MATLAB EQuilibrium Toolbox (MEQ)

• MATLAB EQuilibrium Toolbox (MEQ) is a suite of MATLAB/Octave functions that use short micro-scale bursts for efficient, equation-free macro-scale simulation.
• The toolbox bypasses the need for derived macroscopic equations by aggregating localized simulation patches to compute global system dynamics.
• Its adaptive projective integration and patch dynamics methodology reduces computational demands in fields like atmospheric science, molecular dynamics, and nonlinear diffusion.

The MATLAB EQuilibrium Toolbox (MEQ), referred to as the 'Equation-Free Toolbox', is a suite of functions implemented in MATLAB/Octave designed to facilitate the efficient, system-level simulation of complex, multiscale systems. The toolbox enables users to employ microscopic simulators for macro-scale analyses, bypassing the explicit derivation of macroscopic evolution equations. Its core methodology operates by computing the micro-scale simulator only in short bursts over small, well-separated patches in space and time, thus significantly reducing computational demands while maintaining fidelity to the underlying systems dynamics (Maclean et al., 2020).

1. Fundamental Concepts and Objectives

The MEQ toolbox operationalizes the 'equation-free' approach, which leverages the micro-scale model as a computational black box to conduct macroscopic, system-level explorations. In many scientific fields, the derivation of explicit macroscopic equations is either infeasible or prohibitively complex; micro-scale simulations (e.g., agent-based models, molecular dynamics) often provide the only viable description. The toolbox bridges the gap between micro-scale simulators and macro-scale analysis by orchestrating short, localized simulation bursts and aggregating their results to yield global insights.

The primary objectives of the toolbox are:

• To conduct macro-scale tasks and analyses directly from microscopic simulators without requiring analytical upscaling.
• To reduce computational costs by focusing on localized simulation patches and exploiting temporal bursts.
• To enable researchers to perform system-level investigations where a macroscopic closure is unknown or intractable.

2. Equation-Free Methodology

The equation-free methodology, central to the toolbox, involves organizing computations such that micro-scale models are executed over short time intervals and small spatial regions ('patches'), with each invocation carefully separated in time and space. These micro-simulations are then mapped to estimate the derivative information necessary for macro-scale evolution.

Key methodological aspects include:

• Patch Dynamics: Only a subset of the full spatial domain is simulated, with patches distributed such that they are well-separated, minimizing redundant computation.
• Temporal Bursts: Simulations are executed only over short time windows, sufficient to capture the immediate local dynamics needed for projection to larger scales.
• Black-Box Simulators: The micro-scale model operates as an opaque procedure, with no requirement for analytical tractability or closure approximations.
• System-Level Tasks: By iteratively coupling the results of these localized simulations, the methodology enables predictions and analyses on the macro-scale.

The principal advantage is the avoidance of explicit macro-scale equation derivation, which is often elusive or impossible for many complex systems.

3. Implementation: Algorithms and Toolbox Functions

A novel and efficient algorithm for Projective Integration (PI) forms a cornerstone of the toolbox, operationalized via the function PIG. The algorithm interfaces with adaptive integrators from MATLAB (notably ode45) to achieve both accuracy and efficiency during simulation.

The workflow is as follows:

1. Run the micro-scale simulator for a short 'burst' from the current macro-scale initialization.
2. Extract the time derivative of the macro-scale observables from the burst.
3. Use the extracted derivative to project or integrate forward in macro-time, typically via a standard integrator.

The process can be succinctly represented: $$\text{Function PIG() uses short bursts of a user-provided micro-scale process and computes derivatives needed by a macro-scale integrator.}$$

The function wrappers are designed for flexibility and ease of adaptation to a range of simulator types and problem configurations.

4. Applications and Practical Usage

The toolbox targets scientific and engineering domains where micro-scale simulations dominate, but full-domain computations are computationally infeasible. Example application domains include:

• Atmospheric modeling, where local turbulence and chemical reactions are best described at a micro-scale.
• Molecular dynamics simulations, which are computationally intensive across large domains.
• Nonlinear diffusion systems inspired by fluid dynamics, which served as a use case in the development of the toolbox.

In these scenarios, MEQ enhances overall efficiency by focusing simulation capacity on critical regions and intervals, thereby facilitating analyses that would otherwise be computationally prohibitive.

Application Domain	Micro-scale Model Example	Toolbox Role
Atmospheric science	Turbulence, chemistry	Macro-scale prediction from localized micro-sim runs
Materials science	Molecular dynamics	Patch simulation for property upscaling
Nonlinear diffusion	Discrete diffusion systems	Efficient domain-wide evolution

5. Mathematical Support and Theoretical Foundations

The equation-free functions within the toolbox rest on robust mathematical frameworks:

• Numerical Homogenization: Underpins the justification for substituting micro-scale snapshots for full-domain integration.
• Adaptive Error Control: Integration with MATLAB's ode45 ensures that projection errors are managed within user-specified tolerances.
• A plausible implication is that, by constructing the toolbox functions to exploit these established numerical techniques, the validity of the resulting macro-scale dynamics is preserved, even when simulations are run on significantly reduced spatial and temporal domains.

6. Accessibility, Resources, and Documentation

The MEQ toolbox is openly accessible via GitHub at https://github.com/uoa1184615/EquationFreeGit. The repository provides the suite of MATLAB/Octave functions, accompanied by documentation including example scripts, tutorials, and a Users Manual. The design emphasizes user accessibility, with code and infrastructure suited for rapid adoption in new domains and adaptation to various micro-scale models.

The toolbox's availability and supporting resources facilitate its integration into research workflows, offering researchers in computational science and engineering a pathway to extend the reach of their existing microsimulators to system-level analyses.

In summary, the MATLAB EQuilibrium Toolbox (MEQ) delivers a functional and mathematically supported framework enabling practitioners to utilize existing micro-scale simulators for efficient and accurate macro-scale simulation, circumventing the need for explicit macroscopic modeling (Maclean et al., 2020).