Computing Singularly Perturbed Differential Equations

TL;DR

This paper presents a computational method for efficiently simulating nonlinear differential equations with multiple time scales, capable of handling complex dynamics like relaxation oscillations and initial condition sensitivities, with demonstrated accuracy and computational savings.

Contribution

It introduces a novel coarse-graining approach for nonlinear ODE systems that captures complex behaviors and improves computational efficiency.

Findings

Successfully applied to Hamiltonian and dissipative systems

Accurately captures mixed slow-fast responses and initial condition effects

Achieves significant computational time savings

Abstract

A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the averaging of Hamiltonian as well as dissipative microscopic dynamics whose `slow' variables, defined in a precise sense, can often display mixed slow-fast response as in relaxation oscillations, and dependence on initial conditions of the fast variables. Also covered is the case where the quasi-static assumption in solid mechanics is violated. The computational tool is demonstrated to capture all of these behaviors in an accurate and robust manner, with significant savings in time. A practically useful strategy for initializing short bursts of microscopic runs for the accurate computation of the evolution of slow variables is also developed.