# 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 and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147
##### References:
 [1] M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys. A, 44 (2011), 305205, 14 pp. doi: 10.1088/1751-8113/44/30/305205. [2] W. Gautschi, "Numerical Analysis. An Introduction," Birkhäuser, Boston, 1997. [3] O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Science, 6 (1996), 449-467. doi: 10.1007/BF02440162. [4] E. Hairer, Symmetric projection methods for differential equations on manifolds, BIT, 40 (2000), 726-734. doi: 10.1023/A:1022344502818. [5] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics, 31, $2^{nd}$ edition, Springer-Verlag, Berlin, 2006. [6] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," Springer Series in Computational Mathematics, 8, $2^{nd}$ edition, Springer-Verlag, Berlin, 1993. [7] P. Hartman, "Ordinary Differential Equations," John Wiley & Sons Inc., New York, 1964. [8] V. I. Istrăţescu, "Fixed Point Theory, an Introduction," Mathematics and its Applications, 7, D. Reidel Publishing Co., Dordrecht, Holland, 1981. [9] Toahiaki Itoh and Kanji Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients, J. Comput. Phys., 76 (1988), 85-102. doi: 10.1016/0021-9991(88)90132-5. [10] Robert I. McLachlan, G. R. W. Quispel and Nicolas Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363. [11] R. A. Norton, D. I. McLaren, G. R. W. Quispel, A. Stern and A. Zanna, Projection methods and discrete gradient methods for preserving first integrals of ODEs, preprint, arXiv:1302.2713v1. [12] J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660. doi: 10.2307/2313800. [13] Marco Papi, On the domain of the implicit function and applications, J. Inequal. Appl., 2005 (2005), 221-234. doi: 10.1155/JIA.2005.221. [14] G. R. W. Quispel and H. W. Capel, Solving ODEs numerically while preserving a first integral, Physics Letters. A, 218 (1996), 223-228. doi: 10.1016/0375-9601(96)00403-3. [15] G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045207, 7pp. doi: 10.1088/1751-8113/41/4/045206. [16] G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral, J. Phys. A, 29 (1996), L341-L349. doi: 10.1088/0305-4470/29/13/006.

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##### References:
 [1] M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys. A, 44 (2011), 305205, 14 pp. doi: 10.1088/1751-8113/44/30/305205. [2] W. Gautschi, "Numerical Analysis. An Introduction," Birkhäuser, Boston, 1997. [3] O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Science, 6 (1996), 449-467. doi: 10.1007/BF02440162. [4] E. Hairer, Symmetric projection methods for differential equations on manifolds, BIT, 40 (2000), 726-734. doi: 10.1023/A:1022344502818. [5] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations," Springer Series in Computational Mathematics, 31, $2^{nd}$ edition, Springer-Verlag, Berlin, 2006. [6] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," Springer Series in Computational Mathematics, 8, $2^{nd}$ edition, Springer-Verlag, Berlin, 1993. [7] P. Hartman, "Ordinary Differential Equations," John Wiley & Sons Inc., New York, 1964. [8] V. I. Istrăţescu, "Fixed Point Theory, an Introduction," Mathematics and its Applications, 7, D. Reidel Publishing Co., Dordrecht, Holland, 1981. [9] Toahiaki Itoh and Kanji Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients, J. Comput. Phys., 76 (1988), 85-102. doi: 10.1016/0021-9991(88)90132-5. [10] Robert I. McLachlan, G. R. W. Quispel and Nicolas Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363. [11] R. A. Norton, D. I. McLaren, G. R. W. Quispel, A. Stern and A. Zanna, Projection methods and discrete gradient methods for preserving first integrals of ODEs, preprint, arXiv:1302.2713v1. [12] J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660. doi: 10.2307/2313800. [13] Marco Papi, On the domain of the implicit function and applications, J. Inequal. Appl., 2005 (2005), 221-234. doi: 10.1155/JIA.2005.221. [14] G. R. W. Quispel and H. W. Capel, Solving ODEs numerically while preserving a first integral, Physics Letters. A, 218 (1996), 223-228. doi: 10.1016/0375-9601(96)00403-3. [15] G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045207, 7pp. doi: 10.1088/1751-8113/41/4/045206. [16] G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral, J. Phys. A, 29 (1996), L341-L349. doi: 10.1088/0305-4470/29/13/006.
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