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Equation-free modeling
Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse scale of interest, while accurate models are only given at a finely detailed, microscopic, level of description. The framework empowers one to perform macroscopic computational tasks (over large space-time scales) using only appropriately initialized microscopic simulation on short time and small length scales. The methodology eliminates the derivation of explicit macroscopic evolution equations when these equations conceptually exist but are not available in closed form; hence the term equation-free.
==Introduction==
In a wide range of chemical, physical and biological systems, coherent macroscopic behavior emerges from interactions between microscopic entities themselves (molecules, cells, grains, animals in a population, agents) and with their environment. Sometimes, remarkably, a coarse-scale differential equation model (such as the Navier-Stokes equations for fluid flow, or a reaction-diffusion system) can accurately describe macroscopic behavior. Such macroscale modeling makes use of general principles of conservation (atoms, particles, mass, momentum, energy), and closed into a well-posed system through phenomenological constitutive equations or equations of state. However, one increasingly encounters complex systems that only have known microscopic, fine scale, models. In such cases, although we observe the emergence of coarse-scale, macroscopic behavior, modeling it through explicit closure relations may be impossible or impractical. Non-Newtonian fluid flow, chemotaxis, porous media transport, epidemiology, brain modeling and neuronal systems are some typical examples. Equation-free modeling aims to use such microscale models to predict coarse macroscale emergent phenomena.
Performing coarse-scale computational tasks directly with fine-scale models is often infeasible: direct simulation over the full space-time domain of interest is often computationally prohibitive. Moreover, modeling tasks, such as numerical bifurcation analysis, are often impossible to perform on the fine-scale model directly: a coarse-scale steady state may not imply a steady state for the fine-scale system, since individual molecules or particles do not stop moving when the gas density or pressure become stationary. Equation-free modeling circumvents such problems by using short bursts of appropriately initialized fine-scale simulation.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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