<s>
The	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
methods	I-Algorithm
(	O
LBM	O
)	O
,	O
originated	O
from	O
the	O
lattice	O
gas	O
automata	O
(	O
LGA	O
)	O
method	O
(	O
Hardy-Pomeau-Pazzis	O
and	O
Frisch-Hasslacher-Pomeau	O
models	O
)	O
,	O
is	O
a	O
class	O
of	O
computational	O
fluid	O
dynamics	O
(	O
CFD	O
)	O
methods	O
for	O
fluid	O
simulation	O
.	O
</s>
<s>
A	O
different	O
interpretation	O
of	O
the	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
equation	O
is	O
that	O
of	O
a	O
discrete-velocity	O
Boltzmann	O
equation	O
.	O
</s>
<s>
This	O
timescale	O
determines	O
the	O
kinematic	O
viscosity	B-Application
,	O
the	O
larger	O
it	O
is	O
,	O
the	O
larger	O
is	O
the	O
kinematic	O
viscosity	B-Application
.	O
</s>
<s>
The	O
LBM	O
was	O
designed	O
from	O
scratch	O
to	O
run	O
efficiently	O
on	O
massively	B-Operating_System
parallel	I-Operating_System
architectures	I-Operating_System
,	O
ranging	O
from	O
inexpensive	O
embedded	O
FPGAs	B-Architecture
and	O
DSPs	B-Architecture
up	O
to	O
GPUs	B-Architecture
and	O
heterogeneous	O
clusters	O
and	O
supercomputers	O
(	O
even	O
with	O
a	O
slow	O
interconnection	O
network	O
)	O
.	O
</s>
<s>
LGA	O
suffer	O
from	O
several	O
innate	O
defects	O
for	O
use	O
in	O
hydrodynamic	O
simulations	O
:	O
lack	O
of	O
Galilean	O
invariance	O
for	O
fast	O
flows	O
,	O
statistical	B-General_Concept
noise	I-General_Concept
and	O
poor	O
Reynolds	O
number	O
scaling	O
with	O
lattice	O
size	O
.	O
</s>
<s>
The	O
main	O
motivation	O
for	O
the	O
transition	O
from	O
LGA	O
to	O
LBM	O
was	O
the	O
desire	O
to	O
remove	O
the	O
statistical	B-General_Concept
noise	I-General_Concept
by	O
replacing	O
the	O
Boolean	O
particle	O
number	O
in	O
a	O
lattice	O
direction	O
with	O
its	O
ensemble	O
average	O
,	O
the	O
so-called	O
density	O
distribution	O
function	O
.	O
</s>
<s>
Lattice	B-Algorithm
Boltzmann	I-Algorithm
models	O
can	O
be	O
operated	O
on	O
a	O
number	O
of	O
different	O
lattices	O
,	O
both	O
cubic	O
and	O
triangular	O
,	O
and	O
with	O
or	O
without	O
rest	O
particles	O
in	O
the	O
discrete	O
distribution	O
function	O
.	O
</s>
<s>
For	O
example	O
,	O
D3Q15	O
is	O
a	O
3-dimensional	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
model	O
on	O
a	O
cubic	O
grid	O
,	O
with	O
rest	O
particles	O
present	O
.	O
</s>
<s>
In	O
most	O
Lattice	B-Algorithm
Boltzmann	I-Algorithm
simulations	O
is	O
the	O
basic	O
unit	O
for	O
lattice	O
spacing	O
,	O
so	O
if	O
the	O
domain	O
of	O
length	O
has	O
lattice	O
units	O
along	O
its	O
entire	O
length	O
,	O
the	O
space	O
unit	O
is	O
simply	O
defined	O
as	O
.	O
</s>
<s>
Speeds	O
in	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
simulations	O
are	O
typically	O
given	O
in	O
terms	O
of	O
the	O
speed	O
of	O
sound	O
.	O
</s>
<s>
It	O
is	O
therefore	O
common	O
to	O
raise	O
the	O
lattice	O
Mach	O
number	O
to	O
something	O
much	O
larger	O
than	O
the	O
real	O
Mach	O
number	O
,	O
and	O
compensating	O
for	O
this	O
by	O
raising	O
the	O
viscosity	B-Application
as	O
well	O
in	O
order	O
to	O
preserve	O
the	O
Reynolds	O
number	O
.	O
</s>
<s>
A	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
model	O
for	O
simulation	O
of	O
gas	O
mixture	O
combustion	O
capable	O
of	O
accommodating	O
significant	O
density	O
variations	O
at	O
low-Mach	O
number	O
regime	O
has	O
been	O
recently	O
proposed	O
.	O
</s>
<s>
Starting	O
with	O
the	O
discrete	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
equation	O
(	O
also	O
referred	O
to	O
as	O
LBGK	O
equation	O
due	O
to	O
the	O
collision	O
operator	O
used	O
)	O
.	O
</s>
<s>
The	O
lattice	B-Algorithm
Boltzmann	I-Algorithm
method	I-Algorithm
discretizes	O
this	O
equation	O
by	O
limiting	O
space	O
to	O
a	O
lattice	O
and	O
the	O
velocity	O
space	O
to	O
a	O
discrete	O
set	O
of	O
microscopic	O
velocities	O
(	O
i	O
.	O
e	O
.	O
)	O
.	O
</s>
<s>
The	O
relaxation	O
parameter	O
,,	O
is	O
related	O
to	O
the	O
kinematic	O
viscosity	B-Application
,,	O
by	O
the	O
following	O
relationship	O
:	O
</s>
<s>
For	O
the	O
popular	O
isothermal	O
Lattice	B-Algorithm
Boltzmann	I-Algorithm
methods	I-Algorithm
these	O
are	O
the	O
only	O
conserved	O
quantities	O
.	O
</s>
