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Stochastic numerics for the Boltzmann equation. (English) Zbl 1155.82021
Springer Series in Computational Mathematics 37. Berlin: Springer (ISBN 3-540-25268-1/hbk). xiii, 256 p. (2005).
The first chapter of the book is a review of basics in kinetic theory and Boltzmann equation. Collision transformations, collision kernels, collision integral and moment equations are discussed. A class of piecewise-deterministic Markov processes related to Boltzmann type equations are reviewed in the second chapter. These processes are jump processes that change their state in a deterministic way between jumps. One examines the state space of a process, the construction of sample paths and different types of jump behaviors. The transition to the macroscopic equation is performed heuristically. At this point the authors insert some comments on well known papers related to the history of Boltzmann type processes. Actually they do this for the next chapters too.
The main part of the book is the third chapter where the stochastic algorithms for the Boltzmann equation are developed. The algorithms are based on the Monte Carlo Method introduced by the brilliant scientists J. von Neumann, Stanislaw Ulam and Nicholas Metropolis while working on the Manhattan project in Los Alamos. They developed algorithms for computer implementations, as well as exploring means of transforming non-random problems into random forms that would facilitate their solution via statistical sampling. This work transformed statistical sampling from a mathematical curiosity to a formal methodology applicable to a wide variety of problems. The Monte Carlo method, as it is understood today, encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. However, the idea was not completely new. Buffon’s Needle stated in 1777 is one of the oldest problems in the field of geometrical probability, followed by the works of Lord Rayleigh, Courant, Kolmogorov and Petrowsky. In the present book, the direct simulation Monte Carlo method and an extension of it called stochastic weighted particle method, developed by the authors for the purpose of variance reduction, are applied to the approximation of solutions to the Boltzmann equation. A convergence theorem is proved.
The last chapter is dedicated to numerical experiments performed with the developed algorithms. For instance numerical results are shown for a spatially homogeneous Boltzmann equation, a spatially one-dimensional and a spatially two-dimensional Boltzmann equation. Auxiliary results are contained in two appendices.
The book is well written, clear and as much as possible self-contained.

MSC:
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
65C35 Stochastic particle methods
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
65C05 Monte Carlo methods
65C20 Probabilistic models, generic numerical methods in probability and statistics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76M35 Stochastic analysis applied to problems in fluid mechanics
60K40 Other physical applications of random processes
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