LBM GenericAdvection Ginzburg 2005

Advances in Water Resources 28 (2005) 1171–1195

www.elsevier.com/locate/advwatres

Equilibrium-type and link-type lattice Boltzmann models forgeneric advection and anisotropic-dispersion equation

Irina Ginzburg *

Cemagref, HBAN, Groupement Antony, Parc de Tourvoie, BP 44, 92163 Antony Cedex, France

Received 10 July 2003; received in revised form 10 March 2005; accepted 29 March 2005

Available online 10 May 2005

Abstract

We extend lattice Boltzmann (LB) methods to advection and anisotropic-dispersion equations (AADE). LB methods are advo-

cated for the exactness of their conservation laws, the handling of different length and time scales for flow/transport problems, their

locality and extreme simplicity. Their extension to anisotropic collision operators (L-model) and anisotropic equilibrium distribu-

tions (E-model) allows to apply them to generic diffusion forms. The AADE in a conventional form can be solved by the L-model.

Based on a link-type collision operator, the L-model specifies the coefficients of the symmetric diffusion tensor as linear combination

of its eigenvalue functions. For any type of collision operator, the E-model constructs the coefficients of the transformed diffusion

tensors from linear combinations of the relevant equilibrium projections. The model is able to eliminate the second order tensor of

its numerical diffusion. Both models rely on mass conserving equilibrium functions and may enhance the accuracy and stability of

the isotropic convection–diffusion LB models.

The link basis is introduced as an alternative to a polynomial collision basis. They coincide for one particular eigenvalue con-

figuration, the two-relaxation-time (TRT) collision operator, suitable for both mass and momentum conservation laws. TRT oper-

ator is equivalent to the BGK collision in simplicity but the additional collision freedom relates it to multiple-relaxation-times

(MRT) models. ‘‘Optimal convection’’ and ‘‘optimal diffusion’’ eigenvalue solutions for the TRT E-model allow to remove next

order corrections to AADE. Numerical results confirm the Chapman–Enskog and dispersion analysis.

� 2005 Elsevier Ltd. All rights reserved.

PACS: 76.Rxx; Diffusion and convection

Keywords: Lattice Boltzmann equation; Advection and anisotropic-dispersion equation; Chapman–Enskog expansion; Multiple-relaxation-times

models; BGK model; Numerical diffusion

1. Introduction

Algorithmic advances in numerical methods over thelast decade are significant for important water research

problems (see [22]). This success is mainly due to

increasing efficiency of linear and non-linear solvers

for global systems of equations arising from the discret-

0309-1708/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2005.03.004

* Tel.: +33 140966060.

E-mail address: [email protected]

ization of partial differential equations. The solution

procedure is often advantageous when adapted to a spe-

cific formulation (choice of primary variables) of a givenproblem (e.g., [2]). The coupling of the discretization/

solution methods designed for problems of different

length and time scales (e.g., overland and sub-surface

flow) becomes a difficult task. Intended for solving the

Navier–Stokes equation, the lattice Boltzmann equation

(LBE) was derived by Higuera and Jimenez [16] from

Lattice Gas Automata models of Frisch et al. [7]. Ex-

tended to other problem classes, lattice Boltzmann

mailto:[email protected]

1172 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195

(LB) methods belong to the family of mesoscopic meth-

ods. Each conservation law is related to a microscopic

quantity which is conserved exactly by the collision

operator of the evolution equation. The evolution equa-

tion describes the dynamics of distribution functions

moving with discretized velocities between the nodes ofthe computational grid. The objective of this paper is

to extend the LB methods to advection and aniso-

tropic-dispersion (AADE) equations in a consistent

and generic way. Besides a versatility for coupling flow

and transport problems within an uniform numerical

scheme and its potential flexibility for multi-scale prob-

lems, we recall briefly the computational merits of the

LB methods.LBE methods do not involve any global linear/non-

linear systems of equations. This point is very attractive

for practitioners and novices, because of the simplicity,

and for parallel computation, because of the locality.

As a consequence of the locality of the main operations,

computational efforts per evolution step increase only

linearly with space resolution, giving an opportunity

for realistic three dimensional computations. The meth-od is explicit in time but its time step is relatively inex-

pensive. Designed for regular grids, the method may

accurately fit the boundary conditions on complex

shaped boundaries by a careful computation of the

incoming distribution functions. Stability criteria are re-

lated to equilibrium and collision parameters and may

be enhanced due to some freedom with respect to the

selection of the kinetic components of the method (see[5,20]). In the very first LBE models, the collision oper-

ator is computed as a product of the collision matrix and

a vector which represents the difference between the

local state of the populations and their equilibrium state.

The generalized lattice Boltzmann equation (GLBE)

method of d�Humieres [3] computes the propagation of

the populations in velocity space and their collision in

the moment space spanned by the eigenvectors basisof the collision operator. The GLBE method is numeri-

cally more efficient than the LBE method.

The d-dimensional (d = 2/3) LBE models for convec-

tion–diffusion [6,15,27,29] are constructed in a similar

way to the hydrodynamic models: they are based on a

hydrodynamic-type isotropic equilibrium function but

discard momentum conservation. A similarity of equi-

librium functions enables the moment propagation meth-ods (e.g., [30,21]) to build the tracer transport directly

with the population solutions obtained for flow equa-

tion. In LBE models, the solvability condition of the

evolution equation corresponds to the exact mass con-

servation law. The convection–diffusion equation repre-

sents its second order approximation. Diffusive flux can

be derived locally from the population solution. The

diagonal components of the diffusion tensor arematched by eigenvalues corresponding to the d compo-

nents of the lattice velocity vectors. The eigenvalues

can be redefined locally as a function of the solution,

its gradients, and/or the advection velocity. The vast

majority of such models are based on the BGK operator

[24] resulting in a single diffusion coefficient. Provided

that the equilibrium function contains the conserved

quantity, the BGK collision operator has the same formin both velocity and moment space. A review of recent

applications can be found in [32], where a BGK-type

model for AADE is proposed. Based on the ideas in

[19], the models [32,33] keep the form of the BGK colli-

sion operator but use a specific relaxation parameter for

each pair of populations with opposite velocities. When

relaxation parameters differ, the BGK-type construction

cannot have a mass conserving equilibrium function.Solvability conditions of the evolution equation are then

violated and as a consequence, second order approxima-

tion to the mass conservation property is derived for the

sum of the equilibrium components which differs from

the local mass quantity in the system.

Let us now formulate the main steps of this paper:

• Throughout this paper, ‘‘link’’ means a pair of oppo-site lattice velocities. A link-wise collision operator is

introduced as an alternative to the polynomial-based,

multiple-relaxation-time (MRT) operators [3–5,8,20,

28]. Its basis vectors are easily derived for any sym-

metric velocity distribution. The projections are the

symmetric and the anti-symmetric parts of a pair of

populations with opposite velocities. Link-operators

and MRT collision coincide for one particular configu-ration of eigenvalues associated to symmetric and

anti-symmetric basis vectors. This configuration, suit-

able for both mass and momentum conservation

equations, is called the two-relaxation-time (TRT)

operator. The TRT collision equals the BGK colli-

sion in terms of computational time and simplicity,

but it may benefit from additional collision free-

dom to improve stability, higher order accuracy orprecision of boundary conditions, like the MRT

operators.

• Mass conserving, advection–diffusion equilibrium

distributions require to build the diffusive flux in

terms of the gradient of a specific equilibrium func-

tion rather than as the gradient of the conserved

quantity itself. In particular, the equilibrium function

given in [33] for transport in variably saturated por-ous media and the modified moment propagation

scheme [21] are covered by this extension.

• Based on a link-wise collision operator and an

advection–diffusion equilibrium function, the link-

type (L-) model for AADE is designed as an

improvement and an extension of the BGK-type

models [32,33]. It matches the diffusion coefficients

of the full symmetric tensor as a linear combinationof the eigenvalue functions associated with anti-sym-

metric basis vectors.

Table 2

Number of basis vectors in different models

D2Q5, D3Q7 D2Q9 D3Q13 D3Q15 D3Q19

e0 1 1 1 1 1

{Ca} 2, 3 2 3 3 3

p(e) 1 1 1 1 1

p(xx) 1 1 1 1 1

p(ww) 0, 1 0 1 1 1

{p(ab)} 0, 0 1 3 3 3

fhð1Þa g 0 2 0 3 3

fhð2Þa g 0 0 3 0 3

h(xyz) 0 0 0 1 0

p(e) 0 1 0 1 1

p(xx) 0 0 0 0 1

p(ww) 0 0 0 0 1

Table 3

List of Greek symbols

e Small

parameter

m�q ¼ mðk�q Þ Eq. (38)

mD = m(kD) After Eq. (25)

K(k) Eq. (16) ma = m(ka) Eq. (23)

K2(kD,ke) Eq. (B.2) mð~kÞ Eq. (46)

kk Eq. (6) mth.ð~kÞ; mlbð~kÞ Eqs. (B.1) and (47)

ka Ca-eigen

values

mðrÞn ð~kÞ Eq. (B.1)

k�q Eq. (9) mh, ml., mnl. Eq. (51)

ke, kD Eq. (10) P(0) Eq. (6)

kopte ðkDÞ Eq. (B.4) P, P�, P+, P2, P3, P4 Eq. (A.2)

k0D Eq. (B.6) Peq.;Pn

a Before Eq. (C.3)

kise ðkD; c2s Þ Eq. (B.7) {e,p(xx), p(ww)} 2 P4 Eq. (A.1)

kðuÞe Eq. (B.15) pnak Before Eq. (C.3)

la Eq. (1) r 0ab, r Eq. (54)

mdab Isotropic

tensor

X0;Xna Before Eq. (C.3)

m(k) Eq. (23) xð~kÞ Eq. (46)

Table 1

Equilibrium weights tHp , p ¼ k~Cqk2

D2Q5 D3Q7 D2Q9 D3Q13 D3Q15 D3Q19

tH11

2

1

2

1

3

1

3

1

6

tH21

12

1

8

1

12

tH31

24

I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1173

• Equilibrium functions are then complemented by an

additional part (E-model). It is built utilizing the

Chapman–Enskog analysis performed in both velo-

city and moment spaces. For any type of collision

operator, the E-model represents the coefficients of

the transformed diffusion tensors by linear combina-

tions of extended equilibrium projections. The addi-

tional degree of freedom may enhance the stabilityconditions at high Peclet numbers. In this paper,

the model works with the MRT, TRT and BGK

collisions. A combination of the link-type collision

and expanded equilibrium functions further extends

both E- and L- methods but is not discussed in this

paper.

• The second order tensor of the numerical diffusion,

caused by the linear advection term at equilibrium,is derived. The E-model can match a tensor of anti-

numerical diffusion as a complement to the principal

diffusion tensor. No suitable solution was found for

the L-model in the case of general velocity field and

full anisotropic tensors.

• Based on the eigenmode analysis of the dispersion

equation the form of the higher order corrections to

linear AADE is found in the case of the TRT E-model. The ‘‘optimal convection’’ and ‘‘optimal diffu-

sion’’ equilibrium/eigenvalues configurations enable

the TRT E-model to remove them in convection-

dominant and diffusion-dominant regimes, respec-

tively.

• The numerical study of isotropic and anisotropic

linear advection-dispersion problems is in complete

agreement with the Chapman–Enskog and dispersionanalysis.

• The numerical and analytical analysis of a quasi-lin-

ear diffusion equation illustrates the difference

between the equilibrium and eigenvalue approaches

in the case of solution-dependent diffusion coeffi-

cients. An exact particular time-dependent population

solution is constructed for the equilibrium approach.

• Dirichlet and Neumann (specified flux) boundaryconditions for AADE models are designed in a sepa-

rate paper [11].

The rest of the paper is organized as follows. In Sec-

tion 2 we outline the forms of the AADE equations as

represented by E- and L-models. The physical space is

discretized by a uniform rectangular grid which is trans-

formed into the cubic computational grid after aniso-

tropic rescaling of the AADE. In Section 3 we give the

framework of the GLBE method, introduce link basis

vectors, classify MRT and Link-based collision opera-

tors and derive the form of the generic conservation

law. The E- and L-models are described in Sections 4and 5, respectively. A short overview of them in Section

6, complemented with an outlook of the ‘‘optimal’’

eigenvalue solutions in Section 7.4, is designed as a list

of ‘‘numerical recipes’’. Numerical results are found in

Section 7. Polynomial basis vectors of the D2Q5,

D3Q7, D2Q9, D3Q13, D3Q15 and D3Q19 MRT models

are specified in Appendix A. Eigenmode solutions of dis-

persion analysis of d�Humieres are described in Appen-dix B. Details of the Chapman–Enskog expansion are

sketched in Appendix C. Tables 3 and 4 indicate

definitions of main symbols and variables used in the

paper.

t a ab b c

Table 4

List of symbols

A Eq. (5) f ±, f eq.±, f ne.± Eq. (9)

a(e), a(ab), a(xx), a(ww) Eq. (19) f (0) = f eq., f (1), f (2) Eq. (11)

a(e), a(ab), a(xx), a(ww) Eq. (45) f is., f as. Eq. (19)

b1, b2, b3, b6 Eq. (A.3) h Eq. (2)

bðeÞ; bðabÞ; b5; bðxxÞ; bðxxÞD ; bðxxÞa Eq. (A.4) fhð1Þa ; hð2Þa ; hðxyzÞg 2 P3 Eq. (A.1)

C Any non-zero const ~J Eqs. (1,2,17)

C[d · Q] Eq. (36) Kab Eq. (2)~Cq, ~C�q ¼ �~Cq velocity vectors K(Pi) After Eq. (6)

Ca, a = 1, . . . ,d Eq. (A.1) ~k Eq. (46), Eq. (B.1)

c2s Eq. (19) kT, kL Eq. (56)

c2q Eq. (A.1) L ¼ L0=L; La After Eq. (2)

Dab Eq. (3) {p(e), p(xx), p(ww), p(ab)} 2 P2 Eq. (A.1)

�Deq.ðsÞ, �D Eq. (19), Eq. (25) p ¼ k~Cqk2 Eq. (19)

Dab; Duab Eq. (23), Eq. (30) Q Number of velocities

Dab; Duab Eq. (38) Qm

q Eq. (5)~D Eq. (36) Ra Eq. (B.5)

Dmin, Dmax Eq. (41) Sm Eq. (1)

d Dimension Smn , Sm

0 Eqs. (14) and (23)~d Eq. (57) s Eqs. (1,2,18)

Eu Eq. (31) s0 Eq. (46)

E(r)(mlb) Eq. (48) s1; s01 Eq. (39), Eq. (40)

EðrÞð~U lbÞ Eq. (49) T ¼ T 0=T , t 0 After Eq. (2)

E2L;EL Eq. (55) T(1), Tr(1) Eq. (12)

eq Basis vectors T(2), Tt(2), Tr(2) Eq. (13)

e0 Mass vector, Eq. (A.1) Trt(2), Trr(2) Eq. (C.4)

en Any conserved vector tq; tHp Eq. (19)

eþq , e�q Link basis, before (9) tðeÞq ; tðabÞ

q ; tðxxÞq ; tðwwÞq Eq. (20)

F rð1Þn ; F tð2Þ

n ; F rð2Þn Eq. (15), Eq. (16) ~tq Eq. (60)

f, f eq. Eq. (5) ~U ~J=s~f q;

bfk ; bfkeq.

Eq. (6) u(e), u(ab), u(xx), u(ww) Eq. (31)

f ne. Eq. (7)


2. Generic equations

Greek indices stand for the spatial coordinates x, y,

and so on; the repeated Greek indices correspond to

implicit summations. We develop E-model to compute

approximate solutions for the generic d-dimensional

equation

otsþr ~J ¼ oalaobKab þ Sm; ð1Þand L-model for the AADE in the form

otsþr ~J ¼ oaKabobhþ Sm. ð2ÞHere, sð~r; tÞ is a conserved scalar variable, Sm is a scalar

sink term, ~J and ~l are vector functions, h is a scalar

function, Kab and Kab are components of symmetric

tensors. Any of these functions may depend on sð~r; tÞand any local function, e.g. external advection vector~U . Lattice Boltzmann AADE equations are designedon the cuboid computational grid ð~r; tÞ with space and

time steps equal to one. L ¼ L0=L and T ¼ T 0=T are ra-

tios of the characteristic values for length and time vari-

ables between the physical and the computational grid.

In physical variables, t0 ¼Tt and ~x ¼ L ~r; L ¼LdiagðLx; Ly ; LzÞ, where Lx, Ly, Lz define the relative

scaling factors for every direction. The transformation

from the computational cuboid to orthorhombic discreti-

zation grid is handled by the anisotropic rescaling of the

advection and the diffusion terms:~J ! T�1L ~J ; Kab !T�1L Kab L; Kab ! T�1L Kab L.

We assume below that, rescaled from the physical to

the computational grid, AADE is presented in the form

(1) or (2). In particular, the convection–diffusion equa-tion with constant diffusion tensor Dab,

otsþr ~Us ¼ oaDabobsþ Sm; ð3Þis a particular case of Eqs. (1) and (2), with ~J ¼~Us; laKab ¼ Dabs in the former case and with

hðsÞ ¼ Cs; Kab ¼ Dab=C, in the latter one. Here and be-

low, C means arbitrary non-zero constant. Unlike for

the conventional diffusion LBE models, the diffusive flux

is captured in terms of the gradient of some function,tensorial KabðsÞ or scalar h(s), rather than in terms of

the gradient of the conserved quantity itself. Let us illus-

trate this property using the transport equation [18] in a

variably saturated porous media during transient water

flow:

o ðhcÞ þ r ~Uc ¼ o D ð~UÞo cþ q . ð4Þ


Here, hð~r; tÞ is given water content distribution, cð~r; tÞ isan unknown solute concentration, qc represents a sink

term, Dabð~UÞ is the dispersion coefficient tensor. The

conserved quantity s thus corresponds to hc and ~J to~Uc. The diffusion part is met naturally by the L-model:

hðsÞ ¼ Cs; Kab ¼ Dab=C. The E-model requires an inte-gral transformation when Dabð~UÞ varies in space. For in-

stance, Kab ¼ 1la

R cCDab dc0 when la are all set equal to

some constant. Another example is Richards� equation(e.g, in [2,12,22]) where the conserved quantity is the

moisture content variable and the diffusive flux is ex-

pressed in terms of the pressure head gradient variable

by Darcy law; retention curves relate both variables be-

tween them and the permeability tensor may assume anydegree of anisotropy. The general forms of Eqs. (1) and

(2) are also helpful when an integral transform may

regularize the diffusion term. Their application to iso-

tropic Richards� equation with and without Kirchoff

transform in [10] may serve as an example.

9Þ

3. Chapman–Enskog expansion revisited

3.1. GLBE model

Let us consider a lattice Boltzmann model defined byQ velocities ~Cq on a cubic lattice in d dimensions. The

velocity set is chosen such that it has the same symmetry

group as the cubic lattice; in particular it is invariant un-

der the central symmetry, i.e. if ~Cq is an element of the

set, ~C�q ¼ �~Cq is also an element, and the set is invariant

by any exchange of coordinates. We restrict ourselves to

the models with C3qa ¼ Cqa, a = 1, . . . ,d, and ~C0 ¼~0. In

what follows, the vectors in d-dimensional (physical)space carry ‘‘arrows’’, e.g. ~Cq ¼ fCqa; a ¼ 1; . . . ; dg;q ¼ 0; . . . ;Q� 1. The vectors in population space are

‘‘bold’’, e.g. Ca = {Cqa, q = 0, . . . ,Q � 1}, a = 1, . . . ,d.The LBE models obey the following evolution equation

with source term Qmq ð~r; tÞ:

fqð~r þ ~Cq; t þ 1Þ � fqð~r; tÞ ¼ A ðf � f eq.Þ½ �q þ Qmq ð~r; tÞ.

ð5Þ

The collision matrix Að~r; tÞ is defined by the set P of

its Q eigenvectors, P = {ek}, and corresponding Q

eigenvalues fkkð~r; tÞg. The eigenvalues are restricted to

the interval [�2,0] in order to keep the magnitude of

the eigenvalues of the evolution operator (I + A) less

than 1 [17,31]. The decomposition of the populations fwith respect to the basis vectors is given by f ¼PQ�1

k¼0bfkek. We define the projection w ¼ hwjei of some

vector w on any vector e as hwjei ¼ w ekek2. The coefficients

kekk2 bfk ¼ f ek are usually called ‘‘moments’’. The

GLBE model computes Eq. (5) as:

fqð~r þ ~Cq; t þ 1Þ ¼ ~f qð~r; tÞ;~f qð~r; tÞ ¼ fqð~r; tÞ þ

Xk2KðPð0ÞÞ

kkð bfk � bfkeq.Þekq þ Qm

q .ð6Þ

Here and below, if P is some subset of the eigenvectors,

K(P) denotes the set of their numbers. The summationis restricted to P(0) : P(0) = {ek : kk 5 0}.

The exact form of equilibrium distribution f eq. will

be addressed later but for now we assume that for any

conserved vector en

en ðA f ne.Þ ¼ 0; f ne. ¼ f � f eq.; ð7Þthe equilibrium distribution contains the whole projec-

tion on the en:bfn ¼ hf jeni ¼ hf eq.jeni. ð8ÞWe call bfn a conserved quantity. Note that en does not

have to be one of the eigenvectors of A but for any basis

vector en 62 P(0), the property (7) is met automatically.

We distinguish the following families of basis vectors:

• The MRT basis vectors are orthogonal polynomials

of the velocity components and their projections gen-erally have some physical significance (mass, momen-

tum, etc.). In Appendix A, the polynomial

eigenvectors for the most commonly used hydrody-

namic multiple-relaxation-times (MRT) d-dimen-

sional models with Q velocities are given (DdQq

models, see in [3–5,8,20,28]). The derivation of the

E-model below is valid for all of them. The vectors

are numbered in such a way that the first vector isthe mass vector e0 = 1 and the next d vectors are

the velocity vectors Ca, associated with the eigen-

values ka. When the equilibrium function satisfies

the properties (7) and (8), the eigenvalues associated

with conserved vectors are not necessarily zero and

their exact values are not relevant. The eigenvectors

are divided into symmetric, even-order polynomials

and anti-symmetric, odd-order polynomials (sets P+

and P� in rel. (A.2), respectively). Their eigenvalues

are referred to as ‘‘even’’ and ‘‘odd’’, respectively.

• The BGK basis vectors, eq = dq, have only one non-

zero component: hf jeqi = fq, i.e. the projections coin-

cide with the populations themselves. Mass or

momentum conservative properties (7) and (8) require

all eigenvalues to be equal to each other.

• The Link (L-) basis vectors are constructed for eachpair of opposite velocities ~Cq and ~C�q (called link

throughout the paper): ‘‘even’’ vectors, eþq ¼ dq þ d�q,

q = 0, . . . , (Q � 1)/2, and ‘‘odd’’ vectors, e�q ¼ dq�d�q, q = 1, . . . , (Q � 1)/2. The collision operator is

naturally divided into its symmetric/anti-symmetric

parts:

ðA f ne.Þq ¼ kþq ðf þq � f eq.þq Þ þ k�q ðf �q � f eq.�

q Þ;f �q ¼ ðfq � f�qÞ=2. ð


Even/odd population parts, f �q , take equal (opposite)

values for a pair of populations. Similar relations

take place for equilibrium parts, f eq.�q ¼

ðf eq.q � f eq.

�q Þ=2, and for non-equilibrium parts:

f ne. ± = f ± � f eq.±. When the equilibrium function

satisfies condition (8) for a symmetric conserved vec-tor (e.g., mass vector), the eigenvalues kþq should be

equal to each other. When no anti-symmetric vector

needs to be conserved, each link may keep its own

value k�q . As discussed in Section 5, this enables the

L-model to specify the full diffusion tensor. When

some anti-symmetric vector should be conserved

(e.g, Ca), the eigenvalues k�q should be equal to each

other.• The MRT-L basis combines even MRT-eigenvectors

and odd L-eigenvectors.

The link collision reduces to a two-relaxation-time

(TRT) collision when both mass and momentum conser-

vation laws take place:

ðA f ne.Þq ¼ keðf þq � f eq.þq Þ þ kDðf �q � f eq.�

q Þ;

kþq ¼ ke; k�q ¼ kD. ð10Þ

In the case of the hydrodynamic momentum conser-

vation equation, ke is fixed by the kinematic viscosity

but kD is free. In the AADE case, ke is free and kD is re-lated to the diffusion coefficient. We emphasize that the

computational efforts for the BGK and L-collisions are

equal since even/odd collision counterparts need to be

computed only once for every pair of opposite velocities.

The MRT and MRT-L collisions reduce to the TRT

operator when their even/odd eigenvalues take specific

values, ke/kD, respectively. The TRT collision reduces

to the BGK operator when ke = kD.

3.2. Chapman–Enskog expansion

The macroscopic behavior of Eq. (6) can be obtained

by a standard Chapman–Enskog expansion [7] for a typ-

ical perturbation length e�1, where e is a small parame-

ter. Expanding the distribution f around f (0) = f eq.

results in

f ¼ f ð0Þ þ f ð1Þ þ f ð2Þ þ ; hf ð1Þjeni ¼ 0;

hf ð2Þjeni ¼ 0; . . . ; ð11Þ

where first and second order corrections to equilibrium,

f (1) and f (2), are related to first and second order Taylor

expansions of the evolution equation, eT (1) and e2T (2),

respectively: f (1) = A�1 Æ eT (1) and f (2) = A�1 Æ e2T (2).Here, A�1 is the inverse of A when P(0) = P, otherwise

A�1 is the pseudo-inverse of A. Introducing oa ¼ eoa0

and two time scales, ot ¼ eot1 þ e2ot2 , first order Taylor

expansion is

Tð1Þ ¼ ot1 fð0Þ þ Trð1Þ; Trð1Þ ¼ Cbob0 f

ð0Þ; ð12Þ

The component by component product is assumed

for two vectors of the population space (Cb and

ob0fð0Þ). Following [3], we substitute rel. (12) in to the

second order Taylor expansion. Bearing in mind that

A may vary, we obtain:

e2Tð2Þ ¼ e2ot2fð0Þ þ eT tð2Þ þ eTrð2Þ �Qm;

T tð2Þ ¼ ot1 Iþ 1

2A

� � f ð1Þ; Trð2Þ ¼ Caoa0 Iþ 1

2A

� � f ð1Þ.

ð13Þ

Solvability conditions in rel. (11) are satisfied when ehT(1)jeni = 0, e2hT(2)jeni = 0,. . ., for any conserved vector

en. Note that when condition (8) is satisfied, the eigen-value kn does not influence the derived conservation

relation. Because of rel. (11), Tt(2) Æ en = 0. First and sec-

ond order macroscopic relations come as the approxi-

mations of solvability conditions:

eðot1 fð0Þ þ Trð1ÞÞ en ¼ 0;

e2ðot2 fð0Þ þ Trð2ÞÞ en ¼ Sm

n ; Smn ¼ Qm en.

ð14Þ

Let us recall the notations from Appendix C: pnak de-

notes the projection of the column vector Caek on the

arbitrary vector en : pnak ¼ hCaekjeni; Peq. is a set of vec-

tors on which the equilibrium function is projected.

For a given vector en, Pna ¼ fek : pn

ak 6¼ 0g, and Xna is its

restriction to equilibrium basis vectors, Xna ¼ Pn

a \Peq..Assuming that en is a basis vector one can write the first

order generic macroscopic Eq. (14) with rel. (12) in the

form:

eðkenk2ot1bfnð0Þþ F rð1Þ

n Þ ¼ 0;

F rð1Þn ¼ Trð1Þ en ¼ kenk2

Xk2KðXn

bÞpn

bkob0bfkð0Þ;

ð15Þ

and the second order generic equation (14) with the rel.

(13) as

e2ðkenk2ot2bfnð0ÞþF tð2Þ

n þF rð2Þn Þ¼Sm

n ; KðkÞ¼� 1

2þ1

k

� �;

F tð2Þn ðf ð0ÞÞ¼�kenk2

Xk2KðXn

a\Pð0ÞÞ

pnakoa0KðkkÞot1

bfkð0Þ;

F rð2Þn ðf ð0ÞÞ¼�kenk2

Xj2KðPn

a\Pð0ÞÞ

pnaj

Xk2KðXj

bÞ

pjbkoa0KðkjÞob0

bfkð0Þ

.

ð16Þ

The details to obtain Eqs. (15) and (16) are given by

relations (C.1), (C.3)–(C.5). They also include the situa-

tion where en is not a basis vector. Written in the form of

the projection on the basis vectors, Eq. (16) enable us to


specify relevant equilibrium projections for a selected

conservation law.

4. The equilibrium-type model (E-model)

Relations (15) and (16) show that in case of one con-

served quantity, the terms F rð1Þn and F tð2Þ

n involve only the

first moments of the equilibrium distribution which are

not orthogonal to en. In particular, in case of a mass

conservation relation, en = e0, the terms F rð1Þ0 and F tð2Þ

0

depend only on the equilibrium projections on velocity

vectors. The diffusion part F rð2Þn is built from second or-

der moments of the equilibrium function, which are alsonot orthogonal to the conserved vector. When en = e0, anon-zero contribution to F rð2Þ

0 can only origin from the

equilibrium projections on e0 and second order polyno-

mial basis vectors from the subset P2, P2 = {p(e),p(ab),p(xx),p(ww)} (see Appendix A). The idea of the E-

model is to construct the equilibrium function in the

form of a free projection on all relevant basis vectors

and to fit the projections to the coefficients of the tensorKab in Eq. (1).

4.1. The equilibrium function

The equilibrium function f eq. is divided into its sym-

metric, f eq.+, and anti-symmetric, f eq.�, parts. The anti-

symmetric (odd) part should ensure that the velocity

moment of the equilibrium function is equal to the pre-scribed vector ~J :

J a ¼XQ�1

q¼1

f eq.�q Cqa;

XQ�1

q¼1

f eq.þq Cqa ¼ 0; a ¼ 1; . . . ; d.

ð17Þ

The symmetric part is further divided into the ‘‘isotro-

pic’’ part, f is., and the ‘‘anisotropic’’ part, f as.. The iso-

tropic part is projected on the ‘‘isotropic’’ vectors only:

mass vector e0, energy vector p(e) and, related to the en-ergy square, the vector p(e). These vectors have one spe-

cific value for all velocities with the same magnitude.

The anisotropic part is projected on the vectors from

the subset P2. Since they are orthogonal to the mass vec-

tor e0, the anisotropic equilibrium part has no mass,PQ�1

q¼0 fas.q ¼ 0. The isotropic part contains the sum s,

or mass, of the population solution:

s ¼XQ�1

q¼0

fq ¼XQ�1

q¼0

f eq.þq ¼

XQ�1

q¼0

f is.q ;

XQ�1

q¼0

f eq.�q ¼ 0;

XQ�1

q¼0

f as.q ¼ 0.

ð18Þ

Assuming an arbitrary function �Deq.ðsÞ, we write the

expanded equilibrium function (E-model) as

f eq.q ¼ f eq.þ

q þ f eq.�q ;

f eq.�q ¼ tHp JbCqb;

XQ�1

q¼1

tHp CqaCqb ¼ dab; 8a;b; p ¼ k~Cqk2;

f eq.þq ¼ f is.

q þ f as.q ;

f is.q ¼ tq �D

eq.ðsÞ; tq ¼ c2s t

H

p ; q 6¼ 0; f is.0 ¼ s�

XQ�1

q¼1

f is.q ;

f as.q ¼ c2

s ðaðeÞtðeÞq þ aðabÞtðabÞq þ aðxxÞtðxxÞq þ aðwwÞtðwwÞ

q Þ.ð19Þ

Here, c2s is a free constant, left to keep the historical

notation. The restriction on the weights tHp allows to

match property (17). Their corresponding values can

be found in Table 1. The anti-symmetric equilibrium

part does not contain the mass,PQ�1

q¼0 feq.�q ¼ 0, due to

the anti-symmetry of the velocity components and thesymmetry of the equilibrium weights tHp . Equilibrium

weights tðeÞq ; tðabÞq ; tðxxÞq ; tðwwÞ

q are equal to the components

of the vectors from the subset P2, divided by the con-

stants b(e), b(ab), b(xx) and b(ww), respectively:

tðeÞq ¼pðeÞq

bðeÞ; tðabÞ

q ¼pðabÞ

q

bðabÞ ; tðxxÞq ¼pðxxÞq

bðxxÞ; tðwwÞ

q ¼pðwwÞ

q

bðwwÞ .

ð20ÞThe constants are given by the relations (A.4). Any pro-

jection on fourth order polynomials from the subset P4

may be added to f eq.+ since it does not contribute to thesecond order mass conservation equation. When f as. = 0

we refer to rel. (19) as the advection–dispersion equilib-

rium function. Otherwise, we call it expanded equilibrium

function or E-model. Conventional functions for the iso-

tropic convection–diffusion equation correspond to rel.

(19) with f as. = 0, �Deq.ðsÞ ¼ s; ~J ¼ s~U and, sometimes,

tq = 1/Q, "q.

4.2. First order equation

The first order conservation equation (15) for equilib-

rium distribution (19) is

eðot1sþr0 ~JÞ ¼ 0. ð21Þ

4.3. Second order equation

4.3.1. Pure diffusion

When ~J ¼ 0, f eq.� = 0, then the term F tð2Þ0 in Eq. (16)

vanishes. Provided that Ca are not conserved vectors

and ka are their non-zero eigenvalues, the diffusion term

is obtained from �F rð2Þ0 ðf eq.Þ in Eq. (16)

�F rð2Þ0 ðf eq.Þ ¼ ke0k2

Xj2KðP0

aÞ

p0ajoa0

�X

k2KðXjbÞ

pjbkKðkjÞob0 hf eq.þjeki. ð22Þ


Here, the subset P0a ¼ fej ¼ Cag so that the projection

of their velocity moment on the mass vector is

p0aj ¼

kCak2

ke0k2. Projections pj

bk of the velocity moments Cbekon the vectors ej become hCbekjCai. Substitution of the

projections into rel. (22) and its coupling with Eq. (21)

when ~J ¼ 0 yields a diffusion equation in the form:

ots ¼ oamaobDab þ Sm0 ; ma ¼ c2

sKðkaÞ;

Dab ¼1

c2s

XQ�1

q¼1

CqaCqbf eq.þq .

ð23Þ

Hereafter, mðkÞ ¼ c2sKðkÞ is referred to as a diffusion

combination of the eigenvalue k. Source term Qmq is

usually set equal to tqSm0 ; t0 ¼ 1�

PQ�1

q¼1 tq; Sm0 ¼PQ�1

q¼0 Qmq . We emphasize that the components Dab do

not depend on the equilibrium solution for the rest par-

ticle. Because of the tHp -weights property, the isotropic

equilibrium part f is. yields the isotropic contribution,�D

eq.ðsÞdab. The other part comes from f as. (cf. (19),

(20) and (A.4)):

Dab ¼ aðabÞ; a 6¼ b;

Dxx ¼ �Deq.ðsÞ þ aðxxÞ þ aðeÞ;

Dyy ¼ �Deq.ðsÞ þ bðxxÞD aðxxÞ þ aðwwÞ þ aðeÞ;

Dzz ¼ �Deq.ðsÞ þ bðxxÞD aðxxÞ � aðwwÞ þ aðeÞ.

ð24Þ

The solution for the coefficients is

aðabÞ ¼ Dab; a 6¼ b; aðeÞ ¼ �D� �Deq.ðsÞ; �D ¼

PaDaa

d;

aðxxÞ ¼ bðxxÞa Dxx �1

d � 1ðDyy þDzzÞ

� �; bðxxÞa ¼ 1

1� bðxxÞD

;

aðwwÞ ¼ 1

2ðDyy �DzzÞ. ð25Þ

Relations (A.4) yield: bðxxÞD ¼ � 12; bðxxÞa ¼ 2

3for D3Q13,

D3Q15 and D3Q19. For the D2Q9 model, rel. (24) and

(25) can be used assuming a(ww) = 0, Dzz ¼ 0; bðxxÞD ¼�1; bðxxÞa ¼ 1

2. Let us recall that d is a space dimension,

so that �D is an arithmetical mean value of the diagonalcoefficients Daa. The off-diagonal elements Dab, a 5 b,require that the basis vectors p(ab) are different from

zero. Any velocity model addressed in Appendix A con-

tains them, except for the simplest ones, D2Q5 and

D3Q7, which have a sufficient number of basis vectors

to meet the anisotropy of the diagonal components only.

When off-diagonal elements are non-zero and symmet-

ric, one has to take ka equal to each other: ka = kD,"a, where kD is free parameter. The corresponding

diffusion combination is called mD : mD ¼ c2sKðkDÞ. The

mass conservation relations (23) of the MRT, TRT

and BGK models coincide for ka = kD. A diagonal form

oala(s)oah(s), can be matched by the velocity eigenvalues

of the MRT-model:

maðsÞ ¼ ClaðsÞ; �Deq.ðsÞ ¼ hðsÞ=C; f as. ¼ 0. ð26Þ

When h(s) = s, rel. (19) reduces to the conventional

equilibrium function. The E-model allows to work with

an arbitrary eigenvalue kD. In particular, we will distin-

guish two equilibrium solutions for the diagonal iso-

tropic diffusion form, oamoas, with constant diffusion

coefficient m:

�Deq.ðsÞ ¼ s; c2s ¼

mKðkDÞ

; aðeÞ ¼ 0; ð27Þ

and

�Deq.ðsÞ ¼ s; aðeÞ ¼ s

mmD� 1

� �; mD ¼ c2

sKðkDÞ; 8c2s .

ð28ÞBoth solutions assume a(ab) = 0, a(xx) = 0, a(ww) = 0 in

rel. (19). The projections on the mass vector e0 and the

energy vector p(e) are equal for both distributions. The

last one is controlled by the parameter mKðkDÞ and it is fixed

by the choice of the eigenvalue kD independently of the

c2s value. When a(e) = 0 (Eq. (27)), c2

s and therefore the

projection on the fourth order vector p(e), is also fixed

by the kD. When a(e) 5 0 (Eq. (28)), c2s and the p(e)-pro-

jection are free. The eigenmode analysis in Appendix B

based on the dispersion equation indicates that the iso-

tropic form of the higher order errors, with respect to

arbitrary orientation of wave vector ~k, is simpler

achieved for Eq. (27) than for Eq. (28). At the same

time, according to von Neumann stability analysis sim-

ilar to [20,26], the case a(e)!�s, c2s 6¼ 0, is more favor-

able for high Peclet numbers than the case a(e) = 0,c2s ! 0. A necessary stability condition is m

KðkDÞ 6 1, at

least when c2s ¼ 1=3. The stability interval then includes

the convection-dominant regime, m! 0, confirmed be-

low by simulations in the pure advection case (m � 0,

a(e) = �s). It is remarkable, that m � 0 can formally be

obtained with any eigenvalue kD when a(e) = �s,c2s 6¼ 0. A more precise analysis of stable parameters

ranges (c2s and the eigenvalues) for any specific pair

fm; ~Ug is in progress.

For particular problems (e.g., transport of solute con-

centrations), the positivity of s may be required. The

positivity of s as guaranteed by the positivity of the pop-

ulation solution can be easily checked only for the BGK

model when its eigenvalue k is higher than �1 and

source term is absent. The post-collision values, ~f qð~r; tÞin Eq. (6), are ~f qð~r; tÞ ¼ �kf eq.

q þ ð1þ kÞfqð~r; tÞ. Startingfrom the positive population solution fqð~r; 0Þ and

assuming the condition f eq.q ð~r; tÞ > 0, the population

solution stays positive. The positivity of the equilibrium

function restricts ~J ; c2s�D

eq.ðsÞ and f as. values. In case of

isotropic linear advection–diffusion equation, the solu-

tion (27) covers the modified moment propagation

scheme [21] for transport of the tracer P ð~r; tÞ. It can be

seen as a particular BGK scheme with kD = �1,f as. = 0, s = P, c2

s ¼ 1=3; �Deq.ðsÞ ¼ ð1� D�ÞP , and D*

being the adjusting parameter. The maximum allowed


D* value [21] restricts the Peclet number when the posi-

tivity of f eq.q is postulated. We emphasize that the posi-

tivity of the moving populations is neither a necessary

nor sufficient condition for stability. An estimation of

the effective Peclet numbers requires knowledge of the

numerical diffusion of the scheme which is discussed inthe next section.

4.3.2. Advection–diffusion

When the anti-symmetric equilibrium part f eq.� is

present, the term �F tð2Þ0 contributes to the RHS of Eq.

(16):

� F tð2Þ0 ¼ e2kCak2oa0KðkaÞot1hf

eq.�jCai;ot1f

eq.�q ¼ tHp Cqaot1U as; where ~U ¼~J=s.

ð29Þ

We assume that ot1U as ¼ U aot1sþOðeÞ, then replace

ot1s by �r0 ~J and assume again that U ar0 ~J ¼ob0U aUbsþOðeÞ. The second order correction to the dif-fusion tensor, hereafter also called numerical diffusion,

then takes a form:

�F tð2Þ0 ¼ �e2oa0

ma

c2s

ob0Duabs;D

uab ¼ U aUb. ð30Þ

The components of the numerical diffusion tensor are

proportional to the values of ma. Its diagonal compo-

nents are negative. In particular, when ma = mD andmD� m, the numerical diffusion becomes unbounded un-

less the tensor Duab is removed. With the help of the addi-

tional equilibrium projections sEu, f as.q ! f as.

q þ sEuq ,

one can remove �F tð2Þ0 by adding its counterpart, F tð2Þ

0 ,to the modeled diffusion form. The term Eu has the same

form as f as.:

Euq ¼ uðeÞtðeÞq þ uðabÞtðabÞ

q þ uðxxÞtðxxÞq þ uðwwÞtðwwÞq . ð31Þ

where, similar to Eq. (24):

Duxx ¼ uðxxÞ þ uðeÞ; Du

yy ¼ bðxxÞD uðxxÞ þ uðwwÞ þ uðeÞ;

Duzz ¼ bðxxÞD uðxxÞ � uðwwÞ þ uðeÞ;Du

ab ¼ uðabÞ; a 6¼ b.ð32Þ

The analysis of the obtained projections shows that

sEuq corresponds to ‘‘hydrodynamic’’ non-linear equilib-

rium term. For D2Q9, D3Q15 and D3Q19 models [24]

one can write it as:

sEuq ¼

1

2sU aUbtHp ð3CqaCqb � dabÞ; tH0 ¼ 3�

XQ�1

q¼1

tHq

ð33Þ

XQ�1

q¼1

3tHp C2qaC

2qb ¼ 1; a 6¼ b. ð34Þ

The leading order negative numerical diffusion is thuscanceled by the models which borrow the non-linear

term from the hydrodynamic models [21,23,27]. For a

particular isotropic model, a similar correction was

found by Girand [13] and Grubert [14]. The projection

on the fourth order basis vector p(e), p(e) 2 P4, is free

and can be arbitrarily adjusted (see [5,8]). Provided that

�F tð2Þ0 is removed by adding the term (34) to equilibrium

distribution (19), the second order AADE is the sum of

the Eqs. (21) and (23):

otsþr ~J ¼ oamaobDab þ Sm0 . ð35Þ

It corresponds to Eq. (1) where the diagonal la-part is

built by the eigenvalue functions ma; the coefficients

Kab are connected to the equilibrium parameters

through rel. (24) and the source term represents the mass

of the term Qmq in the evolution equation (6). Similar to

the computation of the stress tensor in the hydrody-

namic models, the diffusive flux �~D can be derived lo-

cally from the non-equilibrium part of the populations:

�~D ¼ C Iþ 1

2A

� � f ne.�. ð36Þ

Here, C[d · Q] is a matrix of the velocity components. In

the case of Eq. (35), Da ¼ maobDab þOðe3Þ. We empha-

size that the total mass flux,~J � ~D, is constant at steady

state.

5. The link-type model for AADE (L-model)

5.1. The BGK-type model

The models discussed in [32,33] use one specific relax-

ation parameter kq, kq ¼ k�q, for each pair of the veloci-

ties, ~Cq and ~C�q : A ¼ diagðkqÞ. With a conventional even

equilibrium part, f þq ðsÞ ¼ tqs; s ¼PQ�1

q¼0 fq, such a BGK-

type relaxation operator does not conserve mass. The

approach [32] consists in replacing s at equilibrium by

the variable m, m ¼PQ�1

q¼0 kqfq=PQ�1

q¼0 kqtq. This enablesthe model to satisfy formally condition (7) but not con-

dition (8) in general. It appears then that s is conserved

but m is not. The AADE are derived, however, for m. A

simple analysis shows that the difference between m and

s depends on the non-equilibrium projection on even

order basis vectors which contains the derivatives, o2na s

and o2n�1a

~J ; n P 1. The model violates condition (8) as

soon as the solution for s is non-linear and/or ~J isnot a constant.

5.2. L-model

Relations (16) suggest that only odd eigenvalues are

needed to recover the coefficients of the diffusion tensor.

With this idea in mind, we examine the collision opera-

tor (9) designed on L-basis. When even eigenvalues kþqare equal (and equal to some free value ke), one can

satisfy both properties (7) and (8) for the mass vector

e0 = 1. The collision operator of link-type L-model is


ðA f ne.Þq ¼ keðf þq � f eq.þq Þ þ k�q ðf �q � f eq.�

q Þ. ð37Þ

The separation of the BGK operator on the symmet-ric and anti-symmetric collision parts enables us to com-

bine mass conserving equilibrium functions and distinct

link eigenvalues k�q . Macroscopic equations can be ob-

tained similarly to Eqs. (15) and (16). Assuming an equi-

librium function (19) with f as. = 0, the second order

mass conservation equation of L-model is:

otsþr ~J ¼ oaDabob�D

eq.ðsÞ þ oaDuabobsþ Sm

0 ;

Dab ¼ 2XðQ�1Þ=2

q¼1

m�q tHp CqaCqb; m�q ¼ mðk�q Þ;

Duab ¼ �

2

c2sUb

XðQ�1Þ=2

q¼1

m�q tHp Cqa

Xc

U cCqc.

ð38Þ

Relations (38) assume that the first (Q � 1)/2 veloci-

ties are anti-parallel to the last ones. The D3Q15 L-

model is illustrated in the next section. The L-model

matches the coefficients of the symmetric diffusion

tensor Dab with the help of the diffusion functions m�q(cf. (39)). The tensor of the numerical diffusion Du is

non-symmetric for the general velocity field when theodd eigenvalues differ (cf. (44)). We have not found a

way of removing Du in general by the L-model. Provided

that Duab is negligible, rel. (36) yields the components of

the diffusive flux �~D of the L-model: Da ¼ Dabob�D

eq.ðsÞ þOðe3Þ.

5.3. D3Q15 model

The derived diffusion tensor Dab in (38) corresponds

to [32] for the D2Q9 model and to [33] for the D3Q19

model. It is claimed in [33], that the D3Q15 model has

not a sufficient number of degrees of freedom to cover

the 3D anisotropic tensor. We demonstrate here that

this model is able to define 6 coefficients of the diffusion

tensor with its 7 free eigenvalues. Assume its seven

velocities ~Cq are: (1,0,0), (0,1,0), (0,0,1), (1,1,1),(�1,1,1), (�1,�1,1), (1,�1,1); the other seven velocities

are anti-parallel and of the same magnitude. Diffusion

combinations corresponding to odd eigenvalues are

labeled as {mxx, myy, mzz, m4, m5, m6, m7}. The rel. (38)

yields:

Daa ¼ 2ðmaatH1 þ s1tH3 Þ; s1 ¼ m4 þ m5 þ m6 þ m7;

Dxy ¼ s2; Dyx ¼ Dxy ; s2 ¼ 2tH3 ðm4 � m5 þ m6 � m7Þ;

Dxz ¼ s3; Dzx ¼ Dxz; s3 ¼ 2tH3 ðm4 � m5 � m6 þ m7Þ;

Dyz ¼ s4; Dzy ¼ Dyz; s4 ¼ 2tH3 ðm4 þ m5 � m6 � m7Þ.ð39Þ

One can parameterize eigenvalue solution with free

parameter s1:

maa ¼tH3tH1

cðDaa � s01Þ; c ¼ 1

2tH3;

s01 ¼s1c; a ¼ 1; . . . ; d;

m4 ¼c4½s01 þ Dxy þ Dxz þ Dyz�;

m5 ¼c4½s01 � Dxy � Dxz þ Dyz�;

m6 ¼c4½s01 þ Dxy � Dxz � Dyz�;

m7 ¼c4½s01 � Dxy þ Dxz � Dyz�.

ð40Þ

When Dab is diagonal, all eigenvalues k�q , q = 4, . . . , 7,are equal to each other. Linear stability conditions

maa P 0, mðk�q ÞP 0, require:

Dmin6 s01 6 Dmax;

Dmax ¼ minfDaa; a ¼ 1; . . . ; dg;

Dmin ¼ maxfDxy ;Dyz;Dxzg.

ð41Þ

The simulations below are done with the following

strategy for the D3Q15 model. We set

s01 ¼ Dmaxð1� 2tH1 Þ if Dmax P Dmin=ð1� 2tH1 Þ. ð42Þ

Otherwise, we define s01 as the arithmetical mean value of

its limits, s01 ¼ ðDmin þ DmaxÞ=2. Assume that the smallest

diagonal coefficient is Daa. Then the choice (42) yieldsmaa = Daa. If Dab is the diagonal tensor, the coefficients

mðk�q Þ, q = 4, . . . , 7, are equal to Daa. In isotropic case

Dab = mdab, solutions (39) and (42) for ‘‘diagonal’’ links

(q = 4, . . . , 7) is

mðk�q Þ ¼ s1=4; s1 ¼ ðm� 2tH1 mDÞ1

2tH3

if maa ¼ mD; a ¼ 1; . . . ; d. ð43Þ

Solution (43) implies that even the isotropic tensor

may be captured with the distinct eigenvalues: one for

the first velocity class ðk~Cqk2 ¼ 1Þ and another one forthe third class ðk~Cqk2 ¼ 3Þ. Stability conditions require

mD < 32m. It is possible however that another choice of

the free eigenvalue combination s01 will modify this sta-

bility condition. Remember here that the E-model is,

conversely, most robust in the interval m� mD. In this

paper, we mainly restrict ourselves to the strategy (42).

It reduces to the conventional solution mD = m when

the first class eigenvalue function mD is set equal to m. Fi-nally, let us illustrate the tensor of numerical diffusion

Duab (see rel. (38) with b = 1, . . . ,d):


Duxb=Ub ¼ �

2

c2s

ðmxxtH1 Ux þ tH3 fm4ðUx þ Uy þ UzÞ

þ m5ðUx � Uy � UzÞ þ m6ðUx þ Uy � UzÞþ m7ðUx � Uy þ UzÞgÞ;

Duyb=Ub ¼ �

2

c2s

ðmyy tH1 Uy þ tH3 fm4ðUx þ Uy þ UzÞ

þ m5ð�Ux þ Uy þ UzÞ þ m6ðUx þ Uy � UzÞþ m7ð�Ux þ Uy � UzÞgÞ;

Duzb=Ub ¼ �

2

c2sðmzztH1 Uz þ tH3 fm4ðUx þ Uy þ UzÞ

þ m5ð�Ux þ Uy þ UzÞ þ m6ð�Ux � Uy þ UzÞþ m7ðUx � Uy þ UzÞgÞ. ð44Þ

We emphasize that for the eigenvalue choice (43) in

the isotropic case, Duab is symmetric and its components

are proportional to mUaUb for any choice kD. This point

is different from the E-model solution (30) where the

components of the numerical diffusion tensor are pro-portional to the used mD value.

6. Preliminary overview: numerical aspects

The E-model for Eq. (1) and the L-model for Eq. (2)

have been introduced. Both models require lattice veloc-

ities with more than one non-zero component to capturethe off-diagonal elements of the symmetric diffusion

form. Therefore, the D2Q5 and D3Q7 velocity sets can-

not handle them. We recall that for any eigenvalue k its

diffusion combination is mðkÞ ¼ c2sKðkÞ, K(k) = �(1/2 +

1/k). Here, k 2 [�2,0] and c2s is a free constant. In partic-

ular, mD = m(kD) and ma = m(ka), ka being the velocity

eigenvalues in MRT basis. Let C be an arbitrary non-

zero constant. Let us now summarize some collisionconfigurations for the two models.

6.1. E-model

Provided that the second order term of the numerical

diffusion is canceled by the equilibrium correction (31)

or that it is negligible, Eq. (35) is equivalent to the Eq.

(1). We assume the equilibrium function (19). In partic-ular, one can fix �D

eq.ðsÞ equal to �D resulting in a(e) = 0.

At high Peclet numbers, however, the choice a(e) 5 0 is

more stable especially when c2s is not too small/high (e.g.,

c2s ¼ 1=3). The typical situations may be solved as

follows:

(1) When Kab is an isotropic tensor, Kab ¼ hðsÞdab,

and the coefficients la(s) are equal, la = l(s), onecan use

(a) The BGK with a constant or variable eigen-

value kD(s). The coefficients of f as. are

computed from rel. (25) where Dab are con-

strained by the condition: mDðsÞoaDaa ¼lðsÞoahðsÞ. In particular, mDðsÞ ¼ ClðsÞ;�D

eq.ðsÞ ¼ hðsÞ=C, f as. = 0.

(b) The MRT with maðsÞ ¼ ClðsÞ and the equilib-

rium as in (a).(2) When Kab is an isotropic tensor, Kab ¼ hðsÞdab,

but the coefficients la(s) are distinct, one can

use:

(a) The BGK with a constant or variable eigen-

value kD(s). The coefficients of f as. are com-

puted from the rel. (25) where a(ab) = 0 and

the diagonal coefficients Daa are constrained

by the condition: mDðsÞoaDaa ¼ laðsÞoahðsÞ.(b) The MRT with maðsÞ ¼ ClaðsÞ; �D

eq.ðsÞ ¼hðsÞ=C and f as. = 0.

(3) When Kab is a fully symmetric tensor and for arbi-

trary coefficients la(s), one can use:

(a) The BGK with a constant or variable eigenvalue

kD(s) . The coefficients of f as. are computed from

rel. (25) whereDab are constrained by the follow-ing condition: mDðsÞobDab ¼ laobKab.

(b) The MRT with ka(s) = kD(s) and the equilib-

rium as in (a).

For all the cases above, we advice to replace the

BGK operator by the TRT operator (10). The stabil-

ity requirement yields mDðsÞ > maxfmaosKabðsÞg. Withthe help of the free even eigenvalue ke, the method

can benefit from the solutions for special ke/kD combi-

nations, especially when the odd eigenvalue kD is

constant. Such solutions are developed in Appendix

B to remove higher order corrections and in [11] to

improve the accuracy of particular boundary

reflections.

In all the cases above, the MRT model keeps alleven eigenvalues as free parameters. Except for the

velocity eigenvalues, the other odd eigenvalues are also

free. Eigenvalues corresponding to the third order

polynomials hð1Þa and hð2Þa (see in Appendix A) are usu-

ally set equal to ka. When the MRT model differs from

the TRT, the collision operator (6) requires the compu-

tation of the equilibrium projections bfkeq.

. The hydro-

dynamic equilibrium distribution is given in a

projected form for D2Q9 in [3,20,28], for D3Q13 in[4], for D3Q15 in [5] and for D3Q19 in [5,28]. The equi-

librium terms f eq.�, f is. from Eq. (19) and sEu from

(31) and (32) are projected on the same eigenvectors

as the hydrodynamic distributions. Projections of the

anisotropic part f as.=c2s (cf. (19) and (20)) on the vec-

tors from the subset P2 are equal to a(e)/b(e), a(ab)/

b(ab), a(xx)/b(xx), a(ww)/b(ww), respectively. The MRT E-

model may match a non-symmetric diffusion formin principle with the help of distinct velocity

eigenvalues ka.


6.2. L-model

Provided that the second order term of the numerical

diffusion is removed or that it is negligible, Eq. (38) fits

Eq. (2) with Kab ¼ CDab; h ¼ �Deq.ðsÞ=C. Based on the

isotropic part of the equilibrium function (19), i.e. withf as. = 0, the link-type collision (37) yields the compo-

nents Dab as a linear function of diffusion combinations

m�q ; m�q ¼ mðk�q Þ; k�q being odd eigenvalues of the colli-

sion operator (37). The L-model requires distinct odd

eigenvalues to achieve any anisotropy between diagonal

coefficients and/or to obtain off-diagonal elements (cf.

(39) and (40)). Stability conditions may restrict the coef-

ficients of the modeled tensor (cf. (41)). When the oddeigenvalues differ, the model cannot easily benefit from

the solutions on the combinations of even/odd eigen-

values. In context of the L-model, we have no solution

to remove the numerical diffusion tensor Du (cf. (38))

in general, i.e. for arbitrary velocity field and full diffu-

sion tensor. Finally, we emphasize that the L-model

can be designed on the MRT-L basis vectors. In this

case, even eigenvalues are not restricted to being equalto each other. This can be helpful to improve the stabil-

ity in analogy to the MRT operators [20,28], as well as

the accuracy similar to the TRT collision above.

7. Numerical results

In Sections 7.1–7.4 we address the linear convection–diffusion equation (3) with constant isotropic and aniso-

tropic diffusion tensors and with constant dispersion

tensor in Section 7.5. The quasi-linear isotropic diffusion

equation is considered in Section 7.6. The equilibrium

function (19) for these problems are proportional to

the conserved quantity s. We set ~J ¼ ~Us; �Deq.ðsÞ ¼ s

and represent f as. as:

f as.q ¼ c2

s sðaðeÞtðeÞq þ aðabÞtðabÞq þ aðxxÞtðxxÞq þ aðwwÞtðwwÞ

q Þ;

aðeÞ ¼ aðeÞ=s; aðabÞ ¼ aðabÞ=s; aðxxÞ ¼ aðxxÞ=s;

aðwwÞ ¼ aðwwÞ=s.

ð45ÞAll computations are done with the D3Q15 velocity

set. For any collision operator, even eigenvalues are all

set equal to a specific value ke.

7.1. Study of a concentration wave

The evolution of a concentration wave sð~r; 0Þ ¼s0 cosð~k ~rÞ in time obeys the solution (see Appendix B):

sð~r; tÞ ¼ s0 cosð~k ð~r � ~UtÞÞ expð�ixð~kÞtÞ;

xð~kÞ ¼ �imð~kÞk2; mð~kÞ ¼Xa;b

Dabkakb

k2.

ð46Þ

The exact solutions for the first coefficients mðrÞn ð~kÞ inthe expansion of mð~kÞ in powers k2n, n = 0,1,2, . . .,

around mðrÞ0 ð~kÞk0 ¼ mð~kÞ are presented in Appendix B.

They are based on an eigenmode analysis of the evolu-

tion equation using the TRT operator and 3D isotro-

pic/anisotropic diffusion tensors.

We restrict ourselves here to the pseudo 3D case:~k ¼ fkx; 0; kzg ¼ kfcos#; 0; sin#g. Full 3D computa-tions are addressed in Section 7.5. The computation

domain has length La for relevant dimensions. Periodic

boundary conditions are used at all ends. The compo-

nents of ~k in rel. (46) should be understood as 2pLa

ka, ka

will be indicated; s0 = 100 in all computations. The effec-

tive value mlbð~k; tÞ is derived from the solution with the

help of the following operations:

mlbð~k; tÞ ¼ logðIðtaÞÞ � logðIðt þ taÞÞk2ta

;

I2ðtÞ ¼ I2s ðtÞ þ I2cðtÞ;

IsðtÞ ¼sinð~k ~U lbÞ

2expð�ixlbtÞ;

IcðtÞ ¼cosð~k ~U lbÞ

2expð�ixlbtÞ.

ð47Þ

Here, Is(t) and Ic(t) are the values of the integralsR Lx

0

R Lz

0sðx; tÞ sinð~k ~rÞ and

R Lx

0

R Lz

0sðx; tÞ cosð~k ~rÞ, respec-

tively. The value of time interval ta depends on model

parameters (we use typically ta � 5, . . . , 50). In all consi-

dered cases, mlbð~k; tÞ reaches the stationary regime. Pro-

vided that previous order errors are absent or removed,

we estimate the relative k2n-error:

EðrÞðmlbÞk2n ¼ mlbð~kÞ � mð~kÞ

mð~kÞk2n; n ¼ 0; 1; 2; 3; . . . . ð48Þ

The total relative error corresponds to n = 0. When

available, the reference dispersion solution for mðrÞn ð~kÞ isused to check the obtained values. In case of advection,the error on the tangential projection of the velocity,

EðrÞð~U lbÞ, is computed as:

EðrÞð~U lbÞ ¼~k ~U lb �~k ~U

~k ~U; tanð~k ~U lb

tÞ ¼ IsðtÞ=IcðtÞ.

ð49ÞThe leading order error can be compared with the exact

solution (B.14).

7.2. Comparison with the dispersion relations without

advection

7.2.1. Isotropic case

We first check the isotropic case Dab = mdab in a

[X · Z] = 502 box. Here, m is a constant diffusion coeffi-

cient. We use the TRT operator (10) and consider two

equilibrium configurations, Eqs. (27) and (28). The


stability interval of the E-model contains the domain

m 6 mD. The leading relative error mðrÞ1 is given by rel.

(B.2). When a(e) 5 0 (case (28)), mðrÞ1 depends upon the

angle #, i.e. the error is anisotropic in general and its

highest value occurs for ~k ¼ f1; 1; 1g. When ke is given

by the ‘‘optimal diffusion’’ solution (B.4), the aniso-tropic part of the leading error vanishes.

7.2.1.1. ‘‘Optimal diffusion’’ solution for ke. When the dif-

fusion tensor is isotropic, mðrÞ1 becomes proportional to

mð~kÞ when ke is equal to optimal solution kopte ðkDÞ com-

puted from rel. (B.4). Computations are first done for

fixed c2s ; c2

s ¼ 1=3 and a(e) = {�0.999, ±0.9, ±0.6, 0, 1,

1.2}. Case a(e) = 0 corresponds to the classical solutionmD = m, i.e. c2

s ¼ mKðkDÞ. The lower limit, a(e) = �0.999, cor-

responds to m/mD = 10�3. Numerical results for E(r)(mlb)/k2 are shown in Fig. 1 as a function of m and for some

different wave vectors ~k. For c2s ¼ 1=3, computations

with two values ke are demonstrated: ke = kD = �1 and

the ‘‘optimal diffusion’’ solution ke ¼ kopte ðkD ¼ �1Þ ¼

0 0.1 0.2 0.3 0.4 0.5Diffusion coefficient ν

0

1

2

3

4

E(r

) ( ν)/κ

2E

(r) ( ν

)/κ2

theory: ν1(r)

( ν, κ) κ ={1,0} κ ={1,2} κ ={1,1}

Isotropic diffusion tensor


0

1

theory: ν1(r)

( ν) κ={1,0} κ={1,2} κ={1,1} κ={1,2}, a

e=0


Fig. 1. Relative error E(r)(m)/k2 in case kD = �1 is compared with the

theoretical solution (B.2) for different wave-vectors. Top: ke =kD = �1. Bottom: ke ¼ kopt

e ð�1Þ ¼ �1.2 according to ‘‘optimal’’ solu-

tion (B.4). Last data (left triangle) is computed with ~k ¼ f1; 2g and

a(e) = 0, c2s ¼ m=KðkDÞ.

�1.2. On the whole, the obtained results agree closely

with the theoretical estimation (B.2). For the first choice,

the error is anisotropic and diverges when a(e)!�1

(m! 0). In the second case, the error is nearly isotropic,

except for the highest ratio a(e) = � 0.999 when the con-

tribution of the next terms in the perturbation series issignificant.

The last data in case ke = �1.2 (bottom picture, left

triangles) corresponds to c2s ¼ m=KðkDÞ leading to

a(e) = 0, mD = m as in case (27). Thus c2s varies between

815

and 13� 10�3. When ke ¼ kopt

e ðkD ¼ �1Þ, mðrÞ1 ð~kÞ coin-

cides for both equilibrium functions (27) and (28) (see

Eq. (B.4)). We find that the numerical solutions are

identical to the solutions above when ~k is parallel toone of the coordinate axes. Otherwise, the results with

mD = m agree better with the theoretical predictions (com-

pare squares and left triangles for ~k ¼ f1; 2g, bottom

picture, the case m = 10�3) since higher order terms do

not diverge when m! 0 and a(e) = 0. We confirm that

for a particular value (B.6), kD ¼ k0D ¼ �3þ

ffiffiffi3

p, the

error E(r)(mlb) scales as k4 for any ~k but the pre-factor

depends on the direction of the wave vector.

7.2.1.2. ‘‘Isotropic solution’’ for ke. Only for the isotropic

tensor, and when a(e) = 0 and c2s ¼ m=KðkDÞ according to

rel. (27), mðrÞ1 ð~kÞ is isotropic for any mD, c2s and ke (cf. rel.

(B.2)). The solution kise ðkD; c2

s Þ given by rel. (B.7) cancels

mðrÞ1 ð~kÞ. For instance, kise ðkD ¼ �1; c2

s ¼ 13Þ ¼ �1. This ex-

plains why the results of the conventional diffusion

BGK model with the relaxation parameter equal to

one are often found to be the most accurate (see [27],

for instance). Fig. 2 shows E(r)(mlb)/k4 for kD = �1,c2s ¼ 2m, ke ¼ kis

e ðkD; c2s Þ. The results agree very closely

with solution (B.9) for mðrÞ2 ð~kÞ. The error diverges when

m!1, but the domain m � mDðc2s � 1Þ lies out of our

interest.


-0.2

-0.1

0

0.1

0.2

0.3

0.4

theory: ν2(r)

( ν, κ) κ ={1,0} κ ={1,2} κ ={1,1}


E(r

) (n)/

k4

Fig. 2. Relative error E(r)(m)/k4 in case kD = �1, ke ¼ kise ðc2s ; kDÞ,

a(e) = 0, c2s ¼ 2m is compared with the theoretical solution (B.9) for

different wave-vectors.

0 0.025 0.05 0.075 0.1Diffusion coefficient

-0.1

-0.05

0

0.05

0.1

E(r

) (Dzz

)/k2

E(r

) (Dxx

)/k2

theory: ν 1(r)

(zz

)TRT-E modeltheory: ν 1

(r)(

zz)

MRT Model

Anisotropic diffusion tensor

0 0.025 0.05 0.075 0.1-0.2

-0.1

0

0.1

0.2

E(r)

(Dxx

)/2

L-model: νxx

L-model: νzz


Diffusion coefficient

k

D

D

Dzz

Dzz

Fig. 3. Top: results of the E-model for E(r)(Dzz)/k2, ~k ¼ f0; 1g in

anisotropic diagonal case Dxx 5 Dzz. Circles: equal odd eigenvalues.

Squares: ma = Daa. Theoretical estimation for leading error is (B.5) in

the first case and (B.2) in the second. Bottom: results of the L-model

for E(r)(Dxx)/k2, ~k ¼ f1; 0g. The figure shows also the diffusion

combinations mzz = Dzz and mxx.


With the particular solution (B.10) for kD and (B.11)

for c2s , we obtain E(r)(mlb) · 105/k6 equal to 2.45, 2.93,

6.16, 7.41, 6.54, 8.0 when ~k ¼ fkx; kzg has the compo-

nents {0,5}, {0,10}, {2,2}, {4,4}, {3,4}, {6,8}, respec-

tively. For this specific choice of the eigenvalue/

equilibrium parameters, the error scales at least as k6

and the obtained values agree with the estimate of mðrÞ3 .

7.2.2. Anisotropic case

We consider again a pseudo 3D case ~k ¼kfcos#; 0; sin#g in [X · Z] = 1002 box and vary the

ratios Dxx/Dzz and Dxx/Dxz. Without loss of generality,

we assume Dxx/Dzz P 1 and Dxz < Dxx. Three configura-

tions of eigenvalues are used. The first set-up corre-sponds to the TRT E-model with mD = max{Dxx,

Dzz} = Dxx (see case 3.a in Section 6). This choice is

based on the stability arguments above. The second

set-up corresponds to the MRT E-model with distinct

velocity eigenvalues ka (ma = Daa) (cf. Eq. (26) and case

(2b) in Section 6). This choice works with the conven-

tional equilibrium, f as. = 0, in case of a diagonal diffu-

sion tensor. The third set-up corresponds to theL-model with eigenvalue choice (40)–(42). Three eigen-

values configurations coincide in the case of the TRT

operator and for the isotropic diffusion tensor only.

Also, kD = � 4/3 is used for simulations in the first case.

In the second case, kx = � 4/3 and kz varies between �4/

3 and �1.9961 when Dxx/Dzz varies from 1 to 28. When

Dab is a diagonal tensor, the eigenvalue kz and ‘‘diago-

nal’’ eigenvalues k�q of the link operator correspond tothe smallest diagonal coefficient (mzz ¼ mðk�q Þ ¼ Dzz,

s1 = 4mzz) and mxx is related to them by the linear func-

tion (40).

7.2.2.1. Experiment 1: diagonal anisotropic tensor. First,

we consider the case where the diffusion tensor is diago-

nal with Dxx = Dyy, Dxx/Dzz = 1, . . . , 28. Simulations are

done for ke ¼ kopte ðkD ¼ �4=3Þ. When the wave vector is

parallel to x (z-) axis, we measure the error for the rele-

vant component Dxx (Dzz), respectively. With the first

set-up, the leading k2-error is linear with respect to

Daa/mD in accordance with rel. (B.5) and it goes to zero

when Daa decreases. When mD corresponds to the highest

diagonal element, the highest relative error value occurs

for this component unless kD ¼ k0D and the leading order

error vanishes.For the second set-up, Daa ¼ c2sKðkaÞ. Then

EðrÞðDaaÞ ¼ f1ðc2s ; ka; keÞ according to rel. (B.2). The error

vanishes for ke ¼ kise ðkaÞ only. Thus the second set-up

can not annihilate mðrÞ1 ðDaaÞ for all a simultaneously.

When ka! 0, g(ka, ke)! 0 and f1ðc2s ; ka; keÞ ! 1.

Within this limit, the first set-up becomes much more

accurate. These observations for E(r)(Dzz) are compared

with the theoretical predictions in Fig. 3, top picture.

The results of the link model are identical here to those

for the second set-up due to the choice of mzz. For the

first set-up, the error is proportional to Daa when mD is

fixed. The second case does not have this advantage

and its error diverges whereas with first set-up it goes

to zero in the most important interval mD� Daa,Daa! 0.

For the two first configurations, the error E(r)(Dxx) is

independent of the ratio Dxx/Dzz when~k is parallel to the

axis. The obtained value (E(r)(Dxx)/k2 = � 0.00695)

agrees with the one (�0.0069) given by the solution

(B.2) and (B.5) for mD = Daa = 1/12. In the case of the

L-model, mðrÞ1 ðDxxÞ takes much higher values (see Fig. 3,

bottom) since ke is not optimal for odd eigenvalues.The figure also shows the corresponding diffusion coef-

ficients mzz ¼ Dzz ¼ m�q and mxx = myy = 3/2Dxx � 1/2Dzz

according to rel. (40). On the whole, the experiment

demonstrates the important role of a special concor-

dance between even/odd eigenvalues for high order

errors.

0 0.2 0.4 0.6 0.8 1a(xz)=Dxz/Dxx

-0.06

-0.04

-0.02

0

0.02

theory: ν1(r)

( ν, κ)

E(r)

( ( ))/ 2

E(r)

(xz

)/κ2

E(r)

(xx

)/κ2


0 0.2 0.4 0.6 0.8 1-6

-5

-4

-3

-2

-1

0

1

E(r)

( ν(κ))/κ2

E(r)

(Dxz

)/k2

E(r)

(Dxx

)/k2


E(r

) /κ2

E(r

) /κ2

a(xz)=Dxz/Dxx

D

D

ν κ κ

Fig. 4. Off-diagonal coefficient Dxz is varied with respect to diagonal

coefficient mD = Dxx = Dyy = Dzz. Top: TRT E-model. Total error

E(r)(mlb)/k2 is compared with the theoretical estimations (B.12).

‘‘Component’’-errors EðrÞðDxzÞk2

, EðrÞðDxxÞk2

are compared to rel. (50). Bottom:

Similar measurements for the L-model. The theoretical solution is not

available.

0 1 2 3 4 5 6 7 8log2( / )

-0.005

-0.003

0

0.003

0.005

E(r

) (κ

)/4

={0,1}={0,2}={1,2}={2,4}


Dxx Dzz

κκκκ

ν

Fig. 5. Relative error E(r)(mlb)/k4 of the E-model in the pseudo 3D

anisotropic case Dyy = mD, Dxx/Dzz = 2n, n = 0,1, . . . , 8, Dxz = 0.8Dyy.

Data: kD ¼ k0D ¼ �3þ

ffiffiffi3

p, ke ¼ kis

e ðc2s ; kDÞ. Leading error mðrÞ1 ð~kÞk2 is

removed according to rel. (B.12).


7.2.2.2. Experiment 2: general anisotropic 2D tensor. We

check the dispersion results (B.12) obtained for the TRT

operator with ke ¼ kise in the most general 2D case.

Without a loss of generality, and according to solution

(24), we set Dxx = mD(1 + a(xx)), Dzz = mD(1 � a(xx)),

Dyy = mD, Dxz = a(xz), Dyx = 0, Dyz = 0, # = h. We firstaddress the anisotropy between the diagonal and the

off-diagonal terms. All diagonal coefficients are equal

to each other and equal to mD = 1/12 whereas the equilib-

rium coefficient a(xz) = Dxz/mD varies between 0 and 1.

The solution approaches the isotropic solution when

Dxz! 0. The reference solution is given by rel. (B.12)

with a(xx) = 0(a = p/2), mð~kÞ ¼ mDð1þ aðxzÞ sin 2hÞ and

the leading absolute error is expected to be equal toc2s f0ðkDÞð1�c2s Þ

mD2ðaðxzÞ sin 2hÞ2k2. The highest error corresponds

to h = p/2. Together with the estimation (48), we mea-

sure the relative error for diagonal and off-diagonalparts separately:

EðrÞðDxxÞk2

¼ c2s f0ðkDÞ

2ð1� c2s ÞðaðxzÞ sin 2hÞ2; mD ¼ Dxx;

EðrÞðDxzÞk2

¼ c2s f0ðkDÞ

2ð1� c2s Þ

aðxzÞ sin 2h.

ð50Þ

Relations (50) assume that the total error is assigned

to one of the components only. The results for total

error plotted in Fig. 4 for ~k ¼ f1; 0; 1g and with

aðxzÞ ¼ f2�2n; 23; 45g; n ¼ 0.5; 1; 2 . . . ; 4, agree very well

with the predictions. The problem occurs when

a(xz)! 0 and the relative error E(r)(Dxz) is not well de-

fined. For comparison, the results of L-model are alsoshown in Fig. 4. The link model is computed with the

eigenvalue configuration (43) (i.e. with equal ‘‘coordi-

nate’’ eigenvalues and different ‘‘diagonal’’ eigenvalues).

The amplitude of the error is an order of magnitude

higher than with the TRT-E model.

Finally, we include anisotropy on the diagonal.

When mD = Dyy is fixed, other components of the tensor

can be computed from their ratios. For a particularvalue kD ¼ k0

D ¼ �3þffiffiffi3

p, the error mðrÞ1 ð~kÞ is equal to

zero for any anisotropic ratios and any ~k provided

that ke ¼ kise ðk

0D; c

2s Þ. Note that kis

e ðk0D; c

2s Þ ¼ kopt

e ðk0DÞ �

�0.9282 for any c2s . The obtained error E(r)(mlb)/k4 is

plotted in Fig. 5.

7.3. Concentration wave with advection

7.3.1. Analysis of the numerical diffusion for the 1D case

We assume one-dimensional advection along x-axis:~U ¼ fU ; 0; 0g; ~k ¼ fkx; 0; 0g and m = Dxx. We distin-

guish the following sources of the difference between mand mlb:

mlb � m ¼ mh þ ml. þ mnl.. ð51ÞThe high order correction mh in cases without advec-

tion was addressed in the previous section. The correc-

0 0,1 0,2 0,3 0,4 0,5Diffusion coefficient

0

1

2

3

4

theory for =0: ν1(r)

( ) κ ={1,0} κ ={1,2} κ ={1,1}

Isotropic diffusion tensor, =0.1, with u

0 0.1 0.2 0.3 0.4 0.5Diffusion coefficient

0

0.2

0.4

0.6

0.8

1

theory =0: ν1(r)

( )=0 κ ={1,0} κ ={1,2} κ ={1,1}

Isotropic diffusion tensor, =0.1, with u

E(r

) (κ

)/2

νE

(r) (

κ )/

2ν

ν

ν, κU

U E

U E

U ν, κ

ν

Fig. 7. When the non-linear advection term Eu is present, the relative

error E(r)(mlb)/k2 of the E-model is compared with the theoretical

solution (B.2) for cases without advection. Top: kD = ke = �1, c2s ¼ 13,

a(e) = m/mD�1. The error E(r)(mlb) for the smallest m = 10�3 is equal to2 2 ~ ~


tion ml. ¼ Duxx is due to the linear equilibrium advection

term whereas mnl. is related to the non-linear equilibrium

term Eu (see rel. (30) for the E-model and rel. (38) for the

L-model). According to solution (30):

Duxx ¼ ml. ¼ �KðkDÞU 2. ð52ÞThe equilibrium term Eu (34) implies that mnl. = � ml.

for the E-model. Without this term, the relative error

E(r)(mlb) is expected to be negative and to behave as

�K(kD)U2/m at the leading order. That means that when

mD is different from m, the relative diffusion error due tothe advection term diverges when m! 0. Otherwise, it is

equal to U 2=c2s . In the case of the D3Q15 L-model and

when mxx = mD, Du has only one non-zero component

(cf. (38,42,44)):

Duxx ¼ ml. ¼ � 2U 2

c2s

ðmDtH1 þ s1tH3 Þ ¼ �U 2

c2s

m. ð53Þ

This means that the leading relative error of the L-

model behaves as U2

c2sfor any m and for any eigenvalues.

7.3.2. Numerical tests

We check these observations using the evolution of

the concentration wave. Fig. 6 shows the obtained rela-

tive error due to advection: E(r)(mlb, U = 0.1) �E(r)

(mlb, U = 0) = ml./m when the non-linear equilibrium

advection term (34) is absent. The E-model behaves

according to the predicted solution (52) when kD is fixed.

The results for the L-model confirm the prediction (53).We study the results of the E-model in the presence of

the non-linear advection term sEu given in rel. (32).

Assuming that the highest error occurs when ~U and ~kare parallel, we focus on this case. The obtained data

for E(r)(mlb)/k2 is plotted in Fig. 7 with two configura-


-6

-5

-4

-3

-2

-1

0

E(r

) (ν)

E-model: theoryE-model: dataL-model: theoryL-model: data x 102

Isotropic diffusion tensor, =0.1, without uU E

Fig. 6. Relative error due to linear advection term in case U = 0.1 is

compared with the theoretical predictions for E(r)(mlb,U) � E(r)(mlb,U = 0) = ml./m for the E-model (rel. (52), kD = �1) and the L-model (rel.

(53)). Note that the error of the L-model is multiplied by a factor 100

for visualization purposes. The relative error of the E-Model is equal

to �29.8402 for m = 10�3.

85.4054k and 129.951k for k ¼ f1; 2g, k ¼ f1; 1g, respectively.

Bottom: kD = �1, c2s ¼ 2m, a(e) = 0, ke ¼ kise ðc2s ; kDÞ, mðrÞ1 ð~k;U ¼ 0Þ ¼ 0.

tions for ke. The first picture is computed with the same

data as in Fig. 1. The second picture is computed with

the ‘‘isotropic’’ set-up used for Fig. 2. When the non-lin-

ear equilibrium term (34) is included, the main part ofthe advection error is removed. The leading error ap-

proaches its value in cases without advection, i.e the the-

oretical solution (B.2). The influence of the higher order

advection terms is strong only for the highest ratio mD/m.Unlike for cases without advection, E(r)(mlb)/k2!1 for

m! 0 even when the wave vector is parallel to the coor-

dinate axis. Also, the optimal isotropic set-up loses its

property to annihilate the k2-error with the choiceke ¼ kis

e . Instead of being equal to zero, the leading error

now behaves as k2U2. A similar behavior is obtained for

a special solution (B.10) and (B.11). For instance,

E(r)(mlb, U = 0.1)/k2 · 103 = {1.44,1.42,�1.0,�1.1} when~k ¼ fkx; kzg has components {0,1}, {0,5}, {1,1}, {2,2},

respectively. This error reduces to 3.8 · 10�4 and


�2.5 · 10�4 when k~Uk is reduced by a factor 2. This

confirms that E(r)(mlb,U) � E(r)(mlb,U = 0) behaves like

U2k2. We find here that the leading error on the tangen-

tial velocity also behaves like k2: EðrÞð~U lbÞ=k2 � 102 ¼f�1.36;�1.38;�1.36;�1.35;�1.31;�1.30g when ~k ¼fkx; kzg is {0,1}, {0,5}, {1,1}, {2,2}, {3,4}, {6,8},

respectively. This solution agrees with the theoretical

predictions (B.14). The leading advection error (B.14)is anisotropic and it is related linearly to c2

smð~kÞ when

kD is fixed. The ‘‘optimal’’ BGK advection solution

(B.15), kD ¼ ke ¼ k0D, removes the k2 part of the advec-

tion error completely. Numerical calculations confirm

that the EðrÞð~UÞ behaves proportionally to k4 for this

BGK set-up.

7.4. Concluding remarks about optimal solutions

Let us conclude with the following remarks. The TRT

E-model is found to be robust for anisotropic constant

diffusion tensors when kD 2 ]�2,0[, c2s > 0 and

a(e) P �1 are chosen so that c2s ðaðeÞ þ 1Þ 6 1. In particu-

lar, the last condition is hold when mD is equal to the

highest diffusion coefficient. When ke ¼ kopte , the aniso-

tropic part of fourth order (k2-) error mðrÞ1 ð~kÞ is canceled

for any anisotropic diffusion tensor, and any values ofc2s and a(e). For an isotropic diffusion tensor, the error

mðrÞ1 ð~kÞ then coincides for the two equilibrium functions,

Eqs. (27) and (28). Only for the particular choice

kD ¼ k0D, k0

D ¼ �3þffiffiffi3

p, ke ¼ kopt

e ðk0DÞ, the error mðrÞ1 ð~kÞ

vanishes when a(e) 5 0.

For the isotropic diffusion tensor, the solution

ke ¼ kise ðkD; c2

s Þ is sufficient to annihilate the mðrÞ1 ð~kÞ errorfor any mD in case of the conventional equilibrium solu-tion (27), i.e. when a(e) = 0. In particular, the BGKmodel

with kD = �1, c2s ¼ 1=3 is free from fourth order error.

When a(e) = 0 for an anisotropic tensor due to the choice�D

eq.ðsÞ ¼ �DðsÞ, the combination fk0D; k

ise g annihilates the

total k2-error on the diffusion term (rel. (B.12)). The

solutions kopte ðk

0DÞ and kis

e ðk0D; c

2s Þ coincide.

The same value kD ¼ k0D removes the k2-part from the

advection error provided that ke is equal to k0D. Both dif-

fusion and advection errors cannot be annihilated simul-

taneously since kopte ðk

0DÞ is not equal to k0

D. One can

suggest that the best accuracy will be achieved when

ke 2 ½k0D; k

opte ðk

0DÞ�. The particular choice depends on

the type of the problem: for convection-dominant prob-

lems, the BGK set-up (kD ¼ ke ¼ k0DÞ would be pref-

erable whereas kopte ðk

0DÞ or kis

e ðk0D; c

2s Þ solutions are

advantageous for problems where diffusion predomi-

nates. We check these observations in the next section.The link model has received less attention since its sta-

bility in highly anisotropic cases is tedious to maintain

and no optimal solution for higher order accuracy has

been established up to now for cases when odd eigen-

values differ.

7.5. Gaussian hill with dispersion tensor

The evolution of the initial Gaussian profile is com-

pared with the exact solution

sð~r; tÞ ¼ s0exp � 1

2

r0ab

krk ðxa � �xaÞðxb � �xbÞh i

ð2pÞd=2krk1=2;

�xa ¼ x0a þ U at; r2

abðtÞ ¼ r2abðt ¼ 0Þ þ 2Dabt.

ð54Þ

Here, krk is the determinant of the diffusion matrix

{rab} and r0ab is the cofactor to the element rab. The

solution is initialized with first order expansion which

corresponds to a Gaussian distribution (54) at t = 0. If

not specially indicated, initial dispersion is uniform,

raa(t = 0) = 4, rab(t = 0) = 0. Provided that the hill is stillfar from the box ends where periodic boundary condi-

tions are used, the results are free from the boundary

errors, but may depend on the initialization error.

Analysis of the moments of the Gaussian distribution,

similar to those above for concentration waves, could

verify the analytical solutions obtained for high order

convection and diffusion errors. We chose here to study

L2-global distributions errors obtained with optimaldiffusion/convection strategies,

E2LðsÞ ¼

Piðsi � sthi Þ

2Piðsthi Þ

2; EL ¼

ffiffiffiffiffiffiE2

L

q. ð55Þ

The summation is taken over all lattice points. Follow-

ing [1], we consider the dispersion tensor in Eq. (3) in

the form:

Dab ¼ k~UkðkT dab þ ðkL � kT ÞuaubÞ; ua ¼U a

k~Uk. ð56Þ

Here, k~Uk is the absolute value of point-wise velocity ~U .

In the streamline coordinate system (x 0,y 0,z 0), the

dispersion tensor is diagonal: Dx0x0 ¼ k~UkkL; Dy0y0 ¼k~UkkT ; Dz0z0 ¼ k~UkkT . We explore the TRT E-modelwith a(e) = 0. The parameter kL is fixed as:

kL ¼ 1

3k~UkKðkDÞ; kD ¼ k0D, so that the isotropic case is

Dab ¼ kLk~Ukdab ¼ m0Ddab, c2s ¼ 1

3. According to rel. (27),

the parameter c2s is derived from the condition a(e) = 0:

c2sKðk

0DÞ ¼ 1

d

PaDaa ¼ k~Uk

d ð2kT þ kLÞ. In this way, c2s var-

ies between 13and 1

9when kT goes from kL to zero. When

the k~Uk varies, the LB dispersion tensor is not modified

because of the construction of kL. This enables us to

examine the convection/diffusion parts of the errors.

Each computation is performed with two strategies for

ke: ‘‘diffusion dominant’’ ke ¼ kopte ðk

0DÞ and ‘‘convection

dominant’’ ke ¼ k0D.

We consider the 2D case ~U ¼ k~Ukfcos h; 0; sin hg in a

[X · Z] = 1002 box, with periodic boundary conditions

along y, and 3D case in 503 box in 3D case. When

velocity is along x-axis, the limit kT! 0, k~UkkL ¼ dmD

corresponds to a diffusion tensor with only one non-

zero element Dxx. When h ¼ p4;Dxx ¼ Dzz ¼ 1

2ðkL þ kT Þ;

0 50 100 150 200TIME STEPS

0

0.005

0.01

0.015

0.02BGK, U=0.1, n=20BGK,U=0.05, n=20OPT, U=0.1, n=20OPT, U=0.05, n=20

Distribution error in 2D

0 50 100TIME STEPS

0

0.005

0.01

0.015

0.02

EL-E

RR

OR

EL-E

RR

OR

BGK, |U|=0.1, n=0BGK, |U|=0.05, n=0OPT, |U|=0.1, n=0OPT, |U|=0.05, n=0BGK, |U|=0.1, n=20BGK, |U|=0.05, n=20OPT, |U|=0.1, n=20OPT, |U|=0.05, n=20

Distribution error in 3D

Fig. 8. Distribution error EL(s) for a Gaussian hill in case of a

dispersion tensor with kL/kT = 2n, k~Uk ¼ 0.05; 0.1. The ‘‘optimal

diffusion’’ solution is labeled ‘‘OPT’’. ‘‘Optimal convection’’ solution

is labeled as ‘‘BGK’’. Top: ~U ¼ k~Ukf1; 0; 0g, n = 20. Bottom:~U ¼ k~Ukf2

3; 13; 23g, n = 0, 20.


Dyy ¼ kT ;Dxz ¼ 12ðkL � kT Þ. When kT/kL! 0, the off-

diagonal coefficient approaches the diagonal coefficient,

i.e another stability limit is concerned. A similar situa-

tion happens for ~U ¼ k~Ukffiffi3

p f1; 1; 1g.The experiments were first carried out for kL = 2nkT,

n = 0, 2, 4, 10, 20, k~Uk ¼ 0.1, each for three velocities in2D, h 2 f0; p

6; p4g, t < 200, and for three velocities in 3D,

~U ¼ k~Ukf1; 0; 0g; ~U ¼ k~Uk3f2; 1; 2g; ~U ¼ k~Ukffiffi

3p f1; 1; 1g; t

< 100. The distribution error appears to be smaller withthe second (‘‘advection’’) strategy in all considered cases

except for h ¼ p4

when kT! 0. When the velocity is

reduced by a factor 2 the error with the diffusion strategy

reduces as well, by a factor 2 in isotropic cases. At the

same time, the error with the ‘‘convection’’ strategy stays

nearly constant. This confirms that in the former case,

when the main diffusion error is removed by

ke ¼ kopte ðk

0DÞ, the rest error ismainly due to advection and

in the latter case, everything is almost reversed. Fig. 8

plots the error EL(s(t)) for h = 0 in 2D and ~U ¼k~Uk3f2; 1; 2g in 3D to illustrate these observations.

Estimations of the errors are confirmed by approxi-

mate contour plots for both theoretical and numerical

solutions in Figs. 9 and 10. The points (x,z) which sat-

isfy the condition s > 10�3 and have max/min x-coordi-nates when z is fixed, are plotted. Fig. 9 demonstrates

the enhanced accuracy of the convection strategy over

the diffusion one in the most favorable case for advec-

tion strategy when diffusion is absent (kL = 0, kT = 0).

We emphasize that here a(e) = �1 and the model does

not encounter difficulties in dealing with pure advection

problems in this test. Fig. 10 illustrates the evolution of

the Gaussian hill for the highest ratio, kL/kT = 220, h = 0and h = p/4 as obtained with the ‘‘convection’’ strategy.

Here, the difference between two strategies is not as sig-

nificant as in the cases of pure advection since the diffu-

sion error of the convection strategy appears as well. On

the whole, numerical results confirm the predictions

with respect to error behavior in convection-dominant

and diffusion-dominant regimes and demonstrate high

accuracy even in extremely anisotropic situations.

7.6. Quasi-linear diffusion equations

Most problems deal with solution-dependent diffu-

sion coefficients. As a test case, let us consider the quasi-

linear diffusion equation

osot� 1

~d þ 1

o2s~dþ1

ox2¼ 0; sðx; 0Þ ¼ 0; ~d > 0; ð57Þ

when time-dependent boundary conditions are applied

sð0; tÞ ¼ ð~dtÞ1~d . ð58Þ

The exact solution of this problem is referred to in the

literature as temperature wave (e.g., [25]):

sðx; tÞ ¼ ½~dðt � xÞ�1~d ; x 6 t;

0; x P t.

(ð59Þ

7.6.1. Particular exact boundary solution

When all eigenvalues are equal to �1, the evolution

equation (6) without source becomes fqð~r þ ~Cq; tþ1Þ ¼ f eq.

q ð~r; tÞ. Let us assume that the boundary node

~rb is shifted by one lattice unit from the last internal

nodes. When the incoming populations carry the equi-

librium solution f eq.q ð~rb; tÞ, the bulk solution is main-

tained exactly. Moreover, when this set-up is satisfiedbut the obtained solution differs from the known refer-

ence solution, that means that the LB equation is not

able to maintain the reference solution exactly.

7.6.2. Equilibrium solution

Here we put Dab ¼ 1ð~dþ1Þ s

~dþ1dab and assume the TRT

collision. According to solution (25), we set �Deq.ðsÞ ¼

s, a(e) = sa(e), aðeÞ ¼ 1mD

1ð~dþ1Þ s

~d � 1:

-20 -10 0 10 20 30 40 50 60 70X

-30

-20

-10

0

10

20

30

40

Z

theory: T=0, T=100, T=200T=100T=200

-40 -30 -20 -10 0 10 20 30 40X

-20

-10

0

10

20

30

40

Z

theory: T=0, T=100, T=200T=100T=200

Fig. 10. Approximate contour plots s > 10�3 for 2D Gaussian

distribution in case of the dispersion tensor with k~Uk ¼ 0.1, kL/

kT = 220. Data is obtained with ‘‘optimal convection’’ strategy. Top

row: h = 0. Bottom row: h = p/4.

-40 -30 -20 -10 0 10 20 30 40X

-30

-20

-10

0

10

20

30

Z

theory: T=0, T=100, T=200T=100T=200

Pure convection with optimal convection solution

-40 -30 -20 -10 0 10 20 30 40X

-30

-20

-10

0

10

20

30

Z

theory: T=0, T=100, T=200T=100T=200

Pure convection with optimal diffusion solution

Fig. 9. Approximate contour plots s > 10�3 for 2D Gaussian distri-

bution in case of pure advection, U = 0.1, h = p/4, kL = kT = 0,

a(e) = �1, c2s � 19. Dotted lines correspond to the numerical solution.

Top row: ‘‘optimal convection’’ strategy. Bottom row: ‘‘optimal

diffusion’’ strategy.


f eq.q ðsÞ ¼ ~tqsþ

1

ð~d þ 1Þpqs

~dþ1; ~tq ¼ tq �c2s

bðeÞpðeÞq ;

pq ¼1

mD

c2sbðeÞ

pðeÞq . ð60Þ

The exact solution (59) in case ~d ¼ 1 yields

ots = �oxs. A Chapman–Enskog expansion of the solu-tion after the front (x < t) takes the form

fq ¼ f eq.q þ f ð1Þq þ f ð2Þq þ f ð3Þq ;

f ð1Þq ¼ 1

kDoxs~tqCqx þ

1

kDsoxspqCqx;

f ð2Þq ¼ 1

kesotspq þ

1

2þ 1

kD

� ��foxðsoxsÞðpqC

2qx �~tqÞ þ otsoxspqCqxg

�f ð3Þq ¼ 1

2þ 1

ke

� �1

kDotsoxspqCqx þ

1

keðotsÞ2pq

� �.

ð61Þ

Since next-order terms vanish, third order expansion

represents the exact solution. It does not introduce

higher order terms to Eq. (57) and therefore, provides

the exact solution (59). Starting from the solution (61)

at time moment t = T0, T0 P l, 0 6 x 6 l, the exact solu-

tion (59) will be obtained at any t and any x provided

that boundary conditions maintain the solution (61)exactly. For this purpose, one can compute explicitly

incoming populations in the form (61) or use a particu-

lar BGK set-up described above. This configuration is

referred to as a ‘‘solution without front’’.

Because of the discontinuity of the first order deriva-

tives, populations ahead of the front (x > t) cannot

transport exact solution behind the front. When we

compute them with rel. (61) (or as the equilibrium solu-tion coming from the node ahead of the front with the

particular BGK set-up above), the solution is referred

to as ‘‘solution with boundary conditions at the front’’.

Since the populations are reset at the nearest to front

nodes, no impact of the solution behind the front on

the solution ahead it appears. Numerical computations

confirm that when both boundary limits are treated

exactly, the obtained solutions are exact.

0 10 20 30 40 50X

0

50

100

Tem

pera

ture

wav

e so

lutio

n, s

(T) t=1000, t=11000

t=3000, t=13000t=5000, t=17000t=7000, t=21000t=9000

Exact solution with equilibrium approach

Fig. 11. Exact temperature wave solution obtained with the equilib-

rium approach in case of constant eigenvalues kD = ke = �1, c2s ¼ 13.

Data: ~d ¼ 1, l = 50, T = 200, L = 1. Start at t0 = 103. Imaginary

boundary nodes are found at x = 0 and x = 49.

0 10 20 30 40 50X

-0.05

0

0.05

0.1

0.15

0.2

Abs

olut

e er

ror

to e

xact

sol

utio

n

t=3000t=5000t=7000t=9000

Equilibrium approach

0 10 20 30 40 50X

-0.05

0

0.05

0.1

0.15

0.2

Abs

olut

e er

ror

to e

xact

sol

utio

n

t=3000t=5000t=7000t=9000

Eigenvalue approach

Fig. 12. The absolute error to the exact solution obtained without the

reset of the populations at boundary nodes after the front. The

simulations run through the whole domain 0 < x < l. Same data as in

picture Fig. 11. Start time t0 = 104. Top: Equilibrium approach.

Bottom: Eigenvalue approach.


In order to improve their stability we apply time/

space scalings (t! t/T, x! x/L). An example is plotted

in Fig. 11. The diffusion coefficient is rescaled by a

pre-factor L2/T and its highest value in the case ~d ¼ 1

is L2t/2T2 at x = 0. The stability condition c2s ð1þ aðeÞÞ

< 1 determines the stable domain as t < KðkDÞ2

T 2

L2 . Related

to highest value of the diffusion coefficient this criterion

has a certain analogy with well known diffusion-domi-

nant criterion for explicit methods (e.g, [25, p. 593]).

With or without front (t0 < Tl/L, or t0 P Tl/L), solu-

tions lose stability approximately at t = K2(kD)T2/2 for

different T and kD values. The analytical form of the

stability curve has not yet be derived.When the population solution is not updated at the

front, the solution is not exact any more. The error is

concentrated however in only two nodes, one just before

and one after the front; other nodes ahead the front con-

tinue to carry zero solution. The absolute error with re-

spect to the exact solution is plotted in Fig. 12. We stress

here that even for a diffusion coefficient equal to zero,

the stability domain of the solution is not altered in thistest.

7.6.3. Eigenvalue solution

As an example, let us consider the TRT operator with

mD ¼ s~d , ke = �1 and conventional equilibrium function�D

eq.ðsÞ ¼ s, f as. = 0. The Chapman–Enskog expansion

yields, sequentially

fq ¼ f eq.q þ f ð1Þq þ f ð2Þq þ ; f eq.

q ¼ tqs

f ð1Þq ¼ 1

kDðsÞoxstqCqx;

f ð2Þq ¼ 1

keðotstq � oxKðkDÞoxstqC

2qxÞ.

ð62Þ

Higher order terms consequently need to maintain a

space/time variation of k�1D ðsÞ. For ~d ¼ 1; 1

kD¼ � s

c2s� 1

2;

osk�1D ! 0 when c2

s !1. Since we do not know the

exact form of the population solution in this case,

boundary populations are only approximated by second

order expansion (62). The obtained solution is not exact

and its relative error with respect to the exact solution is

plotted in Fig. 13 in cases c2s ¼ 1=3 and c2s ¼ 1. In agree-

ment with the predictions, the error is inversely propor-tional to c2

s . The obtained solution does not depend on

the start time t0. Stability conditions for eigenvalues

are satisfied unless the solution s(x, t) is negative. We

emphasize that no loss of stability has been detected

far away from the stability domain of the equilibrium-

type model above. Moreover, when the algorithm runs

across the whole domain 0 < x < l, negative solution val-

ues appear (eigenvalues leave their stability interval]�2,0[) but no direct impact on the solution is detected.

The obtained solution ‘‘with a front’’ is compared to the

equilibrium-type solution in Fig. 12. Although the

amplitude of the error near the front is similar in both

cases, very small (of order 10�5) mass dissipation is

0 10 20 30 40 50X (l.u.)

-2e-05

-1,5e-05

-1e-05

-5e-06

0

Rel

ativ

e er

ror

t=14000t=22000t=34000

0 10 20 30 40 50X (l.u.)

-8e-06

-6e-06

-4e-06

-2e-06

0

Rel

ativ

e er

ror

t=14000t=22000t=34000

Fig. 13. Relative error of the model with variable eigenvalues. Same

data as in picture Fig. 11. Start time t0 = 104. Top: c2s ¼ 13. Bottom:

c2s ¼ 1.


spread ahead of the front. Some significant differenceappears here between two models since the equilib-

rium-type solution stays equal to zero ahead of the

front.

Let us conclude with the following remarks. In case

of constant eigenvalues, the exact population solution

is constructed when the diffusion coefficient is a linear

function of the solution. When ~d 6¼ 1, higher order terms

(e.g otxxs, . . .) may introduce a correction to Eq. (57). Weassume that the population solutions can however be

constructed in the form of a finite series, at least when~d ¼ 1

n ; n ¼ 1; 2; . . . and the higher order corrections

can then be quantified. Regarding stability, it appears

that the eigenvalue approach is more stable when the

diffusion coefficients are unbounded.

8. Conclusion

The generalized lattice Boltzmann equation method

is applied either with conventional polynomial basis vec-

tors or with link vectors. The projections on a pair of

link vectors are the symmetric and the anti-symmetric

parts of a pair of populations with opposite velocities.

Link models are restricted to having only two relaxation

parameters, one for the symmetric and one for the anti-

symmetric collision part when both mass and momen-

tum are conserved. When the momentum conservation

is not needed, the eigenvalues of the anti-symmetric link

basic vectors can be tuned to match the coefficients ofthe full symmetric diffusion tensor. The separation of

the BGK operator on the symmetric and anti-symmetric

collision parts enables us to combine mass conserving

equilibrium functions and distinct link eigenvalues in

the context of the L-model. The L-model improves

and extends the BGK-types models of Zhang et al.

[32,33]. With an alternative, equilibrium-type approach

(E-model), the transform of the diffusion tensor is builtwith the help of the equilibrium projections on the sec-

ond order polynomial basis vectors. The E-model works

with any collision operator. For the two-relaxation-time

(TRT) operator, ‘‘optimal’’ combinations of even/odd

eigenvalues allow to remove fourth order corrections

to advection or to diffusion terms, at least for a linear

convection–diffusion equation. The numerical results

confirm the analytical predictions.Unlike the L-model, the E-model can remove the ten-

sor of the numerical diffusion induced by the advection

equilibrium term in general cases and take advantage of

the TRT configuration even for anisotropic tensors. In

terms of robustness with respect to the high anisotropy

of the diffusion tensor, the E-model seems to surpass

the L-model, at least when both approaches are based

on the present strategies for free parameters and diffu-sion coefficients are constant and continuous. Aniso-

tropic and discontinuous diffusion tensors appear, for

instance, in the case of heterogeneous anisotropic soils

or in the case of non-uniform coordinate transforma-

tions in sub-domains. Preliminary results in [12] for

modeling of variably saturated flow on the anisotropic

layered grids give preference to L-model in case of the

discontinuities. Additional degrees of freedom of thecombination of link-type collision and expanded equi-

librium functions are not yet explored.

The MRT operator and Link-operator coincide in

the two-relaxation-time collision framework. Due to the

specific form of the equilibrium function we use, the

eigenvalues of the TRT collision can be chosen arbi-

trarily. We then distinguish the conventional (eigen-

value) approach when the diffusion coefficient ismatched by the eigenvalue function, and the equilibrium

approach, when the specific equilibrium projection cov-

ers the integral transform of the diffusion function. With

the current stability criteria, the two techniques comple-

ment each other. The eigenvalue approach seems to be

more suitable for diffusion-dominant problems and the

equilibrium approach for advection-dominant (high

Peclet numbers) problems. Future work remains to bedone to go beyond the heuristic arguments employed

for this stability issue.


Acknowledgments

The author is grateful to D. d�Humieres for the eigen-

mode analysis of the TRT E-model and much advice

and to M. Krafczyk for the critical reading of the

manuscript. Thanks go to Y. Nedelec, F. Goeta andJ.P. Carlier for technical support.

Appendix A. Basic vectors

We separate the polynomial basis vectors into the

following families of integer vectors:

e0 ¼ f1g;

fCag ¼ fCqag; a ¼ 1; . . . ; d;

pðeÞ ¼ fQc2q � b2g; c2

q ¼ k~Cqk2;

pðxxÞ ¼ fdC2qx � c2

qg;

pðwwÞ ¼ fC2qy � C2

qzg; d ¼ 3;

fpðabÞg ¼ fCqaCqb; a 6¼ bg;

fhð1Þa g ¼ fðb1c2q � b3ÞCqag; a ¼ 1; . . . ; d;

fhð2Þa g ¼ fðC2qc � C2

qbÞCqag; a 6¼ b 6¼ c; d ¼ 3;

hðxyzÞ ¼ fCqxCqyCqz; d ¼ 3g;

pðeÞ ¼ fa1ðQc4q � b6Þ þ a2ðQc2

q � b2Þg;

pðxxÞ ¼ fðB1c2q � B2ÞðdC2

qx � c2qÞg;

pðwwÞ ¼ fðB3c2q � B4ÞðC2

qy � C2qzÞg; d ¼ 3.

ðA:1ÞThe basis vectors can be obtained by orthogonaliza-

tion of the subsequent order Ca-polynomials followingthe Gram–Schmidt procedure. Not all families (A.1)

are present in each DdQq model [24,4]. Table 2 indicates

the number of non-vanishing basis vectors for D2Q5,

D2Q7, D2Q9, D3Q13, D3Q15 and D3Q19 models. The

mass vector e0 is a zero order polynomial, Ca are first

order ones, P2 = {p(e),p(ab),p(xx),p(ww)} are second order

ones, P3 ¼ fhð1Þa ; hð2Þa ; hðxyzÞg are third order ones and

P4 = {p(e),p(xx),p(ww)} are fourth order ones. In a natu-

ral way, the basis vectors are separated into two subsets:

subset P+ of even-order (or symmetric) polynomial vec-tors and subset P� of odd-order (or anti-symmetric)

polynomial vectors:

P¼P� [Pþ;P� 2 fCa; hð1Þa ; hðxyzÞ; hð2Þa g; a¼ 1; . . . ; d;

Pþ 2 fe0; pðeÞ; pðxxÞ; pðwwÞ; pðabÞ; pðeÞ; pðxxÞ;pðwwÞg.ðA:2Þ

The subset P� contains a number of vectors equal to

half the number of non-zero velocities. The subset P+

has one more vector. The corresponding eigenvalues

are referred to as ‘‘odd’’ and ‘‘even’’, respectively.

The constants b1, b2, b3, b6 are given by the following

relations:

b1 ¼XQ�1

q¼1

C2qa; b2 ¼

XQ�1

q¼1

c2q;

b3 ¼XQ�1

q¼1

c2qC

2qa; b6 ¼

XQ�1

q¼1

c4q. ðA:3Þ

The coefficient a1 in rel. (A.1) can be found from the

orthogonality condition for p(e) and p(e). One can set

a1 = kp(e)k2, a2 ¼ �PQ�1

q¼0 ðQc2q � b2Þ ðQc4q � b6Þ. In prac-

tice, we divide the generic basis vectors (A.1) by some

integers to keep the vectors with the smallest integer

components. Similarly, the coefficients B1 �B4 can be

found from mass conservation and orthogonality condi-tions.Here, they are presented inD3Q19model only, with

B1 = B3 = 3, B2 = B4 = 5. Substituting the basis vectors

into the following relations, one can relate the coefficients

of equilibrium part f as. in (19) to constants of the specific

velocity distribution:

bðeÞ ¼XQ�1

q¼1

pðeÞq C2qa ¼ Qb3 � b2b1; 8a;

bðabÞ ¼XQ�1

q¼1

pðabÞq CqaCqb ¼

XQ�1

q¼0

C2qaC

2qb; a 6¼ b;

b5 ¼XQ�1

q¼1

pðxxÞq C2qb ¼ dbðabÞ � b3; b 6¼ x;

bðxxÞ ¼XQ�1

q¼1

pðxxÞq C2qx ¼ db1 � b3;

bðxxÞD ¼ b5

bðxxÞ; bðxxÞa ¼ 1

1� bðxxÞD

;

bðwwÞ ¼XQ�1

q¼1

pðwwÞq C2

qy ¼ �XQ�1

q¼1

pðwwÞq C2

qz ¼ b1 � bðabÞ.

ðA:4Þ

Appendix B. Dispersion relation

When the equilibrium distribution f eq. is a linear

function of the conserved quantities, the solutions of

the evolution equation (5) can be sought as f 0 expði½~k ð~r � ~UtÞ � xt�Þ, where xð~kÞ ¼ �imð~kÞk2; mð~kÞ ¼

Pa;b

Dabkakb

k2and ~k ¼ ðkx; ky ; kzÞ ¼ kðcos h sin/; sin h sin/;

cos/Þ is a given wave vector. Also, z ¼ expð�iðxþ~k ~UÞtÞ and f0 are the eigenvalues and eigenvectors of

the linear operator N ¼ S�1 ðIþ A ðI� Eeq.ÞÞ; f eq. ¼Eeq. f ; S ¼ diagðexpð~k ~CjÞÞ (note that in this context


i is the complex square root of �1). The TRT model

with the eigenvalues {kD,ke} and the equilibrium func-

tion (19), �Deq.ðsÞ ¼ s, is assumed. The eigenvalue solu-

tions are obtained by D. d�Humieres in an analytical

form for first terms in their expansion (B.1) and

(B.13), for diffusion and advection regimes, respectively.Approximate expressions for the eigenvalues within the

limit of small k for the transport coefficients of the

hydrodynamic equations can be found in [20,31].

B.1. Diffusion solution

First we consider the case without advection. Here

and below, mth. is related to the eigenvalue solution ofthe linear operator N, expanded in powers of k2

mth.ð~kÞ ¼ mð~kÞð1þ mðrÞ1 ð~kÞk2 þ þ mðrÞn ð~kÞk2n þ Þ.ðB:1Þ

At the leading order, the relative error E(r)(m)/k2 is

equal to mðrÞ1 ð~kÞ. For the TRT model, with the help of

Mathematica 5.0, we obtain

mðrÞ1 ð~kÞ ¼ f1ðc2s ; kD; keÞ þ c2

s f2ðkD; keÞmð~kÞmD

� 1

!þ gðkD; keÞGðDab;~kÞ;

f0ðkDÞ ¼ ð6þ 6kD þ k2DÞ=ð6k

2DÞ;

f1ðc2s ; kD; keÞ ¼ c2s f0ðkDÞ þ ð1� c2

s ÞgðkD; keÞ;

f2ðkD; keÞ ¼ f0ðkDÞ � gðkD; keÞ;

gðkD; keÞ ¼ �3

4K2ðkD; keÞ �

2

9

� �;

K2ðkD; keÞ ¼4

3KðkeÞKðkDÞ.

ðB:2Þ

Here, K(k) is defined by (23) and the factor 43in the def-

inition of the function K2(kD,ke) is due to historical nota-

tion (see in [9]). In the isotropic case Dab = mdab, mð~kÞ ¼ m

and mð~kÞmD� 1 in the second term reduces to a(e) according

to rel. (25). However, the function GðDab;~kÞ is aniso-

tropic (depends on the direction of the ~k) even when

{Dab} is an isotropic tensor:

GðDab;~kÞ ¼1

mð~kÞk4ð3mDaðeÞðk4 � k4

aÞ

þX

a

ðDaa � mDð1þ aðeÞÞÞk4a

þ ðmDð1þ aðeÞÞ � mð~kÞÞk4

þ 2X

a6¼b6¼c

Dabkakbðk2a þ k2

b þ 3k2cÞÞ ðB:3Þ

This function vanishes when the diffusion tensor is

isotropic and a(e) = 0. Otherwise, it vanishes when ~k is

parallel to one of the coordinate axes.

B.1.1. ‘‘Optimal diffusion’’ solution

In order to annihilate the anisotropic part of mðrÞ1 ð~kÞ,we look here for the solution of g(kD,ke) = 0, corre-

sponding to K2ðkD; keÞ ¼ 29. The resulting function

kopte ðkDÞ is called the ‘‘optimal diffusion’’ solution for ke:

kopte ðkDÞ ¼ �6ð2þ kDÞ=ð6þ kDÞ; then

mðrÞ1 ð~kÞ ¼ f0ðkDÞmð~kÞ

KðkDÞ. ðB:4Þ

Interesting properties of this solution for fixed kD are

outlined. The leading order relative error EðrÞðmð~kÞÞ is

linearly proportional to mð~kÞ. For any c2s the error takes

the same value when kD is kept. In the isotropic case,

EðrÞðmð~kÞÞ is linearly proportional to c2s ðaðeÞ þ 1Þ. When

the wave vector is parallel to a coordinate axis, say a,EðrÞðmð~kÞÞ reduces to E(r)(Daa) and its value is propor-

tional to Daa/K(kD):

mðrÞ1 ðDaaÞ ¼ c2s f0ðkDÞRa; Ra ¼Daa

mD. ðB:5Þ

Note that mðrÞ1 ¼ 0 for any ~k, c2s and a(e) only when

kD ¼ k0D ¼ �3þ

ffiffiffi3

p; then f 0ðk0

DÞ ¼ 0. ðB:6ÞWhen kD < k0

D, mðrÞ1 ð~kÞ is negative and stability can be

destroyed for large values of k2.

B.1.2. Isotropic case with a(e) = 0

Under these conditions, mðrÞ1 ð~kÞ ¼ f1ðc2s ; kD; keÞ. Then

f1ðc2s ; kD; keÞ ¼ 0 defines a relation between c2s , ke, and

kD in which one parameter can be taken as a function

of the two others, such as for instance

gðkD; keÞ ¼ �c2s

ð1� c2s Þ

f0ðkDÞ; then

kise ðkD; c2

s Þ ¼6kDð2þ kDÞð1� c2

s Þ12c2

s � kDð6þ kDÞð1� 3c2s Þ

. ðB:7Þ

Note that for c2s ¼ 1=3, this relation simplifies to

ke = kD(2 + kD) and that for kD ¼ k0D, the solutions

kopte ðkDÞ and kis

e ðkDÞ coincide for any c2s . Otherwise,

kise ðkDÞ ! kopt

e ðkDÞ when c2s ! 0. In contrast to the opti-

mal set-up above, mðrÞ1 ð~kÞ may be canceled for any kDwhen a(e) = 0 and the tensor is isotropic.

When the wave vector is parallel to a-axis and the

solution (B.7) is used for an anisotropic situation,

E(r)(Daa) is proportional to Ra � 1:

mðr;1Þ1 ðDaaÞ ¼ ðf0ðkDÞ � gðkD; kise ðkDÞÞðRa � 1Þ: ðB:8Þ

It vanishes when mD = Daa or kD ¼ k0D.

The next order has also been computed. Since this

order is relevant only if the previous one is zero, for

the sake of simplicity the formulas will be given only

for a(e) = 0 and ke given by the relation (B.7) as a func-tion of c2

s and mD ¼ c2sKðkDÞ:


mðrÞ2 ðh;/Þ¼�c2s ð3�13c2

sþ15c4s Þ�60ð1�4c2

sþ5c4s Þm2Dþ720m4D

720c2s ð1�c2s Þ

�c4s ð3�6c2

s�2c4s Þþ120m2Dðc4s�6m2DÞ

540c4s ð1�c2

s Þ2

�34cos2/þsin2/sin22hð10�9sin2/Þ

8sin2/.

ðB:9ÞThe anisotropic part vanishes for

c2s ¼

ffiffiffiffiffi15

p 3

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16ð5�

ffiffiffiffiffi15

pÞm2D

q� 1

� ��4 or

mD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2s ðc2

s þffiffiffiffiffiffiffiffi3=5

pð1� c2s ÞÞ=12

q.

ðB:10ÞFinally if mD is fixed by (B.10) and c2

s is

c2s ¼ 2 3� 4

ffiffiffiffiffi15

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi765þ 212

ffiffiffiffiffi15

pq� �� 77; ðB:11Þ

the isotropic part also vanishes, leading to a sixth-order

error in k.

B.1.3. Anisotropic case with kise and a(e) = 0

Let us consider what happens when the solution (B.7)is used in anisotropic situations. When a(e) = 0, the diffu-

sion tensor in two dimensions takes the form:

Dxx = mD(1 + a(xx)), Dyy = mD(1 �a(xx)), Dxy = Dyx =

mDa(xy). Let us introduce the anisotropy parameter ~aand angle a such that: aðxxÞ ¼ ~a cos a and aðxyÞ ¼ ~a sin a.

In these notations, mð~kÞ ¼ mDð1þ ~a cosða� 2hÞÞ and

mðrÞ1 ð~kÞ becomes

mðrÞ1 ð~kÞ ¼c2s f0ðkDÞ

ð1þ ~a cosða� 2hÞÞ

� ð~acos2ða� 2hÞ þ cosða� 2hÞ þ cosðaþ 2hÞÞ

2ð1� c2s Þ

.

ðB:12ÞOnce again, it vanishes when kD ¼ k0

D for any wave vec-

tor ~k.

B.2. ‘‘Optimal convection’’ solution

A similar analysis for the advection term gives

~k ~U th. ¼~k ~Uð1þ uðrÞ1 ð~kÞk2 þ þ uðrÞn ð~kÞk2n þ Þ;ðB:13Þ

where

uðrÞ1 ð~kÞ ¼ gðuÞðkD; keÞð~k ~UÞ2

k2� 1

!þ c2

s fðuÞ2 ðkD; keÞ

mð~kÞmD

;

gðuÞðkD; keÞ ¼ �1

12ð1� 9K2ðkD; keÞÞ;

f ðuÞ2 ðkD; keÞ ¼ �1

4ð1� 3K2ðkD; keÞ � 6K2ðkD; kDÞÞ.

ðB:14Þ

The error uðrÞ1 ð~kÞ vanishes for the BGK configuration

only:

kðuÞe ðkDÞ ¼ kD when K2ðkD; kDÞ ¼1

9. ðB:15Þ

Once again this leads to kD ¼ k0D. The ‘‘optimal convec-

tion’’ solution kðuÞe ¼ k0D is different however from the

‘‘optimal diffusion’’ solution K2ðkD; kopte Þ ¼ 2

9; kopt

e ðk0DÞ �

�0.928203.

Appendix C. Macroscopic equations: details

According to rel. (11)–(13), f (1) and f (2) can be ob-

tained by inversion of the corresponding Taylor expan-

sions, eT(1) and e2T(2) (see also the discussion before

Eq. (12)). In order to do so, we project the Taylor expan-

sion into the moment space. Let us consider the equilib-

rium function in its projected form, f eq. ¼P

k2KðPeq.Þbfkð0Þek; Peq. ¼ fek : bfk

ð0Þ6¼ 0g and X0 = P(0) \ Peq.,

P(0) = {ek: kk 5 0}. We define as pnak the projection of

the column vector Caek on the arbitrary vector

en : pnak ¼ hCaekjeni. For a given vector en, the subset of

basis vectors with a non-zero projection pnak is noted as

Pna;P

na ¼ fek : pn

ak 6¼ 0g. Let Xna be its restriction to the

equilibrium basis vectors, Xna ¼ Pn

a \Peq.. Based on

these definitions, we represent eT(1) as:

eTð1Þ ¼ eX

k2KðPeq.Þekot1

bfkð0Þþ eTrð1Þ;

eTrð1Þ ¼ eCbob0 fð0Þ ¼ e

XQ�1

j¼0

ej

Xk2KðXj

bÞ

pjbkob0

bfkð0Þ

.

ðC:1Þ

We substitute rel. (C.1) into f (1) = A�1 Æ eT (1) and com-

pute ðIþ 12AÞ f ð1Þ ¼ ðA�1 þ 1

2IÞ eTð1Þ. Using the nota-

tion KðkÞ ¼ �ð12þ 1

kÞ, we obtain:

Iþ 1

2A

� � f ð1Þ ¼ �e

Xk2KðX0Þ

KðkkÞekot1bfkð0Þ

ðC:2Þ

¼ �eX

j2KðPð0ÞÞ

ej

Xk2KðXj

bÞ

KðkjÞpjbkob0

bfkð0Þ;

ðC:3Þ

A Taylor expansion in space of this term,

eTrð2Þ ¼ eCaoa0 ðIþ 12AÞ f ð1Þ, is

eTrð2Þ ¼ e2ðTrtð2Þ þ Trrð2ÞÞ;

Trtð2Þ ¼ �XQ�1

j¼0

ej

Xk2KðXj

aÞ

pjakoa0KðkkÞot1

bfkð0Þ;

Trrð2Þ ¼ �XQ�1

m¼0

em

Xj2KðPm

a \Pð0ÞÞ

pmaj

Xk2KðXj

bÞ

pjbkoa0KðkjÞob0

bfkð0Þ

.

ðC:4Þ


and its projection on the conserved vector en becomes:

eTrð2Þ en¼ e2ðF tð2Þn þF rð2Þ

n Þ; F tð2Þn ¼Trtð2Þ en;

F rð2Þn ¼Trrð2Þ en;

F tð2Þn ¼�

XQ�1

j¼0

ej en

Xk2KðXj

aÞ

pjakoa0KðkkÞot1

bfkð0Þ;

F rð2Þn ¼�

XQ�1

m¼0

em en

Xj2KðPm

a \Pð0ÞÞ

pmaj

Xk2KðXj

bÞ

pjbkoa0KðkjÞob0

bfkð0Þ;

ðC:5ÞWhen en is one of the eigenvectors, en 2 P, relations

(C.5) reduce to rel. (16).

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LBM GenericAdvection Ginzburg 2005

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