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A multiscale method for highly oscillatory ordinary differential equations with resonance. (English) Zbl 1198.65110

Summary: A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to \( \varepsilon\) perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C29 Averaging method for ordinary differential equations
37M99 Approximation methods and numerical treatment of dynamical systems

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