Equation-Free Multiscale Computation: enabling microscopic simulators to perform system-level tasks

TL;DR

This paper introduces an equation-free multiscale computational framework that enables microscopic simulators to perform system-level tasks without deriving explicit macroscopic equations, facilitating analysis over large scales.

Contribution

The authors develop a novel equation-free approach that bridges microscopic stochastic simulations with macroscopic modeling tasks without requiring explicit macroscopic equations.

Findings

Successfully applied to chemical kinetics, rheology, and homogenization examples.

Enables large-scale system analysis using short, localized microscopic simulations.

Provides a computational toolkit for multiscale modeling and bifurcation analysis.

Abstract

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, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Our equation-free (EF) approach, 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 mathematics-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 systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.