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The second term in the right-hand side of (7)is the averaging error of the Boolean
discretization in the cell named
m
.
m
=
1
q
(
u
j
(
m
)
−u
m
)
.
(8)
N
Av
(
m
)
From (8)it is clear, that the Boolean discretization error depends on the spa-
tial function smoothness. The largest errors are in those cells, where the spatial
function
u
(
m
)has sharply defined extremes on their averaging areas. Such ex-
tremes may be caused both by the given initial cellular array and by the reaction
function. The errors may be decreased by the appropriate choice of spatial dis-
cretization steps
h
x
,h
y
,h
z
and averaging area size.
3 Approximation of a PDEby a Probabilistic CA
The most general form of PDE representation is as follows
∂u
g
∂t
=
∇
n
u
g
+
f
g
(
u
1
,...,u
l
)
, n
=1
,
2
, g
=1
,...,l,
(9)
where
∇
n
is an
n
-order differential operator,
f
g
(
u
1
,...,u
l
)- an arbitrary func-
tion with the domain and the range in the interval (0,1).
If
n
= 1, the equation system of the type (9)describes, for example, electro-
magnetic process. If
n
= 2, i.e.
∇
2
is a Laplacian, the system (9)represents a
reaction-diffusion phenomenon. When numerical methods of PDE solution are
used, it is precisely these operators which cause the main troubles in providing
stability of computation. Nevertheless, for both above types of differential op-
erators there exist well known cellular automata, whose evolution simulate the
corresponding processes. The examples may be found in [2] where a CA-model is
proposed for elecrtromagnetic field, and in [3,4] where CA-diffusion models are
studied. This brings up to the idea of replacing the finite-difference form of
∇
n
u
by simple and reliable CA-models, which are further referred to as
standard CAs
.
Clearly, this transfers the computation process to the Boolean domain, which
requires to perform the addition of
f
g
(
u
1
,...,u
l
)to a standard CA result in the
Boolean form. Such a procedure is further referred to as
updating
. Its formal
representation is as follows.
After time and space discretization with
t
=
tτ,t
=0
,
1
,...,...
and
x
=
h
x
i, y
=
h
y
j, z
=
h
z
k,
the system (9)takes the form
n
u
g
(
t
+1)=
u
g
(
t
)+
d
∇
u
g
(
t
)+
τf
g
(
u
1
(
t
)
,...,u
l
(
t
))
, g
=1
,...,l.
(10)
n
where
d
∇
u
g
is a finite difference representation of the differential operator,
d
depends on
τ,d,h
. Particularly, in the case of
n
=2
,d
=
τd/h
2
. In the right-
hand side of (10)two first terms are responsible for a process to be simulated
by a standard CA, while the function
f
g
(
u
1
,...,u
l
)should be turned out into
a Boolean form and added to the standard CA result at each iteration of the
simulation process. Since there are
l
variables to be simulated, the naming set
