Lattice-gas cellular automata and lattice Boltzmann models : an introduction / Dieter A. Wolf-Gladrow

In this very well-written book, the author gives a general overview of lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) used to solve numerically some partial differential equations of macroscopic physics (mainly the incompressible Navier-Stokes equation).
The book is quite readable for non-specialists in the subject. Only some familiarity with the equations of fluid mechanics is required of the reader. I think, however, that even specialists will appreciate this work, since many results dispersed in the literature are discussed here in a very enlightening way.
Most of the computations of this book are performed at the "formal'' level, so that the reader need not have a good knowledge of PDE theory or functional analysis. However, some familiarity with complex asymptotic expansions (multi-scale analysis, Chapman-Enskog expansion, etc.) will be helpful when reading it.
After a brief introduction, the author recalls in the second chapter some basic facts about one- and two-dimensional cellular automata. Many illustrations are presented.
Then, in the third chapter, LGCA are defined, and a typical code is presented. The classical "FHP'' model of Frisch, Hasslacher and Pomeau is thoroughly investigated at this point: after having given a code, the author presents the theory allowing one to recover the macroscopic equations from the microscopic dynamics. Numerical simulations are presented and the role of the geometry of the lattice (isotropy of some of the lattice tensors) is emphasized. The rest of the chapter is mainly dedicated to the problem of finding a 3D model having good properties.
The fourth chapter is aimed at recalling some classical facts of statistical mechanics. The Boltzmann equation is introduced and Boltzmann's H-theorem as well as the Chapman-Enskog expansion are presented.
The last chapter is a precise study of LBM models in 2D and 3D, where numerical simulations in a well-defined physical context are given. Some applications outside of the field of fluid mechanics are then explored.
As a conclusion, I wish to say that I very much enjoyed reading this book, and I am sure that it will help all the researchers wishing to discover the field of simulation through lattice gases. (MathSciNet)