Chemical Engineering Science 59 (2004) 1733–1743 www.elsevier.com/locate/ces Equation-free multiscale computations for a lattice-gas model: coarse-grained bifurcation analysis of the NO+CO reaction on Pt(1 0 0) Alexei G. Makeev a , Ioannis G. Kevrekidis b; c ; * a Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow 119992, Russia b Department of Chemical Engineering, E-quad, Olden Street, Princeton University, Princeton, NJ 08544, USA c Program in Applied and Computational Mathematics, Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Received 3 October 2003; received in revised form 16 January 2004; accepted 31 January 2004 Abstract Using the recently developed “coarse timestepper” approach (Proc. Nat. Acad. Sci. USA 97 (2000) 9840) we study a lattice-gas model of the NO+CO/Pt(1 0 0) reaction exhibiting macroscopic bistability and kinetic oscillations. Through numerical continuation and stability analysis, we construct one-parameter coarse bifurcation diagrams and contrast the results of mean-eld dierential equation models with the coarse-grained, expected dynamics of kinetic Monte Carlo (kMC) lattice-gas model simulations. We show how our computational superstructure enables the direct kMC simulator to perform tasks, such as continuation and numerical bifurcation analysis, for which it has not been originally designed. This closure-on-demand approach trades function evaluations with estimation based on short, appropriately initialized kMC simulations. We discuss its scope in complex/multiscale system modeling and simulation. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Multiscale computation; Nonlinear dynamics; Modeling; Kinetic Monte Carlo simulations; Numerical bifurcation analysis; Kinetic oscillations 1. Introduction Many complex multiscale systems are characterized by the emergence of coherent, macroscopic behavior from the interactions of individual “agents”, e.g. cells, molecules, among themselves and with their environment. Macroscopic rules (a description of the system at a coarse-grained, high level) are believed to arise somehow from microscopic ones (a description at a much ner level). For certain problems such as laminar Newtonian uid mechanics, a successful macroscopic description (the Navier–Stokes equations) was known phenomenologically long before it was de- rived from kinetic theory. Similarly, kinetic rate expres- sions based on experiments are traditionally used in simple ordinary dierential equation (ODEs) models of chemi- cal reactors; today we can envision their derivation at a much more microscopic level by computational chemistry. ∗ Corresponding author. Department of Chemical Engineering, Prince- ton University, E-quad, Olden Street, Princeton, NJ 08544, USA. Tel.: +1-609-2582818; fax: +1-609-2580211. E-mail address: yannis@princeton.edu (I. G. Kevrekidis). The models of reaction and transport processes in our textbooks are typically conservation laws (species, momen- tum, energy) closed through constitutive equations (reac- tion rates as a function of concentration, viscous stresses as functionals of velocity gradients). These models are written directly at the scale (alternatively, at the level of complex- ity) at which we are interested in practically modeling the system behavior. What really evolves during an experiment are distributions of colliding and reacting molecules; yet we know from experience that it is possible to write predictive deterministic models for the behavior (over the space/time scales of engineering modeling practice) at the level of con- centrations or velocity elds. Knowing the right level of observation at which we can be predictive for the system behavior, we try to close evolution equations for the sys- tem at this level. The closures may be based on experiment (e.g. through engineering correlations) or on mathematical modeling and approximation at ner, atomistic scales. Over the last few years we have worked on developing a computational framework that, when successful, allows us to bypass the derivation of explicitly closed macro- scopic equations. The basic premise is that information at 0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.01.029