Stabilization of the lattice Boltzmann method by the H theorem: A numerical test
Santosh Ansumali and Iliya V. Karlin*
ETH-Zu ¨rich, Department of Materials, Institute of Polymers, ETH-Zentrum, Sonneggstrasse 3, ML J 27, CH-8092 Zu ¨rich, Switzerland
Received 7 June 2000
For a one-dimensional benchmark shock tube problem, we implement the lattice Boltzmann method based
on the H theorem I. Karlin, A. Ferrante, and H. C. O
¨
ttinger, Europhys. Lett. 47, 182 1999. Results of
simulation demonstrate significant improvement of stability, as compared to realizations without explicit en-
tropic estimations.
PACS numbers: 47.11.+j, 05.20.Dd
I. INTRODUCTION
Since the invention of the lattice-gas model 1, lattice-
based methods for simulations of complex hydrodynamic
phenomena received much attention over the past decade. In
these methods, hydrodynamic equations are not addressed by
a direct discretization procedure, rather, a simple pseudopar-
ticle kinetics is introduced in such a way that the hydrody-
namic equations are obtained on the large space and time
scale. Particularly promising is the well-known lattice Bolt-
zmann method LBM2. It is based on the fully discrete
velocity-space-time kinetic equation of the form,
N x +c, t +1 -N x , t = N x , t . 1
Here N( x , t ) is the b-component vector of populations N
i
of
the pseudoparticles with velocities c
i
, at the sites x of a
lattice at discrete time t. The system of discrete velocities at
any site is formed by the outgoing links of the lattice, and it
also may include the zero vector.
One of the most important problems related to the LBM,
recognized by many authors, is the problem of numerical
stability. For the LBM related to incompressible flow simu-
lations, numerical instabilities preclude so far a study of high
Reynolds number flow situations. Instabilities become even
more annoying for compressible flows 3.
It has been discussed for some time in the literature that
stability of the LBM could be improved if the method could
be equipped with an analog of the Boltzmann H theorem.
Recently, theoretical progress in this direction has been
achieved 4–8. In particular, for the isothermal LBM, the
hydrodynamic fields are the density, =( 1, N), and the av-
erage momentum, u
=( c
, N), where ( • , • ) denotes the
standard scalar product in the b-dimensional space of popu-
lation vectors. In this case, one can construct entropy func-
tions in such a way that its local equilibrium implies the
crucial relation for the stress tensor,
c
c
, N
eq
=c
s
2
+ u
u
,
up to the admissible degree of accuracy of the LBM 6.
Furthermore, it has been suggested how to construct the col-
lision integral based solely on the knowledge of the en-
tropy function, and how to stabilize the updates on the basis
of the discrete-time H theorem 6,8. In the sequel, we term
the LBM based on the H theorem the entropic lattice Boltz-
mann method ELBM.
It is the goal of this paper to test the aforementioned the-
oretical developments for a shock tube problem. This one-
dimensional benchmark problem has been suggested some
time ago 9 for testing various ideas in the LBM. Though
this model is based on a very simple three-velocity lattice, it
provides a stringent test of stability. Implementation of the
ELBM demonstrated a large improvement of stability in this
benchmark problem. In fact, we were able to reach the values
of the kinematic viscosity as low as 10
-12
without any sign
of numerical instabilities. The most important part of the
realization is a robust root-finding procedure which imple-
ments the H theorem.
II. CONSTRUCTION OF THE ELBM
An advantage of the LBM in comparison to the lattice-gas
method is that the Galilean invariance of the Navier-Stokes
equation is easier to control in the former than in the latter.
In order that this advantage should not get lost in the ELBM,
entropy functions should be found for each lattice separately.
We here consider the one-dimensional lattice with spacing c,
and the population vector at each site x has three compo-
nents, N=( N
+
, N
0
, N
-
)
²
, corresponding to velocities c
+
=c , c
0
=0, and c
-
=-c , respectively. For this model, the
entropy function has been found in 6
H =N
0
ln N
0
/4 +N
-
ln N
-
+N
+
ln N
+
. 2
Realizations of the ELBM based on the entropy function 2
result in the one-dimensional Navier-Stokes equation with
the sound speed c
s
= 1/3c , within the accuracy of the order
( u / c
s
)
4
. Construction of the ELBM involves the following
two steps these steps are independent of the choice of the
entropy.
First, we specify bare collision integral in such a way
as to satisfy the admissibility condition, ( , 1) =( , c
) =0,
( , H ) 0, and ( N
eq
) =0. Here H is the gradient of H
in the space of population vectors. The choice of the bare
collision integral is not unique. We here consider three cases,
= g
+
-g
-
exp “ H , g
-
-exp “ H , g
+
, 3a
*Corresponding author. FAX: +41 01 632 10 76. Email address:
ikarlin@ifp.mat.ethz.ch
PHYSICAL REVIEW E DECEMBER 2000 VOLUME 62, NUMBER 6
PRE 62 1063-651X/2000/626/79995/$15.00 7999 ©2000 The American Physical Society