##
**Lattice-gas cellular automata and lattice Boltzmann models. An introduction.**
*(English)*
Zbl 0999.82054

Lecture Notes in Mathematics. 1725. Berlin: Springer. ix, 308 p. (2000).

Lattice-gas cellular automata are have become alternative tool for the numerical analysis of gas dynamics (Navier-Stokes equation). In contrast to the traditional discretization methods, LGCA’s start with a discrete model that has build much of the physically motivated properties of the underlying microsystems, while under suitable scaling is is designed to reproduce the solutions of the desired hydrodynamic equations.

This book is an introduction to this field written from the practitioners point of view, providing all the necessary tools to actually implement these methods in concrete algorithms for the simulation of real world problems. This complements in a welcome way other texts on the subject that often spend more time on the underlying physics or mathematics, but less on the concrete problems facing an actual implementation, such as the precise choice of the lattice, collisions rules and specially adapted programming techniques (indeed, the book almost starts with the question: Fortran or C?!).

The book is divided into 5 Sections and an Appendix. Section 1 discusses the basic issues of simulation of the Navier-Stokes equations and explains the basic ideas of LGCA’s and lattice Boltzmann models.

Section 2 reviews some general aspects of Cellular Automata. Here the reader will find the familiar pictures of Wolfram’s 1D CA’s, Conway’s 2D “game of live”, etc.

Section 3 covers LGCA modelling. The reader is introduced to the classical models like the HPP (Hardy, de Pazzis, Pomeau) and FHP (Frisch, Hasslacher, Pomeau) automata. The role of the choice of the underlying lattice in reproducing the proper symmetries of the solutions, the importance of the incorporation of the appropriate conservation laws in order to obtain the desired macroscopic equations are explained carefully. A large part of this chapter is devoted to the question how in three dimension a suitable lattice can be constructed. At the same time the the programming issues of the numerical implementation are presented in detail.

Section 4 provides the bridge between LGCA’s and lattice Boltzmann models. The Boltzmann equations, \(H\)-theorem, and the Chapman-Enskog expansion linking the Boltzmann equation to Navier-Stokes are presented.

Section 5 then introduces to the various levels of todays highly efficient lattice Boltzmann models.

A large number of exercises of different levels of difficulties are interspersed with the text and help the sudent to actually learn the subject. Anyone wanting to learn how to simulate fluid dynamics problems will find this book most helpful.

This book is an introduction to this field written from the practitioners point of view, providing all the necessary tools to actually implement these methods in concrete algorithms for the simulation of real world problems. This complements in a welcome way other texts on the subject that often spend more time on the underlying physics or mathematics, but less on the concrete problems facing an actual implementation, such as the precise choice of the lattice, collisions rules and specially adapted programming techniques (indeed, the book almost starts with the question: Fortran or C?!).

The book is divided into 5 Sections and an Appendix. Section 1 discusses the basic issues of simulation of the Navier-Stokes equations and explains the basic ideas of LGCA’s and lattice Boltzmann models.

Section 2 reviews some general aspects of Cellular Automata. Here the reader will find the familiar pictures of Wolfram’s 1D CA’s, Conway’s 2D “game of live”, etc.

Section 3 covers LGCA modelling. The reader is introduced to the classical models like the HPP (Hardy, de Pazzis, Pomeau) and FHP (Frisch, Hasslacher, Pomeau) automata. The role of the choice of the underlying lattice in reproducing the proper symmetries of the solutions, the importance of the incorporation of the appropriate conservation laws in order to obtain the desired macroscopic equations are explained carefully. A large part of this chapter is devoted to the question how in three dimension a suitable lattice can be constructed. At the same time the the programming issues of the numerical implementation are presented in detail.

Section 4 provides the bridge between LGCA’s and lattice Boltzmann models. The Boltzmann equations, \(H\)-theorem, and the Chapman-Enskog expansion linking the Boltzmann equation to Navier-Stokes are presented.

Section 5 then introduces to the various levels of todays highly efficient lattice Boltzmann models.

A large number of exercises of different levels of difficulties are interspersed with the text and help the sudent to actually learn the subject. Anyone wanting to learn how to simulate fluid dynamics problems will find this book most helpful.

Reviewer: Anton Bovier (Berlin)

### MSC:

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |

76M28 | Particle methods and lattice-gas methods |

82-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics |

76-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |