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 15213
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.
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