Stochastic Numerics for the Boltzmann Equation

Stochastic numerical methods play an important role in large scale computations in the applied sciences. The first goal of this book is to give a mathematical description of classical direct simulation Monte Carlo (DSMC) procedures for rarefied gases, using the theory of Markov processes as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method. This method includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous convergence results as well as detailed numerical studies are presented.

From the reviews: "The book under review deals with numerical methods for the resolution of the nonlinear Boltzmann equation for rarefied monoatomic gases in 1D and 2D. Because of the high dimensionality of standard kinetic models, the authors privilege the stochastic procedures, namely Direct Simulation Monte Carlo methods (DSMC). Such a method can be investigated mathematically relying on the theory of Markov processes; this in return allows for proposing an extension of DSMC, the so-called Stochastic Weighted Particle Method (SWPM). The outline of the book is classical: Chapter 1 recalls basic features of kinetic models and the Boltzmann equation. Chapter 2 introduces the reader to Markov processes in the context of various Boltzmann models. The main contribution is Chapter 3, where the authors convey the reader to the stochastic algorithms, for which precise convergence results are given in some generality. Finally, Chapter 4 presents numerical results: first for the spatially Boltzmann model, then 1D and 2D simulations are displayed."  (Laurent E. Gosse, Mathematical Reviews) "The main part of the book is … 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. … The book is well written, clear and as much as possible self-contained." (Claudia Simionescu-Badea, Zentralblatt MATH, Vol. 1155, 2009)