Communications in Mathematical Sciences

Volume 1 (2003)

Number 4

Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis

Pages: 715 – 762



C. William Gear

James M. Hyman

Panagiotis G Kevrekidid

Ioannis G. Kevrekidis

Olof Runborg

Constantinos Theodoropoulos


We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/ stochastic) level of description to perform modeling tasks at a coarse (macroscopic/systems) level. These macroscopic modeling tasks, yeilding infomration over long time and large scales, are accomplished through approximately initialized calls to the microscopic simular for only short times and small spatial domains. Traditionally modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive realtions). An arsenal of analytical and numerical techniques for the efficent solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in [1], when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathmatics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, "coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, "closure-on-demand" computational toolkit, bridging microscopic/ stochastic simulation with traditional continuum scientific computation amd numerical analysis. We will breifly survey the application of these "numerical enabling technology" ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

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