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Oscillatory Systems with Three Separated Time Scales: Analysis and Computation

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Numerical Analysis of Multiscale Computations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 82))

Abstract

We study a few interesting issues that occur in multiscale modeling and computation for oscillatory dynamical systems that involve three or more separated scales. A new type of slow variables which do not formally have bounded derivatives emerge from averaging in the fastest time scale. We present a few systems which have such new slow variables and discuss their characterization. The examples motivate a numerical multiscale algorithm that uses nested tiers of integrators which numerically solve the oscillatory system on different time scales. The communication between the scales follows the framework of the Heterogeneous Multiscale Method. The method’s accuracy and efficiency are evaluated and its applicability is demonstrated by examples.

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Acknowledgements

Research are performed under the support of NSF DMS-0714612. Tsai is 499 partially supported by a Sloan Fellowship. We thank Seong Jun Kim for comments, corrections 500 and a careful reading of the manuscript.

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Authors and Affiliations

  1. Bar-Ilan University, Ramat Gan, 52900, Israel

    Gil Ariel

  2. Department of Mathematics and Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, TX, 78712, Austin, USA

    Björn Engquist & Yen-Hsi Richard Tsai

Authors
  1. Gil Ariel
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  2. Björn Engquist
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  3. Yen-Hsi Richard Tsai
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Corresponding author

Correspondence to Gil Ariel .

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Editors and Affiliations

  1. , Department of Mathematics, University of Texas, University Station C 1200 1, Austin, 78712-0257, Texas, USA

    Björn Engquist

  2. Dept. Numerical Analysis &, Computer Science (NADA), Royal Institute of Technology (KTH), Lindstedtsvägen 3, Stockholm, 100 44, Sweden

    Olof Runborg

  3. , Department of Mathematics, University of Texas at Austin, University Station C1200 1, Austin, 78712-0257, Texas, USA

    Yen-Hsi R. Tsai

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Ariel, G., Engquist, B., Tsai, YH.R. (2012). Oscillatory Systems with Three Separated Time Scales: Analysis and Computation. In: Engquist, B., Runborg, O., Tsai, YH. (eds) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21943-6_2

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