volume effect of the particles, the pressure diverges at a maximum density which depends on the particle size distribution. The viscosity diverges at a much smaller density close to the solid-liquid transition density, i.e. a glass transition takes place. The viscosity is also examined for different size distributions and one obtains shear-thickening or shear-thinning as function of the size- and mass-ratio of the particles.

### Hydrodynamic behavior for Lyapunov exponents

M. Mareschal* and Sean McNamara

For sufficiently large systems of hard disks, simulations show that the smallest exponents in the Lyamunov spectrum depend on the system geometry and have similarities with hydrodynamic modes. A theory, based on the collisional invariants and using a generalized Boltzmann equation allows to understand the origin of this behavior.

### DSMC Simulations of Carbon Deposition on Glass Plate and Comparisons with Experiments

M. Ota*, A. Noguchi and S.K. Stefanov+

Numerical approarch has been done by DSMC method for simulating the deposition of carbon particles on a glass plate. The distributions of carbon deposition thicknesses on the glass are revealed by the simulation. The thicknesses by the simulation is also compared with some experimental data are compared with experimental data.

### The Time Relaxed Monte Carlo method: towards an hybrid strategy for multiscale kinetic equations

L. Pareschi*

In this talk we briefly review some recent results concerning the development of Monte Carlo schemes for kinetic equations which are robust in the presence of multiple scales, like the small Knudsen number limit of the Boltzmann equation. The essence of the schemes is to take advantage of the 'apriori' knowledge of the equilibrium states by making use of a suitable expansion in time of the solution. The stability properties and the asymptotic behaviour of the schemes is studied and several results for schemes up to second order accuracy are presented in the Boltzmann case. Higher order recursive extension are also considered.

Next we follow the natural idea of 'hybridization' by coupling finite difference solvers in regions where macroscopic approximations are valid with Monte Carlo solvers for regions characterized by a kinetic regime. In particular we show how this idea can be developed within a general formulation which is not restricted to the Boltzmann case but which can be applied to a wider class of kinetic equations and hyperbolic relaxation systems.

References
-L.Pareschi, R.E. Caflisch, J. Comp. Phys., 154, p.90-116 (1999)
-L.Pareschi, G. Russo, SIAM J. Sci. Comput., 23, 4, p.1253-1273, (2001)
-L.Pareschi, G. Russo, ESAIM: Proceedings, 10, pp.35-75, (2001)
-L.Pareschi, S. Trazzi, preprint (2003)
-L.Pareschi, B.Wennberg, Monte Carlo Meth. and App., 7, 3-4, pp.349-357, (2001).

### Direct simulation of the uniformly heated granular Boltzmann equation

S. Rjasanow*

In the present paper we give an overview of the analytical properties of the steady state solution of the spatially homogeneous uniformly heated granular Boltzmann equation. The asymptotic properties of this distribution (so called tails) are formulated for different models of interaction. A new stochastic numerical algorithm for this problem is presented and tested using analytical relaxation of the temperature. The tails'' of the steady state distribution are computed using this algorithm and the results are compared with the available analytical information.

### On The Validity of the Boltzmann Equation to Describe Rapid Granular Flows

M.J. Ruiz-Montero* and J.J. Brey+

The validity of the Boltzmann equation to describe dilute rapid granular flows beyond the quasi-elastic limit has been questioned since it was first applied to these systems. Nevertheless, its success to describe the behaviour of low density granular flows in very different situations supports clearly its applicability to these systems. This scenario might have changed recently, after the work by H. Nakanishi (Phys. Rev. E Vol. 67, 010301(2003)) in which the behaviour of the velocity distribution of a freely evolving granular system leads him to conclude that the molecular chaos hypothesis and, therefore, the Boltzmann equation, fails from the early stages of the evolution. This conclusion will be investigated by comparing the results of Molecular Dynamics simulations and by using the the DSMC method for a system of freely evolving granular disks in equivalent conditions.

### Time Relaxed Monte Carlo Methods: effective time discretization for large range of mean free path.

G. Russo*

In the talk we discuss Monte Carlo method for the Boltzmann equation, which is based on a new time discretization, called Time Relaxed (TMRC), which is based on the Wild sum expansion of the solution of the Boltzmann equation for Maxwellian molecules. The sum is suitable truncated, and the reminder is substituted by a local Maxwellian. The scheme is generalized to treat Variable Hard Sphere molecules, and can be constructed with the desired order of accuracy in time. The local Mawellian guarantees that the scheme is able to capture the limit of small mean free path, provigind a Monte Carlo kinetic scheme for the Euler equation, in the fluid dynamic limit. Several numerical results confirm the expected robustness of the method.

The time discretization can be effectively used in conjunction with other discretizations (in velocity) of the Boltzmann equation. An example which makes use of a spectral method will be persented.

References
-L.Pareschi, G. Russo, SIAM J. Sci. Comput., 23, 4, p.1253-1273, (2001)
-F.Filbet, G.Russo, Journal of Computational Physics, vol 186/2 pp 457 - 480 (2003)

### Can a system of elastic hard spheres mimic the properties of a granular gas?

A. Astillero and A. Santos*

### Macroscopic boundary condition in vibrofluidized granular media

R. Soto*

The case of a granular media, fluidized by a vibrating plate, is studied in the limit of high frequency $\omega$ and low amplitude $A$ of the forcing. It is shown that in the regime where hydrodynamic equations apply, the vibrating plate can be replaced by a stationary wall that injects energy continuously if the limit is taken such that $A\omega^{5/4}$ is kept constant. At higher frequencies, when kinetic theory must be used, an analysis of Boltzmann equation predicts that the vibrating wall can also be replaced by an energy injecting boundary condition, but now when $A\omega$ is kept constant. In both cases the value of the energy injection rate is obtained.

### Direct Monte Carlo Simulation of Gas Flows through Highly Porous Media

S.K. Stefanov*

The present work deals with some problems of the mathematical formulation and the computer simulation of gas flows through a fine highly porous medium. An important feature of these flows is that the mean pore size of the porous medium is comparable to the mean free path of the molecules moving inside the pores. A well-known fact is that the use of continuum models for mathematical description of this class of flows leads to wrong predictions of the basic macroscopic properties of the gas fluxes through the porous layer. In this case, for a correct analysis the gas motion within the pores must be interpreted as a rarefied gas flow, and consequently, described mathematically on the base of the Boltzmann kinetic equation. On the other side the motion of a single gas molecule through the pore skeleton can be described by using the well-known "dusty gas" model, which considers the porous body skeleton as a system of large, compared with the gas molecules immovable particles randomly distributed within the porous domain. Thus, the velocity distribution function of the molecules moving through the porous body skeleton can be obtained from a linear integral equation describing the scattering of the molecules from the surface of the large immovable particles. In our consideration, both processes of intermolecular collisions and molecular scattering from pore boundaries are incorporated into one kinetic equation. The obtained equation is solved by using the Direct Simulation Monte Carlo (DSMC) method appropriately modified for the aims of our analysis.

### DSMC and the Boltzmann equation

W. Wagner*

In the first part of the talk I give some sketch of the history of the subject. Starting from early work by Leontovich and Kac, the development of stochastic models for the Boltzmann equation is discussed. The main interest here is in proving rigorously the convergence of the system (when the number of particles increases) to the solution of the equation in an appropriate sense. The second part of the talk is devoted to algorithmic and numerical aspects. I discuss some generalization of DSMC called stochastic weighted particle method. The new method contains several degrees of freedom which are used for the purpose of variance reduction. Some results of numerical experiments (calculation of tails of the velocity distribution) are presented.

* Speaker; + Workshop participant