Lattice Boltzmann Method

The Lattice Boltzmann Method (LBM) solves fluid dynamics mesoscopically: instead of tracking fluid parcels, we evolve distribution functions f_i(x,t) on a lattice. For D2Q9 (2D, 9 velocity directions):

f_i(x+e_i, t+1) = f_i(x,t) − (f_i − f_i^eq)/τ (BGK)
f_i^eq = ρ w_i [1 + e_i·u/c_s² + (e_i·u)²/(2c_s⁴) − u²/(2c_s²)]

Macroscopic density ρ = Σf_i and momentum ρu = Σf_i e_i emerge from the distributions. The relaxation time τ = ν/c_s² + ½ controls viscosity. LBM naturally handles complex boundaries (bounce-back), is trivially parallelizable, and recovers Navier-Stokes at low Mach number via Chapman-Enskog expansion. Visualize velocity magnitude, pressure, or vorticity to observe laminar flow, von Kármán vortex streets, and turbulence onset.