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Published 2011 | Submitted
Journal Article Open

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


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.

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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