Abstract
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.
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