Most ways of simulating a fluid start from the top and work down: write the Navier-Stokes equations, the continuum description of pressure and velocity fields, and discretize them onto a mesh with finite differences or finite volumes or finite elements, solving a coupled system of nonlinear PDEs every timestep. The Lattice Boltzmann Method starts from the bottom and works up, and the strange thing is that it arrives at the same destination by refusing to ever write down the equations everyone else considers the actual physics.
Instead of tracking a velocity field, LBM tracks populations — at every point on a fixed lattice, a small set of numbers representing "how much fluid is moving in this particular direction, at this particular speed." The classic 3D scheme, D3Q19, keeps nineteen of these directions per lattice site: rest, the six axis directions, and twelve diagonals. Each timestep does exactly two things to these populations. First, streaming: every population value moves one lattice step in its own direction, deterministically, exactly, no numerical diffusion, because moving a number from one box to an adjacent box is not an approximation — it's just indexing. Second, collision: at each site, the populations that just arrived from different directions relax partway toward a local equilibrium distribution — the Maxwellian a real gas would settle into if left alone — controlled by a single number, the relaxation time, in what's called the BGK approximation (after Bhatnagar, Gross, and Krook, who proposed replacing the ferociously complicated real collision operator with a single relaxation-rate stand-in decades before anyone thought to run it on a lattice).
That's the whole algorithm. Stream, then relax toward equilibrium. Repeat. No pressure Poisson equation to solve, no implicit linear system, no global matrix inversion at all — every lattice site only ever needs to know about its immediate neighbors, which makes LBM almost embarrassingly parallel: it maps onto GPU hardware about as naturally as a fluid solver can, because the update at each cell needs no information from across the domain, ever.
The astonishing part is what falls out of taking the macroscopic moments of these microscopic-looking populations. Sum a cell's nineteen populations and you get the local density. Weight them by their direction vectors and sum, and you get the local momentum — the velocity field. Carry the Chapman-Enskog expansion (a multiscale asymptotic argument, matching timescales order by order) far enough and the discrete streaming-and-relaxing recovers, provably, the incompressible Navier-Stokes equations in the appropriate limit. A method built entirely from bookkeeping simple directional populations on a fixed grid reproduces, at the macroscopic scale, the continuum fluid mechanics everyone else derives from conservation laws directly. The particles were never real — D3Q19 populations aren't molecules, they're a mathematical convenience, chosen for how cleanly they recover the right macroscopic limit — but the fluid that emerges from tracking them has completely forgotten it was ever assembled this way.
Complex geometry, which is a nightmare for a body-fitted mesh solver, becomes almost an afterthought: a solid wall is just a lattice site where the bounce-back rule reverses incoming populations instead of letting them relax normally, no remeshing, no distorted cells near a curved surface, just a different local rule at flagged sites. And multiphase flow — oil and water, liquid and vapor — which classical solvers handle with interface-tracking machinery (level sets, VOF, front tracking, each with its own numerical headaches), LBM handles by letting different fluid species exert a Shan-Chen pseudopotential force on each other, an inter-particle attraction/repulsion added directly into the collision step, and phase separation simply happens, the same way it would happen to real molecules that dislike touching each other, because the rule that produces it is local and symmetric rather than a global interface reconstructed and tracked at every step.
The method's whole appeal is a kind of humility about where the difficulty of fluid dynamics actually lives: not in writing down the right continuum equation, which everyone already knows, but in solving it numerically without the machinery becoming the bottleneck. LBM trades one hard global problem — solving coupled nonlinear PDEs — for one easy local rule repeated at enormous scale, and lets emergence do the rest.
Seed: Lattice Boltzmann Method — D3Q19 streaming/collision, BGK approximation, Chapman-Enskog expansion, Shan-Chen multiphase.