Article

Journal of Scientific Computing

, Volume 54, Issue 2, pp 247-268

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

  • G. ArielAffiliated withBar-Ilan University
  • , B. EngquistAffiliated withDepartment of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin
  • , S. KimAffiliated withDepartment of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin
  • , Y. LeeAffiliated withDepartment of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin
  • , R. TsaiAffiliated withDepartment of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin Email author 

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Abstract

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.

Keywords

Oscillatory dynamical system Averaging