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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

A Multiscale Method for Stiff Ordinary Differential Equations with Resonance

29th March 2007

Author: Richard Tsai (Univ. of Texas at Austin)

Abstract

We introduce a multiscale method to compute the effective behavior of a class of stiff and highly oscillatory ODEs. The oscillations may be in resonance with one another and thereby generate some hidden slow dynamics. Our method relies on correctly tracking a set of slow variables whose dynamics is effectively closed, and is sufficient to approximate the effective behavior of the ODEs. This set of variables is found by our numerical methods. We demonstrate our algorithms by a few examples that include a commonly studied problem of Fermi, Pasta, and Ulam (FPU).