The Lattice Boltzmann method solves fluid dynamics problems through discretization of time and space. Therefore, there are cells representing a small portion of space (Δx) and timesteps (Δt) associated to a tiny period of time. Each cell contains a particle distribution function (PDF). A PDF (set of fi values) contains data about the statistical quantity of particles travelling at a determined point (r vector) and moment (t), at a specific direction (ei vector, considering Δx = eiΔt). There are different types of model, depending if the user wants to simulate problems in 2D or 3D. For 2D problems, the most popular model is the D2Q9 (nine possible directions: one null, four cardinal and four ordinal). Given a PDF, it is possible to determine the macroscopic velocity (u vector) and density (ρ) at that point:

This is the "magic" of the model: establishing a bridge between the microscopic world (particles) and the macroscopic one, it is easier to simulate more complex flows, with complex boundaries and coupled forces. Following the concepts of cellular automata, at each timestep, all cells are updated simultaneously. The update formula is given below:

Note that there is a new function, the equilibrium distribution function (EDF). The EDF determines the PDF at the equilibrium, given a macroscopic velocity and density. The EDF is given by the following formula. Note that there is a new constant in the formula: wi. This value is associated to each direction and, for the D2Q9 model, assumes the following values: 4/9 for the zero velocity, 1/9 for the cardinal directions and 1/36 for the ordinal ones. The c value is given by Δx/Δt.

This is only a small introduction to the LB method. This treatment is applied to the fluid cells. There is a special treatment for the initial conditions and the boundary ones. For better understanding and study, we recommend the reading of the books referenced in the "References" section.