tions. Here, a three-dimensional model with 19 velocities (commonly denoted as
D3Q19
)
will be used. Alternatives are models with 15 or 27 velocities. However, the latter one
has no clear advantages over the 19 velocity model, while the model with 15 velocities
has a decreased stability. The D3Q19 model is thus usually preferable as it requires less
2
The Boltzmann equation itself has been known since 1872. It is named after the Austrian scientist Ludwig
Boltzmann, and is part of the classical statistical physics that describe the behavior of a gas on the microscopic
scale. The LBM follows the approach of cellular automata to model even complex systems with a set of simple
and local rules for each cell [Wol02]. As the LBM computes macroscopic behavior, such as the motion of a
fluid, with equations describing microscopic scales, it operates on a so-called ”mesoscopic” level in between
those two extremes.
Historically, the LBM evolved from methods for the simulation of gases that computed the motion of each
molecule in the gas purely with integer operations.
In [HYP76], there was a first attempt to perform fluid
simulations with this approach. It took ten years to discover that the isotropy of the lattice vectors is crucial for
a correct approximation of the NS equations [FdH
+
87]. Motivated by this improvement, [MZ88] developed
the first algorithm that was actually called LBM by performing simulations with averaged floating point values
instead of single fluid molecules. The third important contribution to the basic LBM was the simplified collision
operator with a single time relaxation parameter [BGK54, CCM92, QdL92].

60
Real Time Physics
Class Notes
D2Q9
D3Q19
DFs of length 1
DFs of length 0
DFs of length
2
Cell boundary
2 Dimensions
9 Velocities
3 Dimensions
19 Velocities
Figure 10.1: The most commonly used LBM models in two and three dimensions.
memory than the 27 velocity model. For two dimensions the D2Q9 model with nine veloc-
ities is the most common one. The D3Q19 model with its lattice velocity vectors
e
1
..
19
is
shown in Fig. 10.1 (together with the D2Q9 model).
The velocity vectors take the fol-
lowing values:
e
1
= (
0
,
0
,
0
)
T
,
e
2
,
3
= (
±
1
,
0
,
0
)
T
,
e
4
,
5
= (
0
,
±
1
,
0
)
T
,
e
6
,
7
= (
0
,
0
,
±
1
)
T
,
e
8
..
11
= (
±
1
,
±
1
,
0
)
T
,
e
12
..
15
= (
0
,
±
1
,
±
1
)
T
, and
e
16
..
19
= (
±
1
,
0
,
±
1
)
T
. As all formulas
for the LBM usually only depend on the so-called particle distribution functions (
DFs
),
all of these two-dimensional and three-dimensional models can be used with the method
presented here. To increase clarity, the following illustrations will all use the D2Q9 model.
For each of the velocities, a floating point number
f
1
..
19
, representing a blob of fluid
moving with this velocity, needs to be stored.
As the LBM originates from statistical
physics, this blob is thought of as a collection of molecules or particles.
Thus, in the
D3Q19 model there are particles not moving at all (
f
1
), moving with speed 1 (
f
2
..
7
) and
moving with speed
√
2 (
f
8
..
19
). In the following, a subscript of
˜
i
will denote the value from
the inverse direction of a value with subscript
i
. Thus,
f
i
and
f
˜
i
are opposite DFs with in-

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