Stochastic Numerics for the Boltzmann Equation [Paperback]

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.

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 booklӽ