Stability and stabilisation of the
lattie Boltzmann method
R. A. Brownlee, A. N. Gorban and J. Levesley
Department of Mathematis, University of Leiester,
Leiester LE1 7RH UK
November 16, 2006
Abstrat
The lattie Boltzmann method (LBM) is known to have stability
deienies. For example, loal blow-ups and spurious osillations are
readily observed when the method is used to model high-Reynolds uid
ow. Beginning from thermodynami onsiderations, the LBM an be
reognised as a disrete dynamial system generated by entropi involution and free-ight and the stability analysis is more natural. In
this paper we solve the stability problem of the LBM on the basis of
this thermodynami point of view. The main instability mehanisms
are identied. The simplest and most eetive reeipt for stabilisation
adds no artiial dissipation, preserves the seond-order auray of
the method, and presribes oupled steps: to start from a loal equilibrium, then, after free-ight, perform the overrelaxation ollision, and
after a seond free-ight step go to new loal equilibrium. Two other
presriptions add some artiial dissipation loally and prevent the
system from loss of positivity and loal blow-up. Demonstration of the
proposed stable LBMs are provided by the numerial simulation of a
1D shok tube and the unsteady 2D-ow around a square-ylinder up
to Reynolds number O(10000).
1
Introdution
A lattie Boltzmann method (LBM) is a disrete veloity method in whih a
uid is desribed by assoiating, with eah veloity vi, a single-partile distribution funtion fi = fi(x; t) whih is evolved by advetion and interation
on a xed omputational lattie.
The method has been proposed as a disretization of Boltzmann's kineti
transport equation:
fi
+ vi rfi = Qi;
(1)
t
where the ollision operator, Qi, is subjet to the fundamental mass, momentum and energy onservation laws. Dutifully, the ompressible Navier{
1
Stokes equations are satised by the disrete population moments provided
the studiously hosen disrete veloities have suÆient symmetry; the Mah
number is suÆiently low and the long time-sale, t, is large ompared
to the time-sale of ollisions (for an histori review see [23℄). Furthermore, the ollision operator an be alluringly simple, as is the ase with
the Bhatnager{Gross{Krook (BGK) operator [5℄, whereby ollisions are desribed by a single-time relaxation to loal entropy maximising equilibria
fi (although other hoies of equilibria are often preferred [23℄). Here, the
relaxation time is proportional to the kinemati visosity of the model.
The overrelaxation disretization of (1) (see, e.g., [4, 11, 19℄) is known
as LBGK and deouples visosity from the time step, thereby suggesting
that LBGK is apable of operating at arbitrarily high-Reynolds number by
making the relaxation time suÆiently small. However, in this low-visosity
regime, LBGK suers from numerial instabilities whih readily manifest
themselves as loal blow-ups and spurious osillations.
To analyse stability, the above histori LBM presription is not immediately useful. However, there is another approah whih arises from thermodynami onsiderations. Central to this alternative presription is the
notion of an entropy maximising or quasiequilibrium manifold in the spae
of distributions and the Ehrenfests' idea of oarse-graining [14, 15℄. In this
new representation, the main element is the disrete (in time) dynamial
system generated by entropi involution and free-ight (advetion). The
disrete veloities appear as approximation nodes in ertain ubatures in
veloity spae, and if the veloities from this set are automorphisms of a
lattie, the LBM in its regular spae-and-time disrete form, as above, is
obtained. The bakground knowledge neessary to disuss the LBM in this
manner is presented in Set. 2. Then, this presription suggests several
soures of numerial instabilities in the LBM and allows several reeipts for
stabilisation. Common to eah reeipt is the desire to stay uniformly lose
to the aforementioned manifold (Set. 3).
In Set. 5 a numerial simulation of a 1D shok tube and the unsteady
2D-ow around a square-ylinder using the present stabilised LBMs are presented. For the later problem, the simulation quantitively validates the experimentally obtained Strouhal{Reynolds relationship up to Re = O(10000).
This extends previous LBM studies of this problem where the relationship
had only been suessfully validated up to Re = O(1000) [1, 3℄.
Set. 6 ontains some onluding remarks as well as pratial reommendations for LBM realisations.
2
Bakground
In this setion, we briey introdue the thermodynami bakground of our
approah, and some notations. Proofs of most statements ould be extrated
2
from [15℄. Historially [23℄, the LBM appeared from the disretization ideas
of:
1. disrete veloity set;
2. lattie spae-and-time representation.
The idea of (symmetri or almost symmetri) overrelaxation was introdued
to deouple visosity from the time step [4, 11, 19, 23℄. This overrelaxation
was transformed into the notion of entropi involution [15, 21, 24℄, and a
new understanding of the LBM was ahieved. In this new representation,
the main element is the disrete dynamial system generated by entropi involution and free-ight. The disrete veloity set arises as ubature approximation nodes for the hydrodynami moments, and when these veloities
are automorphisms of some lattie, the LBM in its regular spae-and-time
disrete form is reovered.
The lattie presription is nie and symmetri, without any dierene between spae and time disretization, but it requires some eort to introdue
thermodynamis and to analyse stability of systems of this kind. On the
ontrary, when we start from thermodynami onsiderations, the entropy
introdution and the stability analysis are very natural, but the ubature
approximation and the spae disretization requires some additional eort.
Here we an nd an analogy to relativisti (quantum) eld theory: the Lagrangian formalism is fully ovariant, but if we would like to enjoy physis
of the Hamiltonian formalism, we should split spae and time, and use a
non-ovariant representation [13℄.
Let us introdue entropi involution in earnest. The starting point is a
onservative kineti equation
df = J (f ):
(2)
dt Here, onservative means that this equation preserves values of a onave
funtional, the entropy, S (f ).
The standard example is the free-ight equation
f
t
+ v rf = 0;
(3)
where f = f (x; v; t) is a single-partile distribution funtion, x is the spae
vetor, v is veloity. The hoie of entropy for (3) is ambiguous; we an
start from any onave funtional of the form
S (f ) =
Z
s(f (x; v; t))f (x; v; t)dxdv
with onave s(f ). The hoie by default is s(f ) = ln f , whih gives the
lassial Boltzmann{Gibbs{Shannon entropy.
3
In addition to the kineti equation (2) we have a xed linear mapping
m : f 7! M to some marosopi variables, for example, M is the set of ve
hydrodynami elds n, nu and E (density{momentum{energy),
Z
Z
Z
M
1
M = n := f dv; Mi = nui := vi f dv;
2 = E := 2 v f dv:
(4)
Further, for eah M , a quasiequilibrium (or onditional equilibrium, or generalised anonial state) fM is dened as a solution of the optimisation
problem
S (f ) ! max; m(f ) = M:
(5)
For eah f , a orrespondent quasiequilibrium state fm f is dened. The set
of all quasiequilibrium states is parameterised by M and referred to as the
quasiequilibrium manifold. The projetor of a point f onto the quasiequilibrium manifold is the following operator:
PS : f 7! fm f :
Let t be the time shift transformation for the initial onservative kineti
equation (2):
t (f (0)) = f (t):
For the free-ight equation (3) we have
t : f (x; v) 7! f (x vt; v):
For a given time , the Ehrenfests' step is a transformation of the quasiequilibrium manifold
Ehr : fM 7! PS ( (fM )):
(6)
The motion starts on the quasiequilibrium manifold, goes time along the
trajetory of the onservative kineti equation (2), and then follows projetion bak onto the quasiequilibrium manifold. Marosopi variables form
oordinates on the quasiequilibrium manifold. In these oordinates,
Ehr : M 7! m( (fM )):
The Ehrenfests' step gives a seond-order in time step approximation to
the solution of the dissipative marosopi equation
dM = m(J(f )) + m((D J(f )) );
(7)
f
f
f f
M
dt
2
where f is the defet of invariane of the quasiequilibrium manifold:
f = J (fM ) DM (fM )m(J (fM ));
(8)
and is the dierene between the vetor-eld J and its projetion on the
quasiequilibrium manifold.
4
0
2
( )
( )
=
M
M
4
M
M
For the free-ight equation and hydrodynami elds M = M (x; t) (4),
the quasiequilibrium distribution is the well known loal Maxwellian
=
m(v u) (v) = n 2kB T
fM
exp
m
2kB T ;
and (7) is the system of ompressible Navier{Stokes equations
X (nui )
n
=
;
2
3 2
t
(nuj )
t
E
t
i
xi
(nui uj )
xi
i
X P uj ui 2Æij
+ 2 x m x + x 3 r u ;
i
i
j
i
X
X
1 (P ui )
(Eui ) 5kB X P T
+ 2 2m x m x ;
m i xi
xi
i
i
i
i
P
= m1 x
j
=
X
where m is partile mass, k is Boltzmann's onstant, T is kineti temperature and P = nkB T is ideal gas pressure [18℄. The dynami visosity is
= P (the kinemati visosity is = where is the thermal veloity for one degree of freedom, = kB T=m). For ! 0, (7) tends to the
onservative marosopi equation
dM = m(J(f )):
(9)
M
dt
For hydrodynamis, this is the (ompressible) Euler equations.
The step with a quasiequilibrium state in the middle gives a seond-order
in time step approximation to the solution of the onservative marosopi
equation (9):
M (0) = m(PS ( = (fM ))) 7! m(PS (= (fM ))) = M ( ):
(10)
In order to deouple visosity and time step, we an ombine (6) with (10):
M (0) = m(PS ( #= (fM ))) 7! m(PS (& #= (fM ))) = M (& + #) = M ( );
where & and # = & . The state M is a mid-point on the trajetory.
This transformation provides a seond-order in time approximation for the
equation:
dM = m(J(f )) + & m((D J(f )) )
(11)
f
f f
f
M
dt
2
for the time step [15℄. For the free-ight equation (3) and hydrodynami
elds (4), the system (11) is the ompressible Navier{Stokes equations with
dynami visosity = & P (the kinemati visosity is = & ).
B
2
2
2 1
2
1
1
2
2
2
+
2
=
M
M
2
2 1
2
5
It is worthwhile to mention that all the points t (fM ) belong to a
manifold that is a trajetory q of the quasiequilibrium manifold due to
the onservative dynamis (2) (in hydrodynami appliations that is the
free-ight dynamis (3)). We all this manifold the lm of non-equilibrium
states [15, 16, 17℄. The defet of invariane f (8) is tangent to q at the
point fM , and belongs to the intersetion of this tangent spae with ker m.
This intersetion is one-dimensional. This means that the diretion of f is seleted from the tangent spae to q by the ondition: derivative of M in
this diretion is zero.
A point f on the lm of non-equilibrium states q is naturally parameterised by (M; ): f = qM; , where M = m(f ) is the value of the marosopi variables, and = (f ) is the time shift from a quasiequilibrium
state: (f ) is a quasiequilibrium state for some (other) value of M . To
the rst-order in ,
qM; = fM + f :
(12)
The quasiequilibrium manifold divides q into two parts, q = q [ q [ q ,
where q = fqM; j < 0g, q = fqM; j > 0g, and q is the quasiequilibrium manifold: q = fqM; g = ffM g.
For eah M and positive s from some interval ℄0; & [ there exist two numbers (M; s) ( (M; s) > 0, (M; s) < 0) suh that
) s:
S (qM; M;s ) = S (fM
The numbers oinide to the rst-order: = + o( ).
We dene the entropi involution as a transformation of q:
IS (qM; ) = qM; :
The pair of points f ; f 2 q onneted by the involution IS (i.e., f =
IS (f )) is dened (in q) by two onditions:
S (f ) = S (f ); m(f ) = m(f ):
The values of entropy and marosopi variables at these points oinide.
Let us hoose an initial marosopi state M , and suppose the initial
mirosopi state f belongs to q [ q in a -small viinity of fM :
m(f ) = M ; f = qM;#;
< # 0:
Then the step
M 7! m(IS ( (IS ( (f )))))
(13)
gives a seond-order in time step approximation to the onservative marosopi equations (9) with time step 2 (the seond appliation of IS in (13)
is added for symmetry and does not eet M ). One shift IS guarantees
rst-order auray only [15℄.
M
M
M
0
+
0
0
0
+
(
)
+
+
+
+
0
0
+
0
0
0
0
0
0
6
+
For modelling the visous motion (11) we an ombine involution and
projetion in the following manner: for f 2 q the point f = IS (f ),
2 [1=2; 1℄, is dened in q by two onditions:
m(f ) = m(f ); S (f ) S (PS (f )) = (2 1) (S (f ) S (PS (f ))):
The point IS (f ) is loser to the quasiequilibrium point PS (f ) than IS (f ).
For = 1 we get the entropi involution: IS = IS , and for = 1=2 we
reeive the operator IS= = PS .
If, for t 2 [0; ℄, the trajetory t (f ) intersets the quasiequilibrium
manifold (i.e., f = qM ;# and < # 0), then, after some initial steps,
the following sequene gives a seond-order in time step approximation
of (11) with & = (1 )= , 2 [1=2; 1℄:
Mn = m((IS )n f ):
(14)
In order to prove this statement we onsider a transformation of the seond
oordinate in qM;# ( < # 0): in linear approximation in # and we
have
(IS )qM;# = qM 0;#0 ;
where
#0 = (2 1)(# + ):
This transformation has a xed point # = (2 1)=(2 ) and
(IS )n qM;# = qM ;# ;
where
#n = # + ( 1)n (2 1)n Æ + o( );
for some Æ. This asymptoti formula is valid for the given 2 [1=2; 1℄ and
! 0, but if 1 is small it has no pratial sense beause relaxation may
be too slow: #n # + ( 1)n exp( 2n(1 ))Æ, and relaxation requires
1=(1 ) steps.
If #n = # + o( ) then the sequene Mn (14) approximates (11) with
& = 2j# j = (1 )= and seond-order auray in time step .
As we have already mentioned, for the transfer from free-ight with
entropi involution to the standard LBGK models we must:
1. transfer to a nite number of veloities with the same marosopi
equations;
2. transfer from spae to a lattie, where these veloities are automorphisms;
and also,
0
1
0
1
1
2
0
0
1 2
0
0
0
n
7
0
0
1
0
0
n
0
0
3. transfer from dynamis and involution on q to the whole spae of
states.
Instead of IS the transformation
I : f 7! PS (f ) + (2 1)(PS (f ) f )
(15)
is used. If, for a given f , the sequene (14) gives a seond-order in time
step approximation of (11), then the sequene
Mn = m((I )n f )
(16)
also gives a seond-order approximation to the same equation.
Entropi LBGK (ELBM) methods [7, 15, 21, 24℄ dier only in the denition of (15): for = 1 it should onserve the entropy, and in general has
the following form:
~
(17)
IE (f ) = (1 )f + f;
with f~ = (1 )f + PS (f ). The number = (f ) is hosen so that
the onstant entropy ondition is satised: S (f ) = S (f~). For LBGK (15),
= 2.
Of ourse, omputation of I is muh easier than that of IS or IE : it
is not neessary to follow exatly the manifold q and to solve the nonlinear
onstant entropy ondition equation. For an appropriate initial ondition
from q (not suÆiently lose to q ), two steps of ELBM with I gives the
same seond-order auray as (14). But a long hain of suh steps an
lead far from the quasiequilibrium manifold and even from q. Here, we see
stability problems arising.
0
0
0
0
0
0
3
0
Stability and reeipts for stabilisation
First of all, if f is far from the quasiequilibrium, the state I (f ) may be
non-physial. The positivity onditions (positivity of probabilities or populations) may be violated. For multi-dimensional and innite-dimensional
problems it is neessary to speify what one means by far. In the previous
setion, f is the whole state whih inludes the states of all sites of the lattie. All the inversion operators with lassial entropies (ones that do not
depend on gradients) are dened for lattie sites independently. Violation of
positivity at one site makes the whole state non-physial. Hene, we should
use here the `1-norm: lose states are lose uniformly, at all sites.
There is a simple reeipt for positivity preservation: to substitute nonpositive I (f ) by the losest non-negative state that belongs to the straight
line
n
o
f + (1 )PS (f )j 2 R
0
0
8
ǻ
Figure 1: Neutral stability and one-step osillations in a sequene of reetions. Bold dotted line { a perturbed motion, { diretion of neutral
stability.
dened by the two points, f and orrespondent quasiequilibrium. Let us all
this reeipt the positivity rule. It has been demonstrated [8℄ (also, independently, in [25℄) that the lassi LBGK model with the positivity rule provides
the same results (in the sense of stability and absene/presene of spurious
osillations) as the entropi LBGK models. This reeipt preserves positivity
of populations and probabilities, but an aet auray of approximation:
to avoid the hange of auray order, the number of sites with this step
should be of the order O(Nh=L) where N is the total number of sites, h is
the step of the spae disretization and L is the marosopi harateristi
length.
The seond problem is non-linearity: for auray estimates we always
use the assumption that f is suÆiently lose to quasiequilibrium. Far from
the quasiequilibrium manifold these estimates do not work beause of nonlinearity (rst of all, the quasiequilibrium distribution, fM , depends nonlinearly on M and hene the projetion operator, PS , is nonlinear). Again we
need to keep the states not far from the quasiequilibrium manifold.
The third problem is a diretional instability that an aet auray:
the vetor f PS (f ) an deviate far from the tangent to q. Hene, we should
not only keep f lose to the quasiequilibrium, but also guarantee smallness
of the angle between the diretion f PS (f ) and tangent spae to q.
One ould rely on the stability of this diretion, but we fail to prove
this in any general ase. The diretional instability hanges the struture of
dissipation terms: the auray dereases to the rst-order in and signiant utuations of the Prandtl number and visosity, et may our. This
arries a danger even without blow-ups; one ould oneivably be relying on
non-reliable omputational results.
Furthermore, there exists a neutral stability of all desribed approximations that auses one-step osillations: a small shift of f in the diretion of
f does not relax bak for = 1, and its relaxation is slow for 1 (for
small visosity). This eet is demonstrated for a hain of mirror reetions
in Fig. 1.
M
9
Three presriptions allow us to improve the situation:
1. Positivity rule.
The tehnial advise is to use this rule in all disrete kineti models.
This rule guarantees positivity of populations and probabilities, and
elementary post-proessing allows one to estimate how these steps affet the whole piture. Tests prove that this rule is as eetive as
entropi methods, and they are muh simpler for realisation (see, [8℄
and Set. 5).
For the stabilisation of LBMs, the entropi version of (17) was proposed
and is used. This approah somehow improves stability, indeed, but annot
erase spurious osillation and large loal deviation from quasiequilibrium [6,
8, 9℄. The H -theorem implies stability of equilibrium in the entropi norm
(that is, a weighted ` -norm, a weighted sum of squared point evaluations)
for isolated systems. For non-isolated systems (e.g., the shok tube, systems
with external ows, et.) the H -theorem (positivity of entropy prodution)
does not guarantee stability in any norm, but an be used to establish ertain
estimates of boundedness with respet to the entropi norm. However, to
suppress loal blow-ups we need estimates in `1-norm, and to suppress
high-frequeny osillations we need boundedness in the Sobolev norm that
depends on derivatives.
2. Ehrenfests' regularisation.
In order to keep the urrent state uniformly lose to the quasiequilibrium manifold we monitor loal deviation of f from the orrespondent quasiequilibrium, and when this deviation is large perform loal
Ehrenfests' steps [18℄,
fj 7! fj;
(18)
where j is the number of the site, fj is the state at this site, and fj is
the orrespondent loal quasiequilibrium (we assume that the entropy
is the sum of nodal values, and the problem of quasiequilibrium (5) is
fully split into loal problems at the sites).
In order to preserve the seond-order of auray, it is worthwhile performing Ehrenfests' steps at only a small number of sites (the number of
sites should be O(Nh=L), where N is the total number of sites, L is the
marosopi harateristi length and h is the lattie step). If only k sites
are required then this onstitutes a omputational ost of O(kN ). Numerial experiments show (see, e.g., [8, 9℄ and Set. 5) that even a small number
of suh steps drastially improves stability.
2
1
1
In our paper [9℄ we used another denition that follows the Euler disretization of the
BGK equation, but, for small visosity this is essentially the same
10
ĬIJ
PS
I0E
QE-manifold
I0
Figure 2: The sheme of oupled steps (19).
3. Coupled steps with quasiequilibrium ends.
Let us take fM as the initial state with given M , then evolve the state
by , apply LBGK reetion I , again evolve by , and nally
projet by PS onto quasiequilibrium manifold:
)))))
M 7! m(PS ( (I ( (fM
(19)
The analysis of entropy prodution easily shows that this step (Fig. 2)
gives a seond-order in time approximation to the shift in time 2
for (11) with & = 2(1 ) , 2 [1=2; 1℄. The stabilisation (restart exatly from a quasiequilibrium point) introdues additional dissipation
of order , and the perturbation of auray is of order . Hene,
the method has the seond-order auray.
It is neessary to stress that the visosity oeÆient is proportional to &
and signiantly depends on the hain onstrution: for the sequene (14)
we have & = (1 )= , and for the sequene of steps (19) & = 2(1 ) (the
proedure for alulating this visosity oeÆient is ontained in Set. 4).
For small 1 the later gives around two times larger visosity (and for
realisation of the same visosity we must take this in to aount).
0
0
2
4
3
Visosity omputation
In this setion, we demonstrate how to ompute visosity for any onstrution of steps on the base of (7) and the representation (12). We ompute the
entropy prodution and ompare it to the entropy prodution in Ehrenfests'
steps.
First of all, for any f , the distribution PS (f ) = fm f is the entropy
maximiser for the given marosopi variables M = m(f ). Hene, by Taylor
expansion,
1
S (f ) = S (PS (f )) + hf PS (f ); f PS (f )iP f + o(kf PS (f )k );
2
( )
S(
11
)
2
where h ; ig is the entropi inner-produt, i.e., the negative of the bilinear
form of the seond dierential of entropy: h'; ig := (Df S (f ))f g ('; ).
In partiular, using (12), we have
S (qM; ) = S (fM ) + hf ; f if + o( ):
2
For the operation I (15) we have
S (I f ) = S (PS (f ))
+ (2 2 1) hf PS (f ); f PS (f )iP f + o(kf PS (f )k ):
In partiular,
) + (2 1) hf ; f if + o( );
S (I qM; ) = S (fM
2
and for the orrespondent entropy gain S we have
S = 2 (1 )hf ; f if + o( ):
Entropy prodution is the ratio of entropy gain to time. For the Ehrenfests'
step (6) in time the entropy gain S ; is
S ; = 2 hf ; f if + o( );
with entropy prodution ; given by the expression
S ; = h ; i + o( ):
; =
(20)
2 f f f
Now, for a oupled step (19) (see Fig. 2)
7! PS ( (I ( (f ))));
fM
M
the free-ight does not hange entropy and the entropy gain is
S = S + S ;
with S = S ; . Thus,
S = 2 (1 )hf ; f if + 2 (1 ) hf ; f if + o( )
= 2 (1 )hf ; f if + o( ):
The orresponding entropy prodution is
S = (1 )hf ; f if + o( ):
=
(21)
After omparison of the two entropy prodution formulas (20) and (21)
we an immediately onlude that the oupled step (19) gives a seond-order
in time approximation of (11) with & = 2(1 ) . For any other variants
of step onstrution the method of visosity omputation is the same: we
estimate the entropy gain up to the seond-order, and nd the orrespondent
value of & .
2
2
=
2
M
M
M
0
0
2
S(
2
2
)
2
2
0
M
M
M
1
2
1
2
M
M
M
Ehr
2
2
Ehr
M
M
M
Ehr
Ehr
Ehr
M
M
M
0
1
2
Ehr 2(1
2
)
2
2
M
M
M
2
2
M
M
2
2
M
M
M
12
M
M
M
M
5
Numerial experiment
To onlude this paper we report two numerial experiments onduted to
demonstrate the performane of the proposed LBM stabilisation reeipts
from Set. 3. The rst test is a 1D shok tube and we are interested in
omparing the Ehrenfests' regularisation (18), the oupled step (19) with
LBGK (15) and ELBM (17).
The seond test is the 2D unsteady ow around a square-ylinder. The
unsteady ow around a square-ylinder has been widely experimentally investigated in the literature (see, e.g., [12, 22, 26℄). We demonstrate that
LBGK (15), with the Ehrenfests' regularisation (18), is apable of quantitively apturing the Strouhal{Reynolds relationship. The relationship is
veried up to Re = 20000 and ompares well with Okajima's experimental
data [22℄.
As we are advised in Set. 3, in all of the experiments, we implement the
positivity rule.
5.1
Shok tube
The 1D shok tube for a ompressible isothermal uid is a standard benhmark test for hydrodynami odes. We will x the kinemati visosity of
the uid at = 10 . Our omputational domain will be the interval [0; 1℄
and we disretize this interval with 801 uniformly spaed lattie sites. We
hoose the initial density ratio as 1:2 so that for x 400 we set n = 1:0 else
we set n = 0:5.
In all of our simulations we use a lattie with spaing h = 1, time step
= 1 and a disrete veloity set fv ; v ; v g := f0; 1; 1g so that the model
onsists of stati, left- and right-moving populations only. The governing
equations for LBGK are then
fi (x + vi ; t + 1) = fi(x; t) + (2 1)(fi (x; t) fi (x; t));
(22)
where the subsript i denotes population (not lattie site number) and f ,
f and f denote the stati, left- and right-moving populations, respetively.
The entropy is S = H , with
H = f log(f =4) + f log(f ) + f log(f );
(see, e.g., [20℄) and, for this entropy, the loal quasiequilibrium state f is
available expliitly:
2n 2 p1 + 3u ;
f =
3
p
n
f = (3u 1) + 2 1 + 3u ;
6n
p
(3
u + 1) 2 1 + 3u ;
f =
6
9
1
2
3
1
2
3
1
1
2
2
3
3
2
1
2
2
2
3
13
where
n :=
X
i
fi; u :=
1 X vi fi:
n
i
For ontrast we are interested in omparison with ELBM (17):
fi (x + vi ; t + 1) = (1 )fi (x; t) + f~i (x; t);
(23)
with f~ = (1 )f + f . As previously mentioned, the parameter, , is
hosen to satisfy a onstant entropy ondition. This involves nding the
nontrivial root of the equation
S ((1 )f + f ) = S (f ):
(24)
Inauray in the solution of this equation an introdue artiial visosity.
To solve (24) numerially we employ a robust routine based on bisetion.
The root is solved to an auray of 10 and we always ensure that the
returned value of does not lead to a numerial entropy derease. We
stipulate that if, at some site, no nontrivial root of (24) exists we will employ
the positivity rule instead.
For the realisation of the Ehrenfests' regularisation of LBGK, whih is
intended to keep states uniformly lose to the quasiequilibrium manifold, we
should monitor non-equilibrium entropy S at every lattie site:
S := S (f ) S (f );
throughout the simulation. If a pre-speied threshold value Æ is exeeded,
then an Ehrenfests' step is taken at the orresponding site. Now, the governing equations beome:
(
f (x; t) + (2 1)(fi (x; t) fi (x; t));
S Æ ,
fi (x + vi ; t + 1) = i
fi (x; t);
otherwise.
(25)
Furthermore, so that the Ehrenfests' steps are not allowed to degrade the
auray of LBGK it is pertinent to selet the k sites with highest S > Æ.
The a posteriori estimates of added dissipation ould easily be performed
by analysis of entropy prodution in Ehrenfests' steps.
The governing equations for the oupled step regularisation of LBGK
alternates between lassi LBGK and Ehrenfests' steps:
(
f (x; t) + (2 1)(fi (x; t) fi (x; t));
N even,
fi (x + vi ; t + 1) = i
fi (x; t);
N odd,
(26)
where N is the umulative total number of time steps taken in the simulation. Of ourse, one is at liberty to ombine the oupled step (26) with
15
step
step
step
14
the Ehrenfests' regularisation (25) to reate another method, and we will do
this as well.
In aordane with the losing remark of Set. 3, as the kinemati visosity of the model uid is xed at = 10 we should take = 1=(2 + 1) 1 2 for LBGK, ELBM and the Ehrenfests' regularisation. Whereas, for
the oupled step regularisation, we should take = 1 .
We observe that the proposed stabilisation reeipts (25) and (26) are
apable of subduing spurious post-shok osillations whereas LBGK and
ELBM fail in this respet (Fig. 3). The oupled step simulation (Fig. 3e)
is strikingly impressive as the sheme introdues zero artiial dissipation.
Furthermore, the small post-shok deviation in Fig. 3e (we believe this phenomenon is unavoidable without adding dissipation) an be eradiated using
Ehrenfests' steps (Fig. 3f and Fig. 3g).
In the example, we have onsidered xed toleranes of (k; Æ) = (4; 10 )
and (k; Æ) = (4; 10 ) only. We reiterate that it is important for Ehrenfests'
steps to be employed at only a small share of sites. To illustrate, in Fig. 4
we have allowed k to be unbounded and let Æ vary. As Æ dereases, the
number of Ehrenfests' steps quikly begins to grow (as shown in the aompanying histograms) and exessive and unneessary smoothing is observed
on the shok and rarefation wave. The seond-order auray of LBGK is
orrupted. In Fig. 5, we have kept Æ xed at Æ = 10 and instead let k vary.
We observe that even small values of k (e.g., k = 1) dramatially improves
the stability of LBGK.
9
3
4
4
5.2
Flow around a square-ylinder
Our seond test is the 2D unsteady ow around a square-ylinder. The
realisation of LBGK that we use will employ a uniform 9-speed square lattie
with disrete veloities
8
0;
i = 0,
>
>
> >
>
<
i = 1; 2; 3; 4,
os (i 1) 2 ; sin (i 1) 2 ;
vi =
>
>
>
>
>
:
p
2 os (i 5) 2 + 4 ; sin (i 5) 2 + 4 ; i = 5; 6; 7; 8.
The numbering f , f ; : : : ; f are for the stati, east-, north-, west-, south-,
northeast-, northwest-, southwest- and southeast-moving populations, respetively. As usual, the quasiequilibrium state, f , an be uniquely determined by maximising an entropy funtional
0
1
8
S (f ) =
X
i
fi log
15
fi ;
Wi
0.2
0.4 x 0.6
x
0.8
1
1
0.3
0.2
0.1
0
0
0.2
0.4 x 0.6
x
0.8
1
1
0.3
0.2
0.1
0
0
0.2
0.4 x 0.6
x
0.8
1
1
0.3
0.2
0.1
0
0
0.2
0.4 x 0.6
x
0.8
1
1
0.3
0.2
0.1
0
0
0.2
0.4 x 0.6
x
0.8
1
1
0.3
0.2
0.1
0
0
0.2
0.4 x 0.6
x
0.8
1
u
1
0.3
0.2
0.1
0
0
n
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
1.1
0.9
0.7
0.5
0
1.1
0.4 x 0.6
x
0.8
u
0.2
n
(a)
0.4 x 0.6
x
0.8
u
0.2
n
(b)
0.4 x 0.6
x
0.8
u
0.2
n
()
0.4 x 0.6
x
0.8
u
0.2
n
(d)
0.4 x 0.6
x
0.8
u
0.2
n
(e)
0.2
0.4 x 0.6
x
0.8
u
0.3
0.2
0.1
0
0
n
(f)
1
0.2 0.4 x 0.6 0.8
1
0.2 0.4 x 0.6 0.8
(g)
Figure 3: Density and veloity prole of the 1:2 isothermal shok tube
simulation after 400 time steps using (a) ELBM (23); (b) LBGK (22);
() Ehrenfests' regularisation (25) with (k; Æ) = (4; 10 ); (d) Ehrenfests'
regularisation (25) with (k; Æ) = (4; 10 ); (e) oupled step regularisation (26); (f) oupled step ombined with Ehrenfests' regularisation with
(k; Æ) = (4; 10 ); (g) oupled step ombined with Ehrenfests' regularisation
with (k; Æ) = (4; 10 ). In this example, no negative population are produed by any of the methods so the positivity rule is redundant. For ELBM
in this example, (24) always has a nontrivial root. Sites where Ehrenfests'
steps are employed are indiated by rosses.
3
4
3
4
16
6
0.9
4
n
ES
1.1
0.7
0.5
0
0.2
0.4
xx
0.6
0.8
15
0.9
10
0.5
0
0.2
0.4
x
x
0.6
0.8
0.9
40
400
100
200
tt
300
400
100
200
t
t
300
400
100
200
tt
300
400
n
20
0.2
0.4
x
x
0.6
0.8
00
1
120
0.9
80
n
ES
1.1
0.5
0
300
ES
60
0.7
40
0.2
0.4
x
x
0.6
0.8
00
1
150
0.9
100
ES
1.1
n
(d)
00
1
0.7
()
200
tt
5
1.1
0.5
0
100
ES
1.1
0.7
(b)
00
1
n
(a)
2
0.7
50
0.5
0
0
0.2 0.4 0.6 0.8
1
0
100
200
300
400
x
t
(e)
Figure 4: LBGK (22) regularised with Ehrenfests' steps (25). Density prole of the 1:2 isothermal shok tube simulation and Ehrenfests' steps histogram after 400 time steps using the toleranes (a) (k; Æ) = (1; 10 );
(b) (k; Æ) = (1; 10 ); () (k; Æ) = (1; 10 ); (d) (k; Æ) = (1; 10 ); (e)
(k; Æ) = (1; 10 ). Sites where Ehrenfests' steps are employed are indiated
by rosses.
3
4
5
7
17
6
1.1
2
n
ES
0.9
0.7
0.5
0
(a)
0.2
0.4
1.1
1.1
x
x
0.6
0.8
00
1
0.9
200
t
t
300
400
100
200
t
t
300
400
100
200
tt
300
400
100
200
tt
300
400
ES
n
(b)
100
2
0.7
0.5
0
1
1
0.2
0.4
x
x
0.6
0.8
00
1
1.1
4
n
ES
0.9
0.7
0.5
0
()
0.2
0.4
x
x
0.6
0.8
2
00
1
1.1
8
n
ES
0.9
0.7
(d)
0.5
0
0.2
0.4
x
x
0.6
0.8
4
00
1
1.1
8
n
ES
0.9
0.7
0.5
0
4
0
0
100
300
200
400
0.2 0.4 0.6 0.8
1
t
x
(e)
Figure 5: LBGK (22) regularised with Ehrenfests' steps (25). Density prole
of the 1:2 isothermal shok tube simulation and Ehrenfests' steps histogram
after 400 time steps using the toleranes (a) (k; Æ) = (1; 10 ); (b) (k; Æ) =
(2; 10 ); () (k; Æ) = (4; 10 ); (d) (k; Æ) = (8; 10 ); (e) (k; Æ) = (16; 10 ).
Sites where Ehrenfests' steps are employed are indiated by rosses.
4
4
4
4
18
4
subjet to the onstraints of onservation of mass and momentum:
q
!
q
2uj + 1 + 3uj v
Y
fi = nWi
2 1 + 3uj
1 uj
j
2
2
2
i;j
(27)
=1
Here, the lattie weights, Wi, are given lattie-spei onstants: W0 = 4=9,
W1;2;3;4 = 1=9 and W5;6;7;8 = 1=36. The marosopi variables are given by
the expressions
n :=
X
i
fi;
X
(u ; u ) := n1 vi fi:
1
2
i
The omputational set up for the ow is as follows. A square-ylinder of
side length L, initially at rest, is emersed in a onstant ow in a retangular
hannel of length 30L and height 25L. The ylinder is plae on the entre
line in the y-diretion resulting in a blokage ratio of 4%. The entre of
the ylinder is plaed at a distane 10:5L from the inlet. The free-stream
veloity is xed at (u1; v1 ) = (0:05; 0) (in lattie units) for all simulations.
On the north and south hannel walls a free-slip boundary ondition
is imposed (see, e.g., [23℄). At the inlet, the inward pointing veloities are
replaed with their quasiequilibrium values orresponding to the free-stream
veloity. At the outlet, the inward pointing veloities are replaed with their
assoiated quasiequilibrium values orresponding to the veloity and density
of the penultimate row of the lattie.
5.2.1
Maxwell boundary ondition
The boundary ondition on the ylinder that we prefer is the diusive
Maxwell boundary ondition (see, e.g., [10℄), whih was rst applied to LBMs
in [2℄. The essene of the ondition is that populations reahing a boundary
are reeted, proportional to equilibrium, suh that mass-balane (in the
bulk) and detail-balane are ahieved. We will desribe two possible realisations of the boundary ondition { time-delayed and instantaneous reetion
of equilibrated populations. In both instanes, immediately prior to the advetion of populations, only those populations pointing in to the uid at a
boundary site are updated. Boundary sites do not undergo the ollisional
step that the bulk of the sites are subjeted to.
To illustrate, onsider the situation of a wall, aligned with the lattie,
moving with veloity u and with outward pointing normal to the wall
pointing in the positive y-diretion (this is the situation on the north wall
of the square-ylinder with u = 0). The time-delayed reetion implementation of the diusive Maxwell boundary ondition at a boundary site
wall
wall
19
(x; y) on this wall onsists of the update
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
with
f (x; y; t) + f (x; y; t) + f (x; y; t)
= :
f (u ) + f (u ) + f (u )
Whereas for the instantaneous reetion implementation,
f (x; y + 1; t) + f (x + 1; y + 1; t) + f (x 1; y + 1; t)
=
:
f (u ) + f (u ) + f (u )
Observe that, beause density is a linear fator of the equilibria (27), the
density of the wall is inonsequential in the boundary ondition and an
therefore be taken as unity for onveniene.
We point out that, although both realisations agree in the ontinuum
limit, the time-delayed implementation does not aomplish mass-balane.
Therefore, instantaneous reetion is preferred and will be the implementation that we employ in the present example.
Finally, it is instrutive to illustrate the situation for a boundary site
(x; y) on a orner of the square-ylinder, say the north-west orner. The
(instantaneous reetion) update is then
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
f (x; y; t + 1) = f (u );
where
= f (x 1; y; t) + f (x; y + 1; t) + f (x 1; y 1; t)
+
f (x + 1; y + 1; t) + f (x 1; y + 1; t)
.
f (u ) + f (u ) + f (u ) + f (u ) + f (u ) :
2
2
wall
5
5
wall
6
6
wall
4
7
wall
2
4
5
1
8
wall
5
wall
6
2
2
wall
3
3
wall
5
5
wall
6
6
wall
7
7
wall
4
wall
5
7
2
wall
6
7
2
5.2.2
8
wall
8
wall
3
wall
5
wall
6
wall
7
wall
Strouhal{Reynolds relationship
As a test of the Ehrenfests' regularisation (18), a series of simulations, all
with harateristi length xed at L = 20, were onduted over a range of
Reynolds numbers
Lu
Re = 1 :
20
0.2
S
0.15
0.1
0.05
101
103
Re
102
104
Figure 6: Variation of Strouhal number as a funtion of Reynolds. Dots
are Okajima's experimental data [22℄ (the data has been digitally extrated
from the original paper). Diamonds are the Ehrenfests' regularisation of
LBGK and the squares are the ELBM simulation from [1℄.
The parameter pair (k; Æ), whih ontrol the Ehrenfests' steps toleranes,
are xed at (L=2; 10 ).
We are interested in omputing the Strouhal{Reynolds relationship. The
Strouhal number S is a dimensionless measure of the vortex shedding frequeny in the wake of one side of the ylinder:
3
S=
Lf!
;
u1
where f! is the shedding frequeny.
For our omputational set up, the vortex shedding frequeny is omputed
using the following algorithmi tehnique. Firstly, the x-omponent of veloity is reorded during the simulation over t = 1250L=u1 time steps.
The monitoring points is positioned at oordinates (4L; 2L) (assuming the
origin is at the entre of the ylinder). Next, the dominant frequeny is
extrated from the nal 25% of the signal using the disrete Fourier transform. The monitoring point is purposefully plaed suÆiently downstream
and away from the entre line so that only the inuene of one side of the
ylinder is reorded.
The omputed Strouhal{Reynolds relationship using the Ehrenfests' regmax
21
ularisation of LBGK is shown in Fig. 6. The simulation ompares well with
Okajima's data from wind tunnel and water tank experiment [22℄. The
present simulation extends previous LBM studies of this problem [1, 3℄ whih
have been able to quantitively aptured the relationship up to Re = O(1000).
Fig. 6 also shows the ELBM simulation results from [1℄. Furthermore, the
omputational domain was xed for all the present omputations, with the
smallest value of the kinemati visosity attained being = 5 10 at
Re = 20000. It is worth mentioning that, for this harateristi length,
LBGK exhibits numerial divergene at around Re = 1000. We estimate
that, for the present set up, the omputational domain would require at
least O(10 ) lattie sites for the kinemati visosity to be large enough for
LBGK to onverge at Re = 20000. This is ompared with O(10 ) sites for
the present simulation.
5
7
5
6
Conlusions
We have presented the main mehanisms of observed LBM instabilities:
1. Positivity loss due to high loal deviation from (quasi)equilibrium;
2. Appearane of neutral stability in some diretions in the zero visosity
limit;
3. Diretional instability.
We found three methods of stability preservation. Two of them, the positivity rule and the Ehrenfests' regularisation, are \salvation" (or \SOS")
operations. They preserve the system from positivity loss or from the loal
blow-ups, but introdue artiial dissipation and it is neessary to ontrol
the number of sites where these steps are applied. In order to preserve
the seond-order of LBM auray, it is worthwhile to perform these steps
on only a small number of sites; the number of sites should not be higher
than O(Nh=L), where N is the total number of sites, L is the marosopi
harateristi length and h is the lattie step. The added dissipation ould
easily be estimated a posterior by summarising the entropy prodution of
the \SOS" steps.
But most eetive is the speial new hoie of ollisions: the oupled
steps (26). These steps alternate between lassi LBM and Ehrenfests' steps,
introdue no artiial visosity, have the seond-order auray, and provide
diretional stability as well as obliterate the eets of neutral stability. Indeed, the shok tube simulation in Fig. 3e is a ompelling demonstration
of the proposed sheme's apabilities. Furthermore, the oupled step introdues no additional omputational ost ompared to lassial LBMs.
The pratial reommendation is to always use the oupled steps, and
to keep the positivity rule and the Ehrenfests' steps as an \SOS" in reserve
22
in order to make rare orretions of positivity loss and of too high loal
deviation from equilibrium.
Aknowledgements
The rst author is grateful to Shyam Chikatamarla for providing the digitally
extrated data used in Fig. 6.
This work is supported by Engineering and Physial Sienes Researh
Counil (EPSRC) grant number GR/S95572/01.
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