Multiscale Modeling with Differential Equations

TL;DR

This paper discusses multiscale modeling approaches for differential equations, illustrating how to effectively capture system behavior across multiple scales without detailed resolution of all small-scale features.

Contribution

It introduces methods for multiscale modeling of differential equations, demonstrating how to simplify complex systems while preserving essential dynamics.

Findings

Multiscale models effectively capture system behavior across different scales.

Simplified models reduce computational complexity.

Examples include cardiovascular systems and shallow water equations.

Abstract

Many physical systems are governed by ordinary or partial differential equations (see, for example, Chapter ''Differential equations'', ''System of Differential Equations''). Typically the solution of such systems are functions of time or of a single space variable (in the case of ODE's), or they depend on multidimensional space coordinates or on space and time (in the case of PDE's). In some cases, the solutions may depend on several time or space scales. An example governed by ODE's is the damped harmonic oscillator, in the two extreme cases of very small or very large damping, the cardiovascular system, where the thickness of the arteries and veins varies from centimeters to microns, shallow water equations, which are valid when water depth is small compared to typical wavelength of surface waves, and sorption kinetics, in which the range of interaction of a surfactant with an air bubble is much smaller than the size of the bubble itself. In all such cases a detailed simulation of the models which resolves all space or time scales is often inefficient or intractable, and usually even unnecessary to provide a reasonable description of the behavior of the system. In the Chapter ''Multiscale modeling with differential equations'' we present examples of systems described by ODE's and PDE's which are intrinsically multiscale, and illustrate how suitable modeling provide an effective way to capture the essential behavior of the solutions of such systems without resolving the small scales.