Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A multiscale method for highly oscillatory ordinary differential equations with resonance


Authors: Gil Ariel, Bjorn Engquist and Richard Tsai
Journal: Math. Comp. 78 (2009), 929-956
MSC (2000): Primary 65L05, 34E13, 34E20
Published electronically: October 3, 2008
MathSciNet review: 2476565
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $ \epsilon$ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65L05, 34E13, 34E20

Retrieve articles in all journals with MSC (2000): 65L05, 34E13, 34E20


Additional Information

Gil Ariel
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: ariel@math.utexas.edu

Bjorn Engquist
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: engquist@math.utexas.edu

Richard Tsai
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: ytsai@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-08-02139-X
PII: S 0025-5718(08)02139-X
Received by editor(s): June 19, 2007
Received by editor(s) in revised form: January 20, 2008
Published electronically: October 3, 2008
Dedicated: In Memory of Germund Dahlquist
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.