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.

[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.

[3]

R. W. R. Darling, Differential Forms and Connections,, Cambridge University Press, (1994). doi: 10.1017/CBO9780511805110.

[4]

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

[5]

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

[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.

[7]

W. Greub, Multilinear Algebra,, $2^{nd}$ edition, (1978).

[8]

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

[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).

[10]

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

[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.

[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.

[13]

C. Meyer, Matrix Analysis and Applied Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (2000). doi: 10.1137/1.9780898719512.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (1997). doi: 10.1137/1.9780898719574.

[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.

show all references

References:
[1]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations,, $2^{nd}$ edition, (2008). doi: 10.1002/9780470753767.

[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.

[3]

R. W. R. Darling, Differential Forms and Connections,, Cambridge University Press, (1994). doi: 10.1017/CBO9780511805110.

[4]

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

[5]

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

[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.

[7]

W. Greub, Multilinear Algebra,, $2^{nd}$ edition, (1978).

[8]

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

[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).

[10]

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

[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.

[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.

[13]

C. Meyer, Matrix Analysis and Applied Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (2000). doi: 10.1137/1.9780898719512.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, Society for Industrial and Applied Mathematics (SIAM), (1997). doi: 10.1137/1.9780898719574.

[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.

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