Journal of Scientific Computing

, Volume 54, Issue 2–3, pp 247–268 | Cite as

A Multiscale Method for Highly Oscillatory Dynamical Systems Using a Poincaré Map Type Technique



We propose a new heterogeneous multiscale method (HMM) for computing the effective behavior of a class of highly oscillatory ordinary differential equations (ODEs). Without the need for identifying hidden slow variables, the proposed method is constructed based on the following ideas: a nonstandard splitting of the vector field (the right hand side of the ODEs); comparison of the solutions of the split equations; construction of effective paths in the state space whose projection onto the slow subspace has the correct dynamics; and a novel on-the-fly filtering technique for achieving a high order accuracy. Numerical examples are given.


Oscillatory dynamical system Averaging 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • G. Ariel
    • 1
  • B. Engquist
    • 2
  • S. Kim
    • 2
  • Y. Lee
    • 2
  • R. Tsai
    • 2
  1. 1.Bar-Ilan UniversityRamat GanIsrael
  2. 2.Department of Mathematics and Institute for Computational Engineering and Sciences (ICES)The University of Texas at AustinAustinUSA

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