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.

中文翻译：

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近似不可压缩流的格子玻尔兹曼方法

这篇综述总结了格子玻尔兹曼方程（LBE）背后的严格数学理论。给出了玻耳兹曼方程的相关性质以及从玻耳兹曼方程推导的LBE。提供了一些重要的LBE模型的摘要。LBE的数值分析将重点放在作为具有适当边界条件的几乎不可压缩的Navier-Stokes方程的求解器的结果上。提供了许多数值结果，以证明格子Boltzmann方法的有效性。