Math 2.0

Fluid mechanics Definition

Lattice Boltzmann method

Builds on BGK collision operatorBoltzmann equationNavier–Stokes equations

Definition

Let Δx>0\Delta x > 0 and Δt>0\Delta t > 0 be the lattice spacing and the time step, and consider the lattice Δx ℤd\Delta x\, \Z^d. Let c0,…,cq−1∈ℝdc_0, \ldots, c_{q-1} \in \R^d be velocities with ciΔt∈Δx ℤdc_i \Delta t \in \Delta x\, \Z^d, and let wi>0w_i > 0 be weights summing to 11. Let cs>0c_s > 0 be a lattice sound speed such that ∑iwici=0\sum_i w_i c_i = 0 and ∑iwiciciT=cs2I\sum_i w_i c_i c_i^T = c_s^2 I. The unknowns are populations fi(x,t)f_i(x, t) at lattice points xx and times t∈Δt ℕt \in \Delta t\, \N, with density ρ=∑ifi\rho = \sum_i f_i and momentum ρu=∑icifi\rho u = \sum_i c_i f_i. The equilibrium populations are fieq=wiρ(1+ci⋅ucs2+(ci⋅u)22cs4−∣u∣22cs2).f_i^{eq} = w_i \rho \left( 1 + \frac{c_i \cdot u}{c_s^2} + \frac{(c_i \cdot u)^2}{2 c_s^4} - \frac{\abs{u}^2}{2 c_s^2} \right). The lattice Boltzmann method with BGK collision and relaxation time τ>Δt/2\tau > \Delta t / 2 is the update fi(x+ciΔt,t+Δt)=fi(x,t)−Δtτ(fi(x,t)−fieq(x,t)).f_i(x + c_i \Delta t, t + \Delta t) = f_i(x, t) - \frac{\Delta t}{\tau} \big( f_i(x, t) - f_i^{eq}(x, t) \big).

Remarks

The update is a collision step, a discrete version of the BGK collision operator, followed by a streaming step to the neighbouring site. The collision conserves ρ\rho and ρu\rho u at each site, since feqf^{eq} has the same zeroth and first moments as ff. A Chapman–Enskog expansion shows that, for small ∣u∣/cs\abs{u}/c_s, the moments approximate the Navier–Stokes equations with kinematic viscosity ν=cs2(τ−Δt/2)\nu = c_s^2 (\tau - \Delta t / 2).

Example: the D2Q9 lattice in two dimensions has Δx=Δt=1\Delta x = \Delta t = 1 and cs2=1/3c_s^2 = 1/3. It uses c0=(0,0)c_0 = (0, 0) with w0=4/9w_0 = 4/9, the four axis vectors (±1,0),(0,±1)(\pm 1, 0), (0, \pm 1) with weight 1/91/9, and the four diagonals (±1,±1)(\pm 1, \pm 1) with weight 1/361/36. Check: the weights sum to 4/9+4/9+4/36=14/9 + 4/9 + 4/36 = 1, and ∑iwici,12=2⋅19+4⋅136=13\sum_i w_i c_{i,1}^2 = 2 \cdot \tfrac19 + 4 \cdot \tfrac1{36} = \tfrac13.

Non-example: the D2Q5 lattice, with only the rest and the four axis velocities, lacks the fourth-order isotropy needed for the Navier–Stokes equations. It is used instead for advection–diffusion problems.

Reading path

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Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods
About this page
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Kinddefinition
WritingW0 stub
MSC76M28, 82C40, 65M75
DefinesLattice Boltzmann method
UsesBGK collision operator, Boltzmann equation, Navier–Stokes equations
Changed3 November 2026

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