LatticeBoltzmann

Lattice Boltzmann is a computational method for simulating fluid dynamics that models a fluid as a set of particle distribution functions evolving on a lattice. The approach is derived from a discretized Boltzmann equation and, in the common BGK (Bhatnagar–Gross–Krook) form, uses a single relaxation time toward an equilibrium distribution determined by local density and velocity. The macroscopic flow variables are obtained as moments of the distributions, and the method is particularly valued for its simplicity and ease of parallelization.

The numerical scheme operates through two basic steps: collision and streaming. At each lattice node, the distribution functions relax toward their local equilibrium f_i^eq, then propagate to neighboring nodes along a set of discrete velocity directions e_i. The standard update in lattice units is f_i(x+e_i, t+1) = f_i(x, t) - (1/τ)(f_i(x, t) - f_i^eq(x, t)) + S_i, with optional forcing S_i. Equilibrium distributions come from a low-Mach expansion of the Maxwell–Boltzmann distribution, often written as f_i^eq = w_i ρ [1 + 3 (e_i · u) + (9/2) (e_i · u)^2 − (3/2) u^2], using lattice-specific weights w_i.

Common lattice choices include D2Q9 in two dimensions and D3Q19 or D3Q27 in three dimensions. LBM has numerous extensions, including multiple-relaxation-time (MRT) models, thermal and multiphase variants, and a variety of boundary conditions (e.g., bounce-back for no-slip walls). It is widely used for flows in complex geometries, porous media, and microfluidic applications, while facing limitations such as compressibility effects at higher Mach numbers and stability challenges at large Reynolds numbers.