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Publication 17-CNA-017

Computing Singularly Perturbed Differential Equations

Sabyasachi Chatterjee
Dept. of Civil & Environmental Engineering
Carnegie Mellon University
Pittsburgh, PA 15215
sabyasac@andrew.cmu.edu

Amit Acharya
Dept. of Civil & Environmental Engineering
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15215
acharyaamit@cmu.edu

Zvi Artstein
Dept. of Mathematics
The Weizmann Institute of Science
Rehovot, Israel
zvi.artstein@weizmann.ac.il

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 accurately initializing short bursts of microscopic runs for the evolution of slow variables is integral to our scheme, without the requirement that the slow variables determine a unique invariant measure of the microscopic dynamics.

Get the paper in its entirety as  17-CNA-017.pdf


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