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
References:
[1]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients,, J. Phys. A, 44 (2011).  doi: 10.1088/1751-8113/44/30/305205.  Google Scholar

[2]

W. Gautschi, "Numerical Analysis. An Introduction,", Birkhäuser, (1997).   Google Scholar

[3]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Science, 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[4]

E. Hairer, Symmetric projection methods for differential equations on manifolds,, BIT, 40 (2000), 726.  doi: 10.1023/A:1022344502818.  Google Scholar

[5]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations,", Springer Series in Computational Mathematics, 31 (2006).   Google Scholar

[6]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems,", Springer Series in Computational Mathematics, 8 (1993).   Google Scholar

[7]

P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons Inc., (1964).   Google Scholar

[8]

V. I. Istrăţescu, "Fixed Point Theory, an Introduction,", Mathematics and its Applications, 7 (1981).   Google Scholar

[9]

Toahiaki Itoh and Kanji Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients,, J. Comput. Phys., 76 (1988), 85.  doi: 10.1016/0021-9991(88)90132-5.  Google Scholar

[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.  doi: 10.1098/rsta.1999.0363.  Google Scholar

[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, ().   Google Scholar

[12]

J. M. Ortega, The Newton-Kantorovich theorem,, Amer. Math. Monthly, 75 (1968), 658.  doi: 10.2307/2313800.  Google Scholar

[13]

Marco Papi, On the domain of the implicit function and applications,, J. Inequal. Appl., 2005 (2005), 221.  doi: 10.1155/JIA.2005.221.  Google Scholar

[14]

G. R. W. Quispel and H. W. Capel, Solving ODEs numerically while preserving a first integral,, Physics Letters. A, 218 (1996), 223.  doi: 10.1016/0375-9601(96)00403-3.  Google Scholar

[15]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/4/045206.  Google Scholar

[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).  doi: 10.1088/0305-4470/29/13/006.  Google Scholar

show all references

References:
[1]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients,, J. Phys. A, 44 (2011).  doi: 10.1088/1751-8113/44/30/305205.  Google Scholar

[2]

W. Gautschi, "Numerical Analysis. An Introduction,", Birkhäuser, (1997).   Google Scholar

[3]

O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Science, 6 (1996), 449.  doi: 10.1007/BF02440162.  Google Scholar

[4]

E. Hairer, Symmetric projection methods for differential equations on manifolds,, BIT, 40 (2000), 726.  doi: 10.1023/A:1022344502818.  Google Scholar

[5]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations,", Springer Series in Computational Mathematics, 31 (2006).   Google Scholar

[6]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems,", Springer Series in Computational Mathematics, 8 (1993).   Google Scholar

[7]

P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons Inc., (1964).   Google Scholar

[8]

V. I. Istrăţescu, "Fixed Point Theory, an Introduction,", Mathematics and its Applications, 7 (1981).   Google Scholar

[9]

Toahiaki Itoh and Kanji Abe, Hamiltonian-conserving discrete canonical equations based on variational difference quotients,, J. Comput. Phys., 76 (1988), 85.  doi: 10.1016/0021-9991(88)90132-5.  Google Scholar

[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.  doi: 10.1098/rsta.1999.0363.  Google Scholar

[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, ().   Google Scholar

[12]

J. M. Ortega, The Newton-Kantorovich theorem,, Amer. Math. Monthly, 75 (1968), 658.  doi: 10.2307/2313800.  Google Scholar

[13]

Marco Papi, On the domain of the implicit function and applications,, J. Inequal. Appl., 2005 (2005), 221.  doi: 10.1155/JIA.2005.221.  Google Scholar

[14]

G. R. W. Quispel and H. W. Capel, Solving ODEs numerically while preserving a first integral,, Physics Letters. A, 218 (1996), 223.  doi: 10.1016/0375-9601(96)00403-3.  Google Scholar

[15]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/4/045206.  Google Scholar

[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).  doi: 10.1088/0305-4470/29/13/006.  Google Scholar

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