Stochastic cellular automaton

Compressibility

Algorithm

Turbulence

Direct simulation Monte Carlo

Bhatnagar–Gross–Krook operator

Statistics

Boundary value problem

Reynolds number

Applied mathematics

Dynamic Monte Carlo method

Mathematical optimization

Solver

Statistical physics

Thermodynamics

Physics

Boltzmann equation

Mathematics

Cellular automaton

HPP model

Mathematical analysis

Monte Carlo method

Mechanics

Lattice Boltzmann methods

Lattice gas automaton

Journal of Computational Physics

This review summarizes the rigorous mathematical theory behind the lattice Boltzmann equation (LBE). Relevant properties of the Boltzmann equation and a derivation of the LBE from the Boltzmann equation are presented. A summary of some important LBE models is provided. Focus is given to results from the numerical analysis of the LBE as a solver for the nearly incompressible Navier-Stokes equations with appropriate boundary conditions. A number of numerical results are provided to demonstrate the efficacy of the lattice Boltzmann method.

The lattice Boltzmann method for nearly incompressible flows