Applied Numerical Mathematics 59 (2009) 2023–2034 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Asymptotic analysis of Complex Automata models for reaction–diffusion systems Alfonso Caiazzo a,∗ , Jean-Luc Falcone c , Bastien Chopard c , Alfons G. Hoekstra b a INRIA Rocquencourt, BP 105, F-78150 Le Chesnay Cedex, France b University of Amsterdam, Section Computational Science, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands c CUI Department, University of Geneva, Switzerland article info abstract Article history: Received 22 July 2008 Received in revised form 27 January 2009 Accepted 6 April 2009 Available online 10 April 2009 Keywords: Complex Automata modeling Reaction–diffusion Lattice Boltzmann method Asymptotic expansion Complex Automata (CxA) have been recently introduced as a paradigm to simulate multiscale multiscience systems as a collection of generalized Cellular Automata on different scales. The approach yields numerical and computational challenges and can become a powerful tool for the simulation of particular complex systems. We present a mathematical framework for CxA modeling to investigate the behavior of the model depending on scale separation and modeling choices. For a simple CxA model for a reaction–diffusion process, we define a Complex Automata model, deriving theoretical error estimates, which are numerically validated.  2009 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction Among the challenges in computational sciences, the field of multiscale simulation has become more and more popular in recent years. Due to the large variety of interesting problems, many approaches for multiscale systems simulations have been and are continuously developed. We focus on the recently introduced Complex Automata (CxA) paradigm [9,10]. Looking at multiscale systems as an ensemble of processes happening on different temporal and spatial scales, the idea behind a Complex Automata is that a multiscale algorithm, designed to simulate such multiscale system, can be replaced by many single scale models, constructed to simulate the relevant sub-processes, choosing appropriately different resolution according to the properties of the original system. According to the original dynamics, the single scale algorithms have to be coupled across the scales using proper coupling templates. Furthermore, we constrain these single scale models to have a specific update paradigm. Namely, we consider numerical methods whose time evolution can be decomposed in a local collision step (when the state of the system is updated using only local information), plus a propagation step, when the new states are communicated through the system. Approaches such as Cellular Automata (CA), lattice Boltzmann methods (LBM), or Agent Based models (ABM) satisfy these assumption. In the context of CA, it has been shown [5] that this update paradigm is equivalent to the more popular choices. Moreover, we observe that many FD schemes can be written in the same fashion. Focusing on this class of algorithm allows us to introduce a special formalism to describe CxA modeling [2,9], to formalize classes of multiscale couplings [10], and it is particularly interesting from the computational point of view, since it can results in efficient numerical schemes and it can be used to design a specific CxA simulation framework [8]. In a few words, a CxA model is a reduction of an original (complicated but accurate) multiscale algorithm to a collection of (simpler but less precise) single scale sub-algorithms. Detailed introduction of the Complex Automata approach and of related * Corresponding author. E-mail addresses: alfonso.caiazzo@inria.fr (A. Caiazzo), Jean-Luc.Falcone@cui.unige.ch (J.-L. Falcone), Bastien.Chopar@cui.unige.ch (B. Chopard), a.g.hoekstra@uva.nl (A.G. Hoekstra). 0168-9274/$30.00  2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2009.04.001