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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Projection methods and discrete gradient methods for preserving first integrals of ODEs

Pages: 2079 - 2098, Volume 35, Issue 5, May 2015      doi:10.3934/dcds.2015.35.2079

 
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Richard A. Norton - Department of Physics, University of Otago, PO Box 56, Dunedin 9054, New Zealand (email)
David I. McLaren - Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia (email)
G. R. W. Quispel - Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia (email)
Ari Stern - Department of Mathematics, Washington University in St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899, United States (email)
Antonella Zanna - Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway (email)

Abstract: In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that each (linear) projection method is equivalent to a class of discrete gradient methods, in both single and multiple first integral cases, and known results for discrete gradient methods also apply to projection methods. Thus we prove that in the single first integral case, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. Our results allow considerable freedom for the choice of projection direction and do not have a time step restriction close to critical points.

Keywords:  Geometric integration, projection, discrete gradients, energy preserving integrators, Hamiltonian systems.
Mathematics Subject Classification:  Primary: 65D30, 65L20; Secondary: 37M99, 65P10.

Received: May 2014;      Revised: September 2014;      Available Online: December 2014.

 References