Stochastic Numerics for the Boltzmann Equation

Abstract

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Cover design: design&production, Heidelberg Typeset by the authors using a Springer L A T E X macro package Printed on acid-free paper 46/3142sz-5 4 3 2 1 0 PrefaceStochastic numerical methods play an important role in large scale computations in the applied sciences. Such algorithms are convenient, since inherent stochastic components of complex phenomena can easily be incorporated. However, even if the real phenomenon is described by a deterministic equation, the high dimensionality often makes deterministic numerical methods intractable. A stochastic procedure, called direct simulation Monte Carlo (DSMC) method, has been developed in the physics and engineering community since the sixties. This method turned out to be a powerful tool for numerical studies of complex rarefied gas flows. It was successfully applied to problems ranging from aerospace engineering to material processing and nanotechnology. In many situations, DSMC can be considered as a stochastic algorithm for solving some macroscopic kinetic equation. An important example is the classical Boltzmann equation, which describes the time evolution of large systems of gas molecules in the rarefied regime, when the mean free path (distance between subsequent collisions of molecules) is not negligible compared to the characteristic length scale of the problem. This means that either the mean free path is big (space-shuttle design, vacuum technology), or the characteristic length is small (micro-device engineering). As the dimensionality of this nonlinear integro-differential equation is high (time, position, velocity), its numerical treatment is a typical application field of Monte Carlo algorithms.Intensive mathematical research on stochastic algorithms for the Boltzmann equation started in the eighties, when techniques for studying the convergence of interacting particle systems became available. Since that time much progress has been made in the justification and further development of these numerical methods.The purpose of this book is twofold. The first goal is to give a mathematical description of various classical DSMC procedures, using the theory of Markov processes (in particular, stochastic interacting particle systems) as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method (SWPM). This VI Preface method has been developed by the authors during the last decade. SWPM includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous results concerning the approximation of solutions to the Boltzmann equation by particle systems are given, covering both DSMC and SWPM. Thorough numerical experiments are performed, illustrating the behavior of systematic...

A kinetic model for epidemic spread

“…Secondly, the particle scheme corresponds numerically to the direct simulation Monte Carlo method, widely used to approximate rarefied gas dynamics. There are several variants of such methods [Bird 1994;Rjasanow and Wagner 2005]. Below, we will deal with an inhomogeneous Kac model [1956] for three species with reactions.…”

“…In the case of the stochastic system, one can apply martingale techniques as in [Wagner 1992], or the hierarchy of equations for the family of the marginals , or coupling techniques [Graham and Méléard 1997]. In the case of the mechanical model, one can resort to the validity techniques for the Boltzmann equation, leading to a short-time result; see [Lanford 1975] and subsequent works [Illner and Pulvirenti 1989;Spohn 1991;Cercignani et al 1994;Gallagher et al 2013;Pulvirenti et al 2014;Pulvirenti and Simonella 2017;Denlinger 2018].…”

Slow nonisothermal flows: Numerical and asymptotic analysis of the Boltzmann equation

“…For small Knudsen numbers, and especially slow flows, the classical DSMC method is extremely expensive. Many specialized stochastic schemes have been proposed in recent years [22,23]. Deterministic methods for the Boltzmann equation are also very promising [24,25].…”

“…At the same time, thanks to the development of computer technology, some applied problems were analyzed numerically using the KGF and kinetic equations: on the basis of model equations [15,16,17,18,19] and the direct simulation Monte Carlo (DSMC) method [20,21].For small Knudsen numbers, and especially slow flows, the classical DSMC method is extremely expensive. Many specialized stochastic schemes have been proposed in recent years [22,23]. Deterministic methods for the Boltzmann equation are also very promising [24,25].…”