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applied to the odd blocks. In [3] the model with
p
=1
/
2 is proved to simulate
diffusion with the value
d
from (10)equal to 3
/
2. Another diffusion coe
%
cient
value may be obtained by appropriate choice of
p,τ
, and
h
[4].
Fire propagation
. Equation (9)with
f
(
u
)=
αu
(1
−u
)is a simplified model
of reaction-diffusion process. With (
α ≤
1
/
2)and the following initial conditions
1if
x,y < K,
0if
x,y ≥ K,
u
(
x,y
)=
K
=
const
, the equation represents an active wave with a center in
x
=
y
=0,
which simulates the propagation of fire, of weeds, of epidemics etc. Simulation
process starts with initial array construction. Let's assume, that it is given as a
2D array
Ω
R
=
{
(
u,m
):
u ∈ R,m ∈ M
}
, where
M
∈ M
is a finite set of names,
given as pairs of Cartesian coordinates, i.e.
M
=
{
(
i,j
):
i
=0
,...,n, j
=
0
,...,m}
. In Fig.1 the first snapshot is the initial cellular array
Ω
R
(0)with
n
=
m
= 200. The black pixels correspond to the cells (1
,
(
ij
)), the white pixels
- to the cells (0
,
(
i,j
)), the gray ones being inside the interval. The initial flash
of fire is given by a black area at the top of the vertical channel formed by two
walls, the right one having a hole with an opened cover. The walls are given by
cells (
c,
(
i,j
))
,c
=
{
0
,
1
}
. The width of walls should be chosen larger than the
radius
ρ
of the averaging area (
ρ
=10
,q
=(2
ρ
)
2
= 400)to avoid hoping the
”ones” over them in the process of averaging.
To begin the computation the initial Boolean array is also required. Hence,
Ω
R
is transformed into
Ω
B
, according to (4), but, since the array has a finite
cardinality and there are cells belonging to the walls, the following border con-
ditions should be taken into account. If in the averaging neighborhood of a cell
(
u,
(
ij
))
∈ Ω
R
(
t
)there occur
r, r >
0
,
cells, which do not belong to
M
, i.e. their
states or names are not in
{
0
,
1
}
or in
M
, respectively, then the probability of
(
v,
(
i,j
)) to be (1
,
(
i,j
)) is as follows.
P
(
v
i,j
=1)
=
u
i,j
q −r
,
(18)
The spatial structure of cell interactions is determined by two neighborhoods:
N
(
i,j
)given as (16)needed for performing BR-diffusion, and
N
Av
(
i,j
)for per-
forming averaging and Boolean discretization.
Each
t
th iteration of CA evolution consists of the following procedures over
Ω
B
(
t
)and
Ω
R
(
t
).
1)The diffusion part is computed by applying the rule (17)both to all even
and all odd blocks, border conditions being as follows. If a cell (
v,
(
i,j
)), such
that either
v
or (
i,j
)do not belong to
{
0
,
1
}
or to
M
, respectively, occurs in
the neighborhood of (
v,
(
i,j
)), then no CA-rule is applied to it. The result of
diffusion part computation is
Ω
S
(
t
+1)=
{
(
w,
(
i,j
))
}
.
2)The function
f
(
v
)for all cells in
Ω
R
(
t
)is computed forming a cellular
array
Ω
F
(
t
)=
{
(
f
i,j
,
(
i,j
))
}
.
3)Updating
Ω
S
(
t
+ 1)is performed according to (15, the result
Ω
B
(
t
+1)
being the initial Boolean array for the next iteration.
