Equation-Free Multiscale Computation: Algorithms and Applications

Abstract

In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form-hence the term equation-free. We selectively discuss basic algorithms and underlying principles and illustrate the approach through representative applications. We also discuss potential difficulties and outline areas for future research.

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

“…In these experiments we used a harmonically excited impact oscillator with control forces applied via electromagnetic actuators as a test specimen and developed methods for tuning the control parameters and for determining stability of steady-state responses. Here, we will provide information about the details of the implementation of the toolbox CONTINEX developed for this type of experiment as well as for equation-free analysis [7,8,9,10,11]. We will motivate the algorithmic developments with new experimental results collected with the set-up used in [1,2].…”

“…Control based continuation in experiments [3,4,5,6] and continuation for equation-free problems [7,8,9,10,11] are two examples of continuation problems that involve noise contaminated zero problems F (u) = 0, a term that will be made more precise in Section 2. In the context of control based continuation, which we discuss with an example in Section 3, the function F is commonly defined as the average of a control signal or the difference between an observed state and a target state.…”

“…The first accepted point u k1 is used as the base point u in the projection condition (7). As a reminder, operating in sample mode means that for all remaining iterations the curve segment class applies an affine-linear projection condition for which the mean of any finite subset of solutions is a solution as well.…”

Multiscale Simulation of Gas-Liquid Flows with Droplets

“…Equation-free modeling 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 [4]. 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.…”

Dynamics of chemical reacting systems in low dimensional lattices via the Equation Free Method

“…Due to the fact that the lifting map (from the macroscopic description to microscopic one) is a one-to-many map we may use a sequence of lift-microscopic run-restrict steps to study the error made by approximating the microscopic system with the macroscopic variables. For a general review on Equation Free Methods (EFM) see [6]. EFM has already been used to get bifurcation results on chemical systems with lateral interactions [7].…”