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Simulating Spatial Dynamics by Probabilistic
Cellular Automata
Olga Bandman
Supercomputer Software Department
ICMMG, Siberian Branch
Russian Academy of Science
Pr.Lavrentieva, 6, Novosibirsk, 630090, Russia
bandman@ssd.sscc.ru
Abstract.
A method is proposed, which is intended for constructing a
probabilistic cellular automaton (CA), whose evolution simulates a spa-
tially distributed process, given by a PDE.The heart of the method is the
transformation of a real spatial function into a Boolean array whose av-
eraged form approximates the given function.Two parts of a given PDE
(a differential operator and a function) are approximated by a combina-
tion of their Boolean counterparts.The resulting CA transition function
has a basic (standard) part, modeling the differential operator and the
updating part modifying it according to the function value.Special at-
tention is paid to the reaction-diffusion type of PDE.Some experimental
results of simple processes simulation are given and perspectives of the
proposed method application are discussed.
1 Introduction
Simulating nonlinear spatial dynamics is the main task of mathematical physics.
By tradition, the process under simulation is represented by a system of partial
differential equations (PDE). It is well known that it is practically impossible
to obtain their analytical solutions (except very simple cases). So, numerical
methods are to be used. Even though numerical analysis is progressing rapidly,
its practical use meets certain di
%
culties. Particularly, implicit schemes of dis-
cretization lead to algorithms, which have no e
%
cient parallel realizations. When
using explicit schemes much effort is to be put to provide computational stability.
Cellular automata models having appeared as an alternative to the PDEs [1],
are free of these shortages. They are absolutely stable, have no rounding off errors
and are less time consuming. Moreover, border conditions in cellular automata
are straightforward, which makes them appropriate to simulate fluids in porous
media. Of course, cellular automata have their own problems, which are not yet
completely solved. Among them the most important are the elimination of the
automata noise and the account of physical parameters (concentration, viscosity,
pressure)in terms of state-transition functions. Moreover, construction of a CA-
model for a given phenomenon simulation is a nonformalized task whose solution
