Methods for Cellular Automata and Evolution Systems in Modelling and Simulation

Cellular automata provide an interesting concept for modelling and simulation. Besides advanced modelling approaches like the Lattice Boltzmann Method cellular automata are also used for simulation in population dynamics, reaction diffusion systems or bio-medicine. While the basic concepts of cellular automata are rather clear and unambiguous, there exists no general mathematical formalism as there is for other modelling approaches like differential equations or stochastic processes. Often such systems are not even labelled as cellular automata due to the historic and diverse connotation of this term. This modelling approach is however only applicable for systems that exhibit a certain topological structure. We interpret the topological concepts of such systems as graphs and vector spaces in order to provide a suitable mathematical framework for analysis. Furthermore a generalisation leads to evolution systems (strongly continuous semigroups, abstract Cauchy problems) on the one hand and stochastic processes on the other. This provides access to a variety of mathematical tools and methods for analysing and validating cellular automaton modelling approaches.