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Equivalent partial differential equations of a lattice Boltzmann scheme
François Dubois
TL;DR
The paper derives the formal macroscopic equations associated with a lattice Boltzmann scheme using equivalent-equation analysis. It obtains Euler equations at first order and compressible Navier–Stokes equations at second order, while noting that a diffusive scaling yields incompressible Navier–Stokes equations.
Problem
The paper examines which macroscopic fluid equations arise formally from a lattice Boltzmann scheme.
Source: 6) Taylor expansion at first order · p. 6, 7) Equivalent equation at second order · p. 8Method
Equivalent-equation expansions are used to derive the scheme’s associated partial differential equations through second order.
Source: 7) Equivalent equation at second order · p. 8, 8) Equivalent equation at second order · p. 10Results
The scheme yields Euler equations at first order and Navier–Stokes equations at second order.
Source: 6) Taylor expansion at first order · p. 6, 7) Equivalent equation at second order · p. 8, 8) Equivalent equation at second order · p. 10Takeaways & Limitations
Under diffusive scaling, the formal limit is the incompressible Navier–Stokes equations; under the paper’s scaling, it is compressible Navier–Stokes.
Source: 8) Equivalent equation at second order · p. 10In the general lattice case, the integer J and nondimensionalized vectors e_j are not precisely specified.
Source: Abstract · p. 2
Abstract
from arXiv · showhide
We show that when we formulate the lattice Boltzmann equation with a small time step $\Delta$t and an associated space scale $\Delta$x, a Taylor expansion joined with the so-called equivalent equation methodology leads to establish macroscopic fluid equations as a formal limit. We recover the Euler equations of gas dynamics at the first order and the compressible Navier-Stokes equations at the second order. 1) Discrete geometry $\bullet$ We denote by d the dimension of space and by L a regular d-dimensional lattice. Such a lattice is composed by a set L 0 of nodes or vertices and a set L 1 of links or edges between two vertices. From a practical point of view, given a vertex x, there exists a set V (x) of neighbouring nodes, including the node x itself. We consider here that the lattice L is parametrized by a space step $\Delta$x > 0. For the fundamental example called D2Q9 (see e.g. Lallemand and Luo, 2000), the set V (x) is given with the help of the family of vectors (e j) 0$\le$j$\le$J defined by J = 8, (1.1) e j = 0 0 , 1 0 , 0 1 , --1 0 , 0 --1 , 1 1 , --1 1 , --1 --1 , 1 --1 and the vicinity (1.2) V (x) = { x + $\Delta$x e j , 0 $\le$ j $\le$ J } .
Abstract
The lattice is represented by neighboring nodes separated by a space step; in the general case, the number and directions of the lattice vectors are left unspecified.
Abstract
- The vicinity V(x) is defined as the set of nodes of the form x + Δx e_j, with index j ranging from 0 through J.Source: Abstract · p. 1
- In the general case, the integer J and the nondimensionalized vectors e_j are not precisely defined.The stated condition is that each x + Δx e_j is another lattice node when x is a lattice node.Source: Abstract · p. 2
2) Lattice Boltzmann framework
The framework relates lattice displacements to particle velocities and represents particle states through distributions and moments, including density and momentum as conservative variables.
2) Lattice Boltzmann framework
- The local velocities v_j are defined so that Δt v_j = Δx e_j, linking each lattice direction to its displacement over one time step.A lattice node shifted by Δt v_j remains a lattice node.Source: 2) Lattice Boltzmann framework · p. 2
- The conservative variables W comprise density ρ and momentum q, and the first d + 1 moments m_i are identical to those variables.The framework also allows conservation of total energy to be included.Source: 2) Lattice Boltzmann framework · pp. 2–3
- The moment vector is constructed from the particle distribution, with the matrix M assumed invertible for the inverse relation.When the matrix of moment coefficients has full rank, the conservative moments W are independent variables.Source: 2) Lattice Boltzmann framework · pp. 2–3
3) Collision step
Collision is a local operation in moment space: conservative components remain fixed, while non-conservative components are relaxed using fixed ratios tied to time-step and relaxation-time scales.
3) Collision step
- The collision step is local in space and is defined in moment space, with conservative moment components unchanged by construction.The conservative variables are density and momentum.Source: 3) Collision step · p. 3
- The numerical collision scheme is exactly the explicit Euler scheme for the continuous-in-time relaxation equation.The proposition identifies the collision update with explicit Euler time discretization.Source: 3) Collision step · p. 3
- For non-conservative moments, the ratios s_k between the time step and relaxation times are fixed, although the scheme does not correctly approximate those physical constants.The relaxation times are assumed fixed in the analysis.Source: 3) Collision step · pp. 3–4
4) Advection step
The advection update streams each post-collision particle population to a neighboring node and is identified with an explicit upwind scheme for the advection equation.
4) Advection step
- The advection step sends particles with velocity v_j from x to the neighboring node x + v_jΔt, carrying the post-collision density there.The arrival-node form rewrites the update using the upstream position x − v_jΔt.Source: 4) Advection step · pp. 4–5
- The advection scheme is the explicit upwind scheme for the particle advection equation.The Courant-Friedrichs-Lewy number is defined as σ_j = 1 by the lattice velocity relation, and the cited passage states that upwind is exact when σ_j equals unity.Source: 4) Advection step · p. 5
5) Equivalent equation at zero order
The section fixes the lattice Boltzmann scheme’s parameter regime and begins deriving its equivalent equations through a Taylor expansion in the time step.
5) Equivalent equation at zero order
- With the lattice geometry, matrix, equilibrium function, and selected ratios fixed, the scheme depends on the single parameter ∆t.The fixed ratios are s_k for nonequilibrium moments with k ≥ d + 1.Source: 5) Equivalent equation at zero order · p. 5
- The zero-order analysis expands the advection relation with respect to ∆t to derive an equation defined from the conservative variables W.The section introduces this as a Taylor expansion at zero order.Source: 5) Equivalent equation at zero order · p. 6
- Using relations (2.14) and (3.1) on one hand, and (2.12) and (3.9) on the other, the authors deduce (5.1) and (5.2) from (5.7) and (5.8).Source: 5) Equivalent equation at zero order · p. 6
6) Taylor expansion at first order
The first-order Taylor analysis yields mass and momentum conservation, identified as the Euler equations of gas dynamics.
6) Taylor expansion at first order
- The derivation expands the advection relation one step further and introduces the second-order moment.The section denotes partial derivatives in time and space by ∂t and ∂β.Source: 6) Taylor expansion at first order · p. 6
- At first order, the lattice Boltzmann scheme gives conservation of mass and momentum, the Euler equations of gas dynamics.The proof takes moments of the first-order expansion, using that the zeroth moment is density and the first moments are momentum components.Source: 6) Taylor expansion at first order · pp. 6–7
- The conservation-defect lemma supplies properties used to establish the first-order conservation result.The proof applies the moment identities and relations involving θk to obtain the mass and momentum equations.Source: 6) Taylor expansion at first order · pp. 7–8
7) Equivalent equation at second order
The second-order expansion establishes Navier–Stokes-type equations, with viscosities linked to the scheme’s relaxation ratios.
7) Equivalent equation at second order
- At second order, the scheme yields expansions identified as the Navier–Stokes equations of gas dynamics.The result is stated for the scheme and conservation defect θk defined in (6.5).Source: 7) Equivalent equation at second order · p. 8
- The scheme approximates a Navier–Stokes-type equation at second-order accuracy, with viscosities µk.The stated viscosity form is tied to the D’Humières relations; the parameters sk are chosen to satisfy them when viscosities are physically imposed and ∆t is constrained by ∆x.Source: 7) Equivalent equation at second order · p. 8
- The proof derives the expansions by taking moments of a second-order Taylor expansion and using microscopic conservation.It takes moments for indices 0 ≤ i ≤ d and divides the resulting relation by ∆t.Source: 7) Equivalent equation at second order · p. 8
- The second-order mass balance is established by showing that the additional term vanishes to second-order accuracy.The proof separately treats momentum conservation to establish the corresponding second-order relation.Source: 7) Equivalent equation at second order · pp. 9–10
8) Equivalent equation at second order
The equivalent-equation approach matches the formal Chapman–Enskog expansion through second order, yielding Euler-type equations at first order and Navier–Stokes-type equations at second order. It requires no a priori two-time multiple scaling, and the chosen scaling determines whether the Navier–Stokes equations are compressible or incompressible.
8) Equivalent equation at second order
- Through second order, the equivalent partial differential equations match the formal Chapman–Enskog expansion.Source: 8) Equivalent equation at second order · p. 10
- The first-order equations are Euler-type, while the second-order equations are Navier–Stokes-type.Source: 8) Equivalent equation at second order · p. 10
- The framework derives the Navier–Stokes equations without a priori formal two-time multiple scaling.Source: 8) Equivalent equation at second order · p. 10
- The scaling Δt/Δx = constant gives the stated compressible result, whereas diffusive scaling Δt/Δx^2 = constant leads to incompressible Navier–Stokes equations.Source: 8) Equivalent equation at second order · p. 10
- Both scaling cases use the Taylor formula for a single infinitesimal parameter.Source: 8) Equivalent equation at second order · p. 10