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From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials

Tao, Molei and Owhadi, Houman and Marsden, Jerrold E. (2011) From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials. Applied Mathematics Research eXpress, 2011 (2). pp. 242-280. ISSN 1687-1200. http://resolver.caltech.edu/CaltechAUTHORS:20120215-101206163

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Abstract

We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1093/amrx/abr008DOIUNSPECIFIED
http://amrx.oxfordjournals.org/content/2011/2/242.abstractPublisherUNSPECIFIED
http://arxiv.org/abs/1006.4659arXivUNSPECIFIED
Additional Information:© 2011 The Author(s). Published by Oxford University Press. Received July 9, 2010; Revised April 12, 2011; Accepted May 16, 2011. Advance Access publication June 30, 2011. We sincerely thank Charles Van Loan for a stimulating discussion and Sydney Garstang for proofreading the manuscript. We are also grateful to two anonymous referees for precise and detailed comments and suggestions. This work was supported by the National Science Foundation [CMMI-092600].
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NSFCMMI-092600
Record Number:CaltechAUTHORS:20120215-101206163
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120215-101206163
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29301
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:20 Mar 2012 23:02
Last Modified:26 Dec 2012 14:50

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