# American Institute of Mathematical Sciences

March  2014, 34(3): 1147-1170. doi: 10.3934/dcds.2014.34.1147

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

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

Received  November 2012 Revised  February 2013 Published  August 2013

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.
Citation: Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147
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