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Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. (English) Zbl 1086.65066
Summary: 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.
Traditional modeling approaches first involve the derivation of macroscopic evolution equations balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually partial differential equations) is then brought to bear on the problem. Our equation-free 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, kinetic Monte Carlo or molecular dynamics 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 and numerical analysis.
We briefly 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.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
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