RAPID COMMUNICATIONS
PHYSICAL REVIEW E 86, 065701(R) (2012)
Accuracy of the viscous stress in the lattice Boltzmann equation with simple boundary conditions
Wen-An Yong ()
*
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
Li-Shi Luo ()
†
Department of Mathematics & Statistics and Center for Computational Sciences, Old Dominion University, Norfolk, Virginia 23529, USA
and Computational Science Research Center, Beijing 10084, China
(Received 28 August 2012; published 7 December 2012)
Based on the theory of asymptotic analysis, we prove that the viscous stress tensor computed with the lattice
Boltzmann equation (LBE) in a two-dimensional domain is indeed second-order accurate in space. We only
consider simple bounce-back boundary conditions which can be reduced to the periodic boundary conditions by
using the method of image. While the LBE with nine velocities on two-dimensional square lattice (i.e., the D2Q9
model) and with the Bhatnagar-Gross-Krook collision model is used as an example in this work, our proof can
be extended to the LBE with any linear relaxation collision models in both two and three dimensions.
DOI: 10.1103/PhysRevE.86.065701 PACS number(s): 47.11.−j, 05.70.Ln, 05.20.Dd
The lattice Boltzmann equation (LBE) [1,2] is a numerical
method for computational fluid dynamics (CFD) derived
from the Boltzmann equation and kinetic theory [3,4]. The
lattice Boltzmann (LB) method has become popular recently
perhaps because it is rather simple to write an LB code
with a second-order accuracy in space [5–8], and because
of its successful applications to simulate complex fluids such
suspensions [9–14] and interfacial dynamics [15–19]. There
have been observations that the LBE is better than its second-
order finite-difference (FD) counterparts based on direct
discretizations of the Navier-Stokes equations, because of its
low numerical dissipation and better isotropy [20–22]. Dellar
[23,24] showed that the LBE with the multiple-relaxation-time
(MRT) collision model [25] is stable and free of spurious
vortices observed in other FD schemes for a double-shear
flow in two dimensions (2D). Peng et al. [26] demonstrated
that the LBE can accurately compute the vorticity in decaying
homogeneous isotropic turbulence in three dimensions (3D).
More recently, Kr¨ uger et al. [27] observed that the deviatoric
stress tensor computed with the LBE for the decaying Taylor-
Green vortex flow in 2D is second-order accurate. These
circumstantial evidence indicates that the kinetic modes in the
LBE, which have no counterparts in the Navier-Stokes based
CFD schemes, may have benefits in terms of accuracy and
stability. In this Rapid Communication, we will use the theory
of asymptotic analysis [5,7,28–30] to prove that the viscous
stress tensor computed with the LBE is indeed second-order
accurate in space.
We usually designate an LBE in d -dimensional space
and with q := (N + 1) discrete velocities, V :={ c
i
|i =
0,1,...,N }, as the Dd Qq model. The LBE involves q real
distribution functions {f
i
|i = 0,1,...,N } corresponding to q
discrete velocities { c
i
} and evolves on a d -dimensional lattice
δx Z
d
and discrete time t
n
:= nδt , n ∈ N
0
, where δx and δt
are the lattice constant (or grid spacing) and the time step
size, respectively, and N
0
:={0,1,...}. The discrete velocity
*
wayong@tsinghua.edu.cn
†
lluo@odu.edu; http://www.lions.odu.edu/∼lluo
set V includes the zero velocity c
0
= 0 and is symmetric, i.e.,
V =−V . The unit of the velocities { c
i
} is c := δx/δt . The
LBE can be concisely written in the following q -dimensional
vector form:
f ( x
j
+ cδt,t
n+1
) = f ( x
j
,t
n
) + J[f ( x
j
,t
n
)] + F( x
j
,t
n
), (1)
where the following notations have been used:
f ( x
j
+ cδt,t
n+1
):= [f
0
( x
j
,t
n+1
),f
1
( x
j
+ c
1
δt,t
n+1
),...,
f
N
( x
j
+ c
N
δt,t
n+1
)]
†
,
f ( x
j
,t
n
):= [f
0
( x
j
,t
n
),f
1
( x
j
,t
n
),...,f
N
( x
j
,t
n
)]
†
,
J( x
j
,t
n
):= [J
0
( x
j
,t
n
),J
1
( x
j
,t
n
),...,J
N
( x
j
,t
n
)]
†
,
F( x
j
,t
n
):= [F
0
( x
j
,t
n
),F
1
( x
j
,t
n
),...,F
N
( x
j
,t
n
)]
†
,
where † denotes the transpose, and J and F denote the
collision and external forcing terms, respectively. Commonly,
the collision term in the LBE, J, is the linear relaxation
model with constant relaxation rates, which is based on
the linearized Boltzmann collision operator about the local
Maxwellian equilibrium [3,4,20,21,25]. The equilibrium in
the LBE is usually a linear function of the flow density ρ and
polynomial of the flow velocity u, which can be obtained by
truncating the Taylor expansion of the Maxwellian equilibrium
at u = 0[3,4].
To be concrete, we will use the D2Q9 model in the
ensuing discussion. The discrete velocities in the D2Q9
model are c
0
= (0,0), c
1
= (1,0)c =−c
3
, c
2
= (1,0)c =−c
4
,
c
5
= (1,1)c =−c
7
, and c
6
= (−1,1)c =−c
8
. For the sake
of simplicity, we will use the Bhatnagar-Gross-Krook (BGK)
collision model [31]:
J
i
=−
1
τ
f
i
− f
(eq)
i
(ρ,u)
, (2)
where the equilibria for incompressible flows (cf., e.g.,
Ref. [32]) are given by
f
(eq)
i
(ρ,u) = w
i
ρ +
ˆ c
i
· u
θ
+
1
2
ˆ c
i
· u
θ
2
−
u · u
θ
,
(3a)
065701-1 1539-3755/2012/86(6)/065701(4) ©2012 American Physical Society