A quasi-Monte Carlo method for the Boltzmann equation

Lécot, Christian

doi:10.1090/s0025-5718-1991-1068812-4

Cited by 10 publications

(4 citation statements)

References 13 publications

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“…Now we specify the assumptions of Theorem 3.22 for the reduction measure (3.230). 120 3 Stochastic weighted particle method Remark 3.36. Assume that there exist k γ ≥ 1 and δ ∈ (0, 1) such that (cf.…”

Section: General Constructionmentioning

confidence: 99%

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Stochastic Numerics for the Boltzmann Equation

Rjasanow¹,

Wagner²

2005

Springer Series in Computational Mathematics

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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...

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Section: General Constructionmentioning

confidence: 99%

“…[152], [150]). Further studies concerning low discrepancy sequences in the context of the Boltzmann equation were performed in [118], [119], [120]. Stochastic algorithms for generalized Boltzmann equations, including multicomponent gases and chemical reactions, were studied, e.g., in [55], [129].…”

Section: Further Referencesmentioning

confidence: 99%

Stochastic Numerics for the Boltzmann Equation

Rjasanow¹,

Wagner²

2005

Springer Series in Computational Mathematics

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“…Sequences of quasi-random numbers have been applied in the past with some success to the numerical evaluation of high-dimensional integrals [17], in particular in problems of financial mathematics [18]. Other applications have been made to the generation of quasi-random paths of stochastic processes in [19,20], to the Boltzmann equation [21], and to a simple system of diffusion equations in R d [22]. A failure in applying them to the solution of stochastic differential equations has been pointed out in [23], but it has been shown in [24] that a careful implementation allows for a successful use of them.…”

Section: Introductionmentioning

confidence: 99%

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Acebrón

Busico²,

Lanucara³

et al. 2005

Journal of Computational Physics

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“…Monte Carlo and quasi-Monte Carlo methods are also successfully applied to several classes of nonlinear equations (see [18], [19], [27], [2], [4]). In case of second order hyperbolic equations, äs the wave equation, the Situation is quite different.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo difference schemes for the wave equation

Ermakov¹,

Wagner²

2002

Monte Carlo Methods and Applications

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The paper is concerned with Monte Carlo algorithms for iteration processes. A recurrent procedure is introduced, where Information on various iteration levels is stored. Stability in the sense of boundedness of the correlation matrix of the component estimators is studied. The theory is applied to difference schemes for the wave equation. The results are illustrated by numerical examples. 2000 Mathematics Subject Classification: 65C05

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A quasi-Monte Carlo method for the Boltzmann equation

Cited by 10 publications

References 13 publications

Stochastic Numerics for the Boltzmann Equation

Stochastic Numerics for the Boltzmann Equation

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Monte Carlo difference schemes for the wave equation

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