Source: Scholarpedia
The lattice Boltzmann equation (LBE) is a minimal form of Boltzmann kinetic equation which is meant to simulate the dynamic behaviour of fluid flows without directly solving the equations of continuum fluid mechanics. Instead, macroscopic fluid dynamics emerges from the underlying dynamics of a fictitious ensemble of particles, whose motion and interactions are confined to a regular space-time lattice (Benzi and Succi, 1992; Chen and Doolen, 1998). Technically, the distinctive feature of LBE is a dramatic reduction of the degrees of freedom associated with the velocity space. In fact, particle velocities are restricted to a handful of discrete values v⃗ =c⃗ i , i=0,b , by assuming that at each site the particles can only move along a finite number of directions. By endowing this set with sufficient symmetries to fulfill the basic conservation laws of mass, momentum and energy, the LBE can be shown to quantitatively reproduce the equations of motion (Chapman and Cowling, 1952) of continuum fluid mechanics, in the limit of long wavelengths as compared to the lattice scale. The idea of simplified Boltzmann equations with discrete speeds can be traced to the pioneering work of Broadwell, back in the 60’s (Broadwell, 1964). However, to the best of our knowledge, these discrete-velocity Boltzmann equations were mostly intended to provide simpler, sometimes even analytically tractable, model equations for rarefied gas dynamics, but never meant as a computational alternative for the numerical solution of the Navier-Stokes equations of continuum fluid dynamics. The major conceptual twist in this direction had to wait another 20 years, with the advent of the celebrated Frisch-Hasslacher-Pomeau (FHP) lattice gas automaton (Frisch et al., 1986; Wolfram, 1986, Wolf-Gladrow, 2000). Originally developed in response to the major pitfalls of the Lattice Gas Cellular Automaton approach (Frisch et al., 1986; Wolf-Gladrow, 2000), the LBE rapidly developed into a vigorous self-standing research subject (Mc Namara and Zanetti, 1988; Higuera and Jimenez, 1989; Higuera et al., 1989; Chen et al., 1992). The distinctive features of LBE as a computational solver for fluid problems are its space-time locality, and the fact that information travels along the straight lines defined by the (constant) particle velocities associated with the lattice, rather than along the space-time dependent material lines defined by the flow speed. Due to these properties, the LB approach counts today an impressive array of applications across virtually all fields of fluid dynamics and allied disciplines, such as biology and material science (Succi, 2001, 2008).
