The Lattice Boltzmann Method (LBM) is a fully discrete mesoscopic method capable of simulating incompressible fluid flow. The method was discovered in 1988, further developed in the first half of the 1990’s and reached full maturity in the last decade. The original LBM was extended in many ways, in the direction of multiphase flows, thermal flows, suspension flows, and to include turbulence, chemical reactions and solute transport. The basic theory of LBM for single-phase flow is well established, although even today research papers are being published that address subtle issues in the theory. Moreover, understanding of the embedding of the LBM in the broader context of the continuous Boltzmann equation, the macroscopic Navier Stokes equations, and microscopic theories is by now well understood. In this document the Lattice Boltzmann Method will first be introduced, followed by a theoretical demonstration that the model indeed mimics incompressible hydrodynamics. Next, the embedding of the method in both microscopic – and macroscopic theories will be clarified.
Krüger, T. (2004). The Lattice Method Boltzmann Principles and Practice. Physics (pp. 1–7).