# American Institute of Mathematical Sciences

May  2015, 35(5): 2079-2098. doi: 10.3934/dcds.2015.35.2079

## 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, Australia 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  May 2014 Revised  September 2014 Published  December 2014

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.
Citation: Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079
##### References:
 [1] J. C. Butcher, Numerical Methods for Ordinary Differential Equations,, $2^{nd}$ edition, (2008). doi: 10.1002/9780470753767. Google Scholar [2] 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 [3] R. W. R. Darling, Differential Forms and Connections,, Cambridge University Press, (1994). doi: 10.1017/CBO9780511805110. Google Scholar [4] W. Gautschi, Numerical Analysis. An Introduction,, Birkhäuser, (1997). Google Scholar [5] O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Science, 6 (1996), 449. doi: 10.1007/BF02440162. Google Scholar [6] V. Grimm and G. R. W. Quispel, Geometric integration methods that preserve Lyapunov functions,, BIT, 45 (2005), 709. doi: 10.1007/s10543-005-0034-z. Google Scholar [7] W. Greub, Multilinear Algebra,, $2^{nd}$ edition, (1978). Google Scholar [8] E. Hairer, Symmetric projection methods for differential equations on manifolds,, BIT, 40 (2000), 726. doi: 10.1023/A:1022344502818. Google Scholar [9] 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 [10] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems,, Springer Series in Computational Mathematics, 8 (1993). Google Scholar [11] T. Itoh and K. 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 [12] R. I. McLachlan, G. R. W. Quispel and N. 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 [13] C. Meyer, Matrix Analysis and Applied Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (2000). doi: 10.1137/1.9780898719512. Google Scholar [14] R. A. Norton and G. R. W. Quispel, Discrete gradient methods for preserving a first integral of an ordinary differential equation,, Discret. Contin. Dyn. S., 34 (2014), 1147. doi: 10.3934/dcds.2014.34.1147. Google Scholar [15] J. M. Ortega, The Newton-Kantorovich theorem,, Amer. Math. Monthly, 75 (1968), 658. doi: 10.2307/2313800. Google Scholar [16] M. Papi, On the domain of the implicit function and applications,, J. Inequal. Appl., 2005 (2005), 221. doi: 10.1155/JIA.2005.221. Google Scholar [17] 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 [18] G. R. W. Quispel and C. Dyt, Solving ODE's numerically while preserving symmetries, Hamiltonian structure, phase space volume, or first integrals,, in Proceedings of the 15th IMACS World Congress (ed. A. Sydow), 2 (1997), 601. Google Scholar [19] 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 [20] 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 [21] J. C. Simo, N. Tarnow and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics,, Comput. Methods Appl. Mech. Engrg., 100 (1992), 63. doi: 10.1016/0045-7825(92)90115-Z. Google Scholar [22] L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (1997). doi: 10.1137/1.9780898719574. Google Scholar [23] G. Zhong and J. E. Marsden, Lie-Poisson Hamiltonian-Jacobi theory and Lie-Poisson integrators,, Physics Letters A, 133 (1988), 134. doi: 10.1016/0375-9601(88)90773-6. Google Scholar

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##### References:
 [1] J. C. Butcher, Numerical Methods for Ordinary Differential Equations,, $2^{nd}$ edition, (2008). doi: 10.1002/9780470753767. Google Scholar [2] 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 [3] R. W. R. Darling, Differential Forms and Connections,, Cambridge University Press, (1994). doi: 10.1017/CBO9780511805110. Google Scholar [4] W. Gautschi, Numerical Analysis. An Introduction,, Birkhäuser, (1997). Google Scholar [5] O. Gonzalez, Time integration and discrete Hamiltonian systems,, J. Nonlinear Science, 6 (1996), 449. doi: 10.1007/BF02440162. Google Scholar [6] V. Grimm and G. R. W. Quispel, Geometric integration methods that preserve Lyapunov functions,, BIT, 45 (2005), 709. doi: 10.1007/s10543-005-0034-z. Google Scholar [7] W. Greub, Multilinear Algebra,, $2^{nd}$ edition, (1978). Google Scholar [8] E. Hairer, Symmetric projection methods for differential equations on manifolds,, BIT, 40 (2000), 726. doi: 10.1023/A:1022344502818. Google Scholar [9] 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 [10] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems,, Springer Series in Computational Mathematics, 8 (1993). Google Scholar [11] T. Itoh and K. 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 [12] R. I. McLachlan, G. R. W. Quispel and N. 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 [13] C. Meyer, Matrix Analysis and Applied Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (2000). doi: 10.1137/1.9780898719512. Google Scholar [14] R. A. Norton and G. R. W. Quispel, Discrete gradient methods for preserving a first integral of an ordinary differential equation,, Discret. Contin. Dyn. S., 34 (2014), 1147. doi: 10.3934/dcds.2014.34.1147. Google Scholar [15] J. M. Ortega, The Newton-Kantorovich theorem,, Amer. Math. Monthly, 75 (1968), 658. doi: 10.2307/2313800. Google Scholar [16] M. Papi, On the domain of the implicit function and applications,, J. Inequal. Appl., 2005 (2005), 221. doi: 10.1155/JIA.2005.221. Google Scholar [17] 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 [18] G. R. W. Quispel and C. Dyt, Solving ODE's numerically while preserving symmetries, Hamiltonian structure, phase space volume, or first integrals,, in Proceedings of the 15th IMACS World Congress (ed. A. Sydow), 2 (1997), 601. Google Scholar [19] 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 [20] 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 [21] J. C. Simo, N. Tarnow and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics,, Comput. Methods Appl. Mech. Engrg., 100 (1992), 63. doi: 10.1016/0045-7825(92)90115-Z. Google Scholar [22] L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (1997). doi: 10.1137/1.9780898719574. Google Scholar [23] G. Zhong and J. E. Marsden, Lie-Poisson Hamiltonian-Jacobi theory and Lie-Poisson integrators,, Physics Letters A, 133 (1988), 134. doi: 10.1016/0375-9601(88)90773-6. Google Scholar
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