LBM

LBM stands for the lattice Boltzmann method, a computational approach for simulating fluid flow and related transport phenomena. It models the evolution of mesoscopic particle distribution functions on a regular lattice, instead of directly solving the macroscopic Navier–Stokes equations.

At each lattice node, a set of discrete velocity directions carries populations f_i. The method alternates between collision, which relaxes f_i toward a local equilibrium f_i^eq, and streaming, which propagates populations to neighboring nodes. The relaxation is controlled by a relaxation time parameter, or by multiple relaxation rates in advanced variants.

Macroscopic quantities are obtained as moments of f_i: density rho and momentum rho u. In lattice units, the kinematic viscosity is related to the relaxation time tau; nu = c_s^2 (tau - 0.5) for the standard BGK implementation. A common choice of velocity sets is D2Q9 in two dimensions and D3Q19 in three dimensions; c_s^2 is a fixed lattice constant (for example, 1/3 in D2Q9). The Navier–Stokes equations are recovered in the limit of small Mach numbers.

Boundary conditions include no-slip via bounce-back rules, and more advanced schemes for velocity boundaries. LBM can handle complex geometries and is well suited to multiphase, porous-media, and microscale flows; it also supports coupling to particles and thermal effects in various extensions.

Variants such as multiple-relaxation-time (MRT) and entropic LBM improve stability; multiphase models (Shan–Chen, free-energy) and thermal LBM extend the method to heat transfer and phase change. History traces to lattice gas automata in the 1980s; the BGK form popularized in the 1990s.