An Analyzable Method for Constructing a Cellular Automaton from a Continuous System

We propose an analyzable method of constructing a cellular automaton (CA) which simulates a given partial differential equation (PDE). We follow the study by Kawaharada and Iima who proposed a procedure for empirical construction of a CA from a given dataset. Their procedure is applicable for any spatiotemporal dataset, including numerical solutions of a PDE, in principle. However, the resultant CA is hardly identified a priori in a theoretical manner. An advantage of our proposed is that it is capable of being analyzed mathematically. The key is to design a minimal set of numerical experiments for collecting the spatiotemporal dataset for use in this procedure. We apply the proposed method to numerical solutions of three PDEs: the diffusion equation, the advection equation, and the Burgers equation. We discuss the difference with the existing method and the asymptotic convergence of the local rule depending on the amount of data, exploiting the advantage of the proposed method.