Advection-diffusion processes can be simulated by the Lattice Boltzmann method. Two formulations have been proposed in the literature. We show that they are not fully correct (only first order accurate). A new formulation is proposed, which is shown to produce better results, both from the point of view of the Chapman-Enskog expansion or when comparing simulations with an exact time-dependent solution of the advection-diffusion equatio

Chopard, B.

Falcone, J. L.

Latt, J.

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Eur. Phys. J. Special Topics 171, 245–249 (2009)c© EDP Sciences, Springer-Verlag 2009DOI: 10.1140/epjst/e2009-01035-5THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICSRegular ArticleThe lattice Boltzmann advection-diﬀusionmodel revisitedB. Chopard1,a, J.L. Falcone1,b, and J. Latt2,c1 University of Geneva, Switzerland2 Tufts University, USAAbstract. Advection-diﬀusion processes can be simulated by the Lattice Boltz-mann method. Two formulations have been proposed in the literature. We showthat they are not fully correct (only ﬁrst order accurate). A new formulation isproposed, which is shown to produce better results, both from the point of viewof the Chapman-Enskog expansion or when comparing simulations with an exacttime-dependent solution of the advection-diﬀusion equation.1 IntroductionAdvection-diﬀusion describes the diﬀusion of a quantity ρ subject to an external drift u. It canbe expressed by the diﬀerential equation∂tρ+∇(uρ) = D∇2ρ (1)where u(r, t) is a given function of space and time and D the diﬀusion coeﬃcient.Advection-diﬀusion processes are common, for instance, in ﬂuid ﬂow (passive scalar trans-ported by a velocity ﬁeld u(r, t)) or also in case of the diﬀusion of charged particles in presenceof an electric ﬁeld E = u.The lattice Boltzmann (LB) method [1] provides a mesoscopic description of a physicalsystem in terms of density distribution function fi(r, t). The fi(r, t) describe, on a discretespace-time world, the density of ﬂuid particles entering site r, at time t, with velocity vi,i = 0, . . . q − 1. In the so-called DdQq lattice (d-dimensional lattice with q velocities), thevelocities vi are chosen so that, in one time step Δt, a particle can reach one of its q − 1possible nearest neighbors. The velocity v0 = 0 describes particles at rest.The quantity ρ is computed from the fi by the relation ρ =∑q−1i=0 fi.The so-called single-time relaxation LB model readsfi(r +Δtvi, t+Δt) =(1− 1τ)fi +1τfeqi (2)where feqi is a given function of ρ (and possibly higher moments of fi). The quantity τ is arelxation time, which can be adjusted to tune the transport coeﬃcients.By choosing an adequate expression of feqi , the LB approach has been successfully used todescribe many phenomena such as complex ﬂows, reaction-diﬀusion processes or wave equa-tion [2]. Advection-diﬀusion processes have also been addressed within the LB framework.a e-mail: Bastien.Chopard@unige.chb e-mail: Jean-Luc.Falcone@unige.chc e-mail: Jonas.Latt@gmail.com246 The European Physical Journal Special TopicsHowever, two diﬀerent expressions for feqi can be found in the literature. For instance in [3–5],we ﬁndfeqi = tiρ(1 +1c2svi · u)(3)where ti and c2s are some constants such thatq−1∑i=0ti = 1q−1∑i=0tiviαviβ = c2sδαβ (4)where greek indices label spatial coordinates.In other papers (see for instance [6–8]), the following expression is used for feqifeqi = tiρ(1 +1c2svi · u+ 12c4sQiαβuαuβ)(5)where Qiαβ is deﬁned as Qiαβ = viαviβ − c2sδαβ (here we follow Einstein summation conventionover repeated greek indices.)In this paper we show that none of these expressions guarantees that the LB model repro-duces the advection-diﬀusion equation (1) to second order accuracy in Δt, the time step, andΔr, the lattice spacing.The paper is organized as follows. In section 2, we derive the PDE associated with eq. (2)for both expressions (3) and (5). We show the presence of a term absent in eq. (1). We indicatehow it can be eliminated by modifying eq. (2) by adding a body force. In section 3 we validatethe theoretical ﬁndings with a time-dependent advection-diﬀusion problem which can be solvedanalytically and simulated with the LB approach. Some conclusions are drawn in section 4.2 The Lattice Boltzmann advection-diﬀusion model2.1 The Chapman-Enskog solutionLet us now assume that the local equilibrium is choosen asfeqi = tiρ(1 +1c2svi · u+ η 12c4sQiαβuαuβ)(6)where η is a quantity which is either 1 or 0, whether we are interested in expressions (3) or (5)of feqi .Then, a ﬁrst-order multiscale Chapman-Enskog expansion (see [2,9]) applied on eq. (2) givesthe following equation for ρ∂tρ+∇(uρ) = D∇2ρ+ Dc2s∂t∇(uρ) + ηDc2s∂α∂βρuαuβ . (7)Note that the above derivation is second order accurate in Δt and Δr. With respect to eq. (1)the proposed LB approaches contain the following unwanted term ΔΔ =Dc2s[∂t∇(uρ) + η∂α∂βρuαuβ ] . (8)Discrete Simulation of Fluid Dynamics 2472.2 Case of a linear local-equilibriumFor η = 0, feq is linear in u. Eq. (8) reduces to Δ = (D/c2s)∂t∇(uρ). To interprete this term,let us assume that u is constant. By taking the gradient of eq. (7) and neglecting all third orderderivatives, we obtain∂t∇ρ+ u∇2ρ = 0. (9)Thus, the error term can be written as Δ = −(D/c2s)u2∇2ρ and this means that eq. (7) become∂tρ+ u · ∇ρ = D(1− u2c2s)∇2ρ. (10)Therefore, the error terms amounts to a undesired correction to the diﬀusion coeﬃcient, pro-portional to the square of the advection velocity. From eq. (4), cs is parameter of the LB modelwhich has the unit of a velocity, i.e. it scales as c¯s(Δr/Δt), where c¯s is a pure number. Thequantities D and u are assumed to be given by the physical problem. As such they are indepen-dent of the chosen time and space discretization Δr and Δt. Thus, the correction (10) to thediﬀusion coeﬃcient scales as (Δt/Δr)2. Thus, reducing Δr while keeping Δt constant increasesthe error and should be avoided. If we take both the limits Δt→ 0 and Δr → 0, with Δr2/Δtconstant (natural limit for a diﬀusion process), then the correction to the diﬀusion coeﬃcientis O(Δr). The scheme is thus ﬁrst order accurate. But, if we take Δt→ 0, Δr → 0 and Δr/Δtconstant, then the error is O(1) and the scheme is zeroth order accurate.2.3 Case of a non-linear local-equilibriumLet us now consider the case η = 1, corresponding to the non-linear expression of feq in u.Eq. (8) can be written asΔ =Dc2s∂α [∂tρuα + η∂βρuαuβ ] =Dc2s∂α [(uα∂tρ+ ρ∂tuα) + η (uα∂βρuβ + ρuβ∂βuα)]=Dc2s∂α [uα (∂tρ+ η∂βρuβ) + ρ (∂tuα + ηuβ∂βuα)] . (11)Since ρ obeys eq. (7) ∂α[∂tρ+η∂βρuβ ] can be neglected as it amounts to a third order derivative.Therefore, the error term isΔ =Dc2s∂α [ρ(∂tuα + uβ∂βua)] . (12)In many applications of the advection-diﬀusion equation, the advection is due to a ﬂuid ﬂow.In the case u obeys the incompressible Navier-Stokes equation, the term Δ can be furtherdeveloped. We then have(∂tuα + uβ∂βua) = − 1ρf∂αpf + ν∂2βuαwhere ρf and pf are the density and pressure of the advecting ﬂuid and ν its viscosity.We can neglect ν∂2βuα because it will contribute to a third order derivative. ThenΔ = −Dc2s∂α[ρρf∂αpf]. (13)248 The European Physical Journal Special Topics2.4 A proposed correctionIn the case we consider feq linear in u (i.e. eq. (3)), only the linear component is present in theunwanted term Δ. It can be canceled by adding a termFi =vic2sti(1− 12τ)∂t(ρu) (14)on the right hand side of the LB evolution equation (2) as a “body force” (see [9]). This recoverssecond order accuracy in time and space. This body force term can be obtained, to the requiredlevel of accuracy, by the ﬁnite diﬀerence expression∂t(ρu) ≈ ρ(t)u(t)− ρ(t− 1)u(t− 1).When considering the non-linear form (5), the full expression of Δ cannot be canceled througha “body force”, unless u has additional properties. Thus, in general, the approach would be ﬁrstto cancel the non-linear contribution in feq and then apply the “linear” correction discussedabove. However, when the external velocity ﬁeld is solution to the incompressible Navier-Stokesequations, the unwanted term described by eq. (13) can be cancelled in a simple way with thefollowing correction termFi = −vic2sti(1− 12τ)ρρf∇pf . (15)3 Numerical resultsEq. (1) can be solved analytically in the case of a periodic 1D system of length L, where thedriving velocity isu(x, t) = u0 cos(νt). (16)Note that this advection ﬁeld is solution of the Navier-Stokes equation in a periodic domain,when a time-dependent pressure gradient (or body force) ∇p ∝ sin(νt) is applied.If we consider for instance the initial condition ρ(x, 0) = ρ0 + ρ1 cos kx, for x ∈ [0, L] andk = (2πn)/L, then the time-dependent solution of the periodic 1D advection-diﬀusion equationis easily found to beρ = ρ0 + ρ1e−k2Dt[cos kx cos(u0(k/ν) sin νt) + sin kx sin(u0(k/ν) sin νt)]. (17)We can now compare the result of simulations against the analytical solution. After one periodT = 1ν= NtΔt, where Nt is the number of time steps in one period, the error is computedas 2 = (∑t∑x |ρsimulation − ρanalytical|2)/(NxNt) with Nx the number of grid points. Fromeq. (10) we expect that the errors will grow as u0 increases, as shown in ﬁg. 1. We observethat the proposed corrections improve the quality of the simulations. However, when u obeysNavier-Stokes, the non-linear feq performs better than the linear feq. In this case, it is alsofound that the value of c2s aﬀects the accuracy (c2s = 1/3 being the best here) whereas, wheneq. (3) is used, cs can be chosen much more arbitrarily.Fig. 1 also illustrates how accuracy changes when reﬁning the space, with ﬁxed Δt. Asdiscussed above, reﬁning space without reﬁning time eventually degrades the precision.4 ConclusionTwo diﬀerent expressions of feq are found in the literature to model an advection-diﬀusionprocess with the LB method. We have shown that none of them is fully correct. However,adding a simple correction term, acting as a body force, improves the accuracy of the LBadvection-diﬀusion model to second order in Δt and Δr. The two formulations should be usedDiscrete Simulation of Fluid Dynamics 24910−3 10−2 10−1 100 10110−410−2100102u0 in physical unitserrorεLinear BGKLinear with correctionNon−linear BGKNon−linear with correction10110−310−210−1100101Number of lattice sites NerrorεLinear BGKLinear with correctionNon−linear BGKNon−linear with correctionFig. 1. Error when comparing LB simulations with the analytical solution of the advection-diﬀusionequation. Left: the amplitude u0 of the external velocity term is varied, whereas the number of latticesites N = 40 and the time discretization δt = 0.002 are constant. Right: the number of lattice sites isvaried, whereas the velocity amplitude u0 = 8.0 and the time discretization δt = 0.002 are constant.The curves for the corrected linear model and for the uncorrected non-linear model are almost identical.in diﬀerent situations. If the advecting ﬁeld u is solution to the Navier-Stokes eq. then thenon-linear version of feq should be used, with a correction term proportional to ∇pf . If u doesnot obey Navier-Stokes, the linear feq should be preferred, with a correction term proportionalto ∂t(ρu). Our study also shows the importance of correctly taking the limit Δt → 0 andΔr → 0. For instance reﬁning the space while keeping the time discretization constant canactually degrade the quality of the simulation instead of improving it [9].We thank Alexander Wagner for pointing us the role of the u2 term in the local equilibrium distribution,when the advection ﬁeld u is solution of the Navier-Stokes equation. We also thank him for sharingwith us his calculations, which are partly reproduced above. The work presented here was supportedby the Swiss National Science Foundation (FN 200020–1007931, PBGe2–117144) and the Europeanproject COAST.References1. S. Succi, The Lattice Boltzmann Equation, For Fluid Dynamics and Beyond (Oxford UniversityPress, 2001)2. B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems (Cambridge University Press,1998)3. E.G. Flekkoy, Phys. Rev. E 47, 4247 (1993)4. Z.L Guo, B.C. Shi, N.C. Wang, J. Scient. Comp. 14, 291 (1999)5. S. Suga, Int. J. Mod. Phys. C 17, 1563 (2006)6. R.G.M. Van der Sman, M.H. Ernst, J. Comp. Phys. 160, 766 (2000)7. A. Wagner (private communication)8. B. Shi, Lattice Boltzmann simulation of some nonlinear complex equations, Computanional ScienceICCS 2007, vol. 4487 of LNCS (Springer Berlin, Heidelberg, 2007), p. 8189. J. La¨tt, Hydrodynamic Limit of Lattice Boltzmann equations, Ph.D. thesis, University of Geneva,Switzerland, 2007 http://www.unige.ch/cyberdocuments/theses2007/LattJ/meta.html

The lattice Boltzmann advection-diffusion model revisited

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The lattice Boltzmann advection-diffusion model revisited

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