Download - 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
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)
1174 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1175
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
1176 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1177
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
1178 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1179
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
1180 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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):
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1181
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.
1182 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1183
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 0.1 0.2 0.3 0.4 0.5Diffusion coefficient ν
0
1
theory: ν1(r)
( ν) κ={1,0} κ={1,2} κ={1,1} κ={1,2}, a
e=0
Isotropic diffusion tensor
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 0.1 0.2 0.3 0.4 0.5Diffusion coefficient ν
-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}
Isotropic diffusion tensor
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
Anisotropic diffusion tensor
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.
1184 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
Anisotropic diffusion tensor
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
Anisotropic diffusion tensor
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}
Anisotropic diffusion tensor
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).
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1185
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 ~ ~
1186 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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-
0 0.1 0.2 0.3 0.4 0.5Diffusion coefficient ν
-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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1187
�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
0D�. 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.
1188 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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.
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1189
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.
1190 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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.
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1191
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.
1192 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1193
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:
1194 I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195
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
I. Ginzburg / Advances in Water Resources 28 (2005) 1171–1195 1195
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).
References
[1] Bear J. Hydraulics of groundwater. New York: McGraw-Hill;
1979.
[2] Celia M, Bouloutas E, Zabra R. A general mass-conservative
numerical solution for the unsaturated flow equation. Water
Resour Res 1990;26:1483–96.
[3] d�Humieres D. Generalized lattice Boltzmann equations. AIAA
rarefied gas dynamics: theory and simulations. Prog Astronaut
Aeronaut 1992;59:450–548.
[4] d�Humieres D, Bouzidi M, Lallemand P. Thirteen-velocity three-
dimensional lattice Boltzmann model. Phys Rev E
2001;63:066702-1–7.
[5] d�Humieres D, Ginzburg I, Krafczyk M, Lallemand P, Luo LS.
Multiple-relaxation-time lattice Boltzmann models in three
dimensions. Phil Trans R Soc Lond A 2002;360:437–51.
[6] Flekkoy EG, Oxaal U, Feder J, Jossang T. Hydrodynamic
dispersion at stagnation points—simulations and experiments.
Phys Rev E 1995;52(5):4952–62.
[7] Frisch U, d�Humieres D, Hasslacher B, Lallemand P, Pomeau Y,
Rivet JP. Lattice gas hydrodynamics in two and three dimensions.
Complex Syst 1987;1:649–707.
[8] Ginzburg I, Steiner K. Lattice Boltzmann model for free-surface
flow and its application to filling process in casting. J Comp Phys
2003;185:61–99.
[9] Ginzburg I, d�Humieres D. Multi-reflection boundary condi-
tions for lattice Boltzmann models. Phys Rev E 2003;68:
066614-1–30.
[10] Ginzburg I, Carlier J-P, Kao C. Lattice Boltzmann approach to
Richards� equation. In: Miller CT, editor. Proceedings of the
CMWR XV, June 13–17, Computational methods in water
resources. Chapel Hill, NC, USA: Elsevier; 2004. p. 583–97.
[11] Ginzburg I. Generic boundary conditions for lattice Boltzmann
models and their application to advection and anisotropic-
dispersion equations, in press, doi:10.1016/j.advwatres.2005.
03.009.
[12] Ginzburg I. Variably saturated flow described with the anisotropic
lattice Boltzmann methods. Comp Fluids, in press, in Proceeding
of ICMESS 2004, Braunschweig.
[13] Giraud L. Fluides visco-elastique par la methode de Boltzmann
sur reseau, PhD thesis, Universite Pierre et Marie Curie, Paris,
1997.
[14] Grubert D. Using the FHP-BGK-model to get effective dispersion
constants for spatially periodic model geometries. Int J Mod Phys
C 1997;8(4):817–25.
[15] Grubert D. Effective parameter interpretation and extrapolation
of dispersion simulations by means of a simple two-velocity
model. Transport Por Med 1999;37:153–67.
[16] Higuera FJ, Jimenez J. Boltzmann approach to lattice gas
simulations. Europhys Lett 1989;9:663–8.
[17] Higuera FJ, Succi S, Benzi R. Lattice gas dynamics with enhanced
collisions. Europhys Lett 1989;9:345–9.
[18] Huang K, Simu·nek J, Van Genuchten M. A third-order
numerical scheme with upwind weighting for solving the sol-
ute transport equation. Int J Numer Meth Eng 1997;40:1623–
37.
[19] Jawerth B, Lin P, Sinzinger E. Lattice Boltzmann models for
anisotropic diffusion of images. J Math Imaging Vis 1999;
11:231–7.
[20] Lallemand P, Luo LS. Theory of the lattice Boltzmann method:
Dispersion, dissipation, isotropy, Galilean invariance, and stabil-
ity. Phys Rev E 2000;61:6546–62.
[21] Merks RMH, Hoekstra AG, Sloot PMA. The moment propaga-
tion method for advection–diffusion in the lattice Boltzmann
method: validation and Peclet number limits. J Comp Phys
2002;183:563–76.
[22] Miller CT, Abhishek C, Sallerson AB, Prins JF, Farthing MW. A
comparison of computational and algorithmic advances for
solving Richards� equation. In: Miller CT, editor. Proceedings of
the CMWR XV, June 13–17, Computational methods in water
resources, vol. 2. Chapel Hill, NC, USA: Elsevier; 2004. p.
1131–45.
[23] Palmer B, Rector DR. Lattice-Boltzmann algorithm for simulat-
ing thermal flow in compressible fluids. J Comp Phys 2000;
161:1–20.
[24] Qian Y, d�Humieres D, Lallemand P. Lattice BGK models for
Navier–Stokes equation. Europhys Lett 1992;17:479–84.
[25] Tichonov A, Samarsky A. Equations of mathematical
physics. Moscow: Nauka; 1977.
[26] Van der Sman RGM, Ernst MH. Diffusion lattice Boltzmann
scheme on an orthorhombic lattice. J Stat Phys 1999;94(1/2):
203–17.
[27] Van der Sman RGM, Ernst MH. Advection–diffusion lattice
Boltzmann scheme for irregular lattices. J Comp Phys 2000;
160(2):766–82.
[28] Verberg R, Ladd AJC. Accuracy and stability of a lattice-
Boltzmann model with subgrid scale boundary conditions. Phys
Rev E 2001;65:016701-1–16.
[29] Wolf-Gladrow D. Lattice gas cellular automata and lattice
Boltzmann models: an introduction. Lecture notes in mathemat-
ics, vol. 1725. Springer; 2000. ISBN 3-540-66973-6.
[30] Warren PB. Electroviscous transport problems via lattice Boltz-
mann. Int J Mod Phys C 1997;8(4):889–98.
[31] Worthing RA, Mozer J, Seeley G. Stability of lattice Boltzmann
methods in hydrodynamic regimes. Phys Rev E 1997;56:2243–
53.
[32] Zhang X, Bengough AG, Crawford JW, Young IM. A lattice
BGK model for advection and anisotropic dispersion equation.
Adv Water Resour 2002;25:1–8.
[33] Zhang X, Bengough AG, Deeks LK, Crawford JW, Young IM. A
novel three-dimensional lattice Boltzmann model for solute
transport in variably saturated porous media. Water Resour Res
2002;38:1167–77.