Discrete gradient methods for preserving a first integral of an ordinary differential equation

Richard A. Norton; G. R. W. Quispel

doi:10.3934/dcds.2014.34.1147

Article Contents

2014, Volume 34, Issue 3: 1147-1170. Doi: 10.3934/dcds.2014.34.1147

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Discrete gradient methods for preserving a first integral of an ordinary differential equation

Richard A. Norton^1, and
G. R. W. Quispel^2,

1.
Department of Physics, University of Otago, PO Box 56, Dunedin 9054
2.
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086

Received: November 2012

Revised: February 2013

Published: March 2014

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Abstract

In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large class of discrete gradient methods that the numerical solution exists and is locally unique, and that for arbitrary $p\in \mathbb{N}$ we may construct a method that is of order $p$. In the proofs of these results we also show that the constants in the time step constraint and the error bounds may be chosen independently from the distance to critical points of the first integral.
In the case when the first integral is quadratic, for arbitrary $p \in \mathbb{N}$, we have devised a new method that is linearly implicit at each time step and of order $p$. A numerical example suggests that this new method has advantages in terms of efficiency.

Keywords:

Mathematics Subject Classification: Primary: 65D30; Secondary: 65L20, 37M99, 70B10.

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