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

Richard A. Norton; David I. McLaren; G. R. W. Quispel; Ari Stern; Antonella Zanna

doi:10.3934/dcds.2015.35.2079

Article Contents

2015, Volume 35, Issue 5: 2079-2098. Doi: 10.3934/dcds.2015.35.2079

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

1.
Department of Physics, University of Otago, PO Box 56, Dunedin 9054
2.
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086
3.
Department of Mathematics, Washington University in St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899
4.
Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen

Received: April 30, 2014

Revised: August 31, 2014

Published: May 2017

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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.

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Mathematics Subject Classification: Primary: 65D30, 65L20; Secondary: 37M99, 65P10.

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