Lattice Boltzmann Method

Nine Arrows of Maxwell–Boltzmann — Why LBM's Equilibrium Is a Polynomial

Cutting Maxwell–Boltzmann at Hermite order two builds the D2Q9 polynomial, and that same cut leaves the Mach 0.3 ceiling in place.

2026.05.25

Entropy Stops the Blow-Up — Karlin's Entropic Lattice Boltzmann

When low viscosity pushes τ toward 1/2, BGK overshoots equilibrium. Entropic LBM stops at the α that leaves discrete H unchanged.

2026.06.10

Surviving the Low-Viscosity Regime Where BGK Blows Up — MRT Lattice Boltzmann Collision

At nu=0.004 the shear rate s_nu is 1.953. BGK copies that rate onto every ghost mode; the MRT rates in the listing keep s_e at 1.64 and s_q at 1.7.

2026.06.26

In Lattice Boltzmann, the Wall Isn't on the Node — Bounce-Back and Zou-He Boundaries

Half-way bounce-back puts the wall a half-cell off the node. Couette speed is U(j+1/2)/ny. Height ny-1 leaves friction bias that refinement never clears.

2026.07.20

Where Half the Force Goes in LBM — Four Forcing Schemes and 1−1/(2τ)

Steady Poiseuille gives the same error for five LBM forcing schemes. At τ=0.6, the half-shift alone inflates the amplitude by a factor of 1.83.

2026.07.27

Raise the Velocity, Lose 27% of Your Diffusion — The Extra Flux in LBM Convection-Diffusion

Set D through tau in a lattice Boltzmann convection-diffusion model and it still spreads at D(1-u²/cs²). Where Chapman-Enskog drops the extra flux, and the source term that erases it.

2026.08.14

I Plugged in Pressure Directly and a Still Interface Started to Shiver — Two Forms of Nonideal LBM Forcing

Pressure-form and chemical-potential forcing agree for van der Waals in the continuum, then split on the lattice where the Gibbs-Duhem defect lives.

2026.08.21

Three at a wall, five at a corner — counting the populations an LBM boundary node loses

After streaming, empty populations at an LBM node equal the solid cells around it: three on a flat wall, five in a concave corner, one on a convex corner.

2026.08.24

I Dropped the −1/2 and the Viscosity Came Back Six Times Too Large — the Δt/2 Left Behind by LBM's Discretisation

One trapezoidal step and one change of variables leave a Δt/2 in viscosity, velocity, and stress. A scalar ODE and a D2Q9 lattice measure all three.

2026.08.26

ρ and u Agree to the Last Digit While the Strain Rate Swells 45% — Nonequilibrium Rescaling at an LBM Refinement Interface

Copying distribution functions across an LBM AMR level interface keeps ρ and u exact but inflates the stress by up to 2x. I measured the rescale factor.

2026.09.02

Doubling τ Left Pr at 1.0000 — The Two Constants Locked Inside Thermal LBM

A single-distribution thermal LBM ties viscosity and thermal diffusivity to one τ, locking Pr at 1. I measured on D2Q9 what a second distribution unlocks.

2026.09.04

27 of 1,681 Voxels Flipped Inside Out — Ray Parity and Link Cutting in STL Voxelization

Turning an STL into an LBM voxel grid takes four stages: octree culling, SAT overlap tests, ray parity, and the per-link cut fraction q, verified in code.

2026.09.07

Pushing the Frame to 0.2 Cut the Viscosity by 5.97% — The Missing Third Moment in LBM Equilibria

On D2Q9, Σf c³ equals ρu for any distribution you can write. The missing ρu³ breaks Galilean invariance and makes measured viscosity depend on the frame.

2026.09.16

One node pushed the error from 0.21% to 76% — local relaxation time in power-law LBM

One centreline node asks for infinite tau at n=0.6. Cap it near 4 and L2 error stays near 0.21%; leave it and that node drives the error to 76%.

2026.09.21

Giving every species its own tau threw the total mass off by 16.8% — correction velocity in multicomponent LBM

Different taus make Fick fluxes miss zero. Over 2000 steps a three-species LBM drops the worst mass defect from 0.168 to 0.0005.

2026.09.23

Moving refine one line below stream cost a 7.2x error — the recursive time cycle of LBM AMR

Moving refine below stream on the same LBM grid raises the L2 error from 0.657% to 4.704%, a factor of 7.2, and the excess sits on the seam.

2026.09.25

At tau = 0.51 I read the viscosity 51x too high — the half step left behind by the discrete LBE

At tau=0.51 the lattice reads nu=0.00333, not 0.17. A second-order expansion drops Delta t/2 from tau: that half step is a factor of 51.

2026.09.28

Filling the Third Moment Honestly Made the Error 2.8x Worse — What a Forcing Term Actually Owes

On D2Q9, filling the force third moment raises error from 0.053% to 0.150%. The lattice locks half the tensor, and it arrives only at Burnett order.

2026.09.30

Degree six returned 9 where 15 was required — the three Hermite roots that fixed D2Q9

Where c_s² = 1/3 and the weights 4/9, 1/9, 1/36 actually come from, traced all the way back to the degree of a Gauss–Hermite quadrature rule.

2026.10.05

A Motionless Droplet Melted to Half Its Mass — The Curvature Term Conservative Allen–Cahn Erases

At zero velocity a droplet lost 49.4% of its mass under classical Allen–Cahn. The conservative form held mass to 1e-14 by cancelling one curvature term.

2026.10.09

Giving Conduction Its Own τ Left Viscous Heating Stuck at Pr = 1 — The Quasi-Equilibrium g* in Two-Population LB

In Couette flow a plain two-τ thermal LB keeps θ_max at 1.000 for every Pr. Adding the quasi-equilibrium g* recovers 0.498, 0.710, 1.000, and 1.997.

2026.10.19

Changing only τ changed the density of a gas at rest — the (τ−½)² stress of Shan–Chen velocity-shift forcing

Apply the same force to a resting D1Q3 gas column and only Shan–Chen forcing gives a τ-dependent density. The shifted equilibrium velocity leaves a fake (τ−½)²F²/ρ stress.

2026.10.23

Velocity came out one sixth at τ=0.6 — dropping the half-force F/2 from Guo forcing

Keep the Guo source term but drop F/2 from the LBM velocity and the effective body force shrinks by (1−1/2τ). On Kolmogorov flow at τ=0.6 the amplitude is 0.166 of exact.

2026.10.28