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Energy-preserving Runge-Kutta methods

Published online by Cambridge University Press:  08 July 2009

Elena Celledoni ,
Robert I. McLachlan ,
David I. McLaren ,
Brynjulf Owren ,
G. Reinout W. Quispel  and
William M. Wright
Elena Celledoni
Affiliation:
Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway.
Robert I. McLachlan
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand. r.mclachlan@massey.ac.nz
David I. McLaren
Affiliation:
Mathematics Department, La Trobe University, VIC 3086, Australia.
Brynjulf Owren
Affiliation:
Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway.
G. Reinout W. Quispel
Affiliation:
Mathematics Department, La Trobe University, VIC 3086, Australia.
William M. Wright
Affiliation:
Mathematics Department, La Trobe University, VIC 3086, Australia.
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Abstract

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.

Keywords

B-seriesHamiltonian systemsenergy-preserving integratorsRunge-Kutta methods.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 645 - 649
Copyright
© EDP Sciences, SMAI, 2009

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References

Calvo, M.-P., Iserles, A. and Zanna, A., Numerical solution of isospectral flows. Math. Comput. 66(1997) 1461–1486. CrossRef
E. Celledoni, R.I. McLachlan, B. Owren and G.R.W. Quispel, Energy-preserving integrators and the structure of B-series. Preprint.
Chartier, P., Faou, E. and Murua, A., An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575–590. CrossRef
Cooper, G.J., Stability of Runge-Kutta methods for trajectory problems. IMA J. Numer. Anal. 7 (1987) 1–13. CrossRef
Faou, E., Hairer, E. and Pham, T.-L., Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44 (2004) 699–709. CrossRef
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin, 2nd Edition (2006).
Iserles, A. and Zanna, A., Preserving algebraic invariants with Runge-Kutta methods. J. Comput. Appl. Math. 125 (2000) 69–81. CrossRef
McLachlan, R.I., Quispel, G.R.W. and Turner, G.S., Numerical integrators that preserve symmetries and reversing symmetries. SIAM J. Numer. Anal. 35 (1998) 586–599. CrossRef
McLachlan, R.I., Quispel, G.R.W. and Robidoux, N., Geometric integration using discrete gradients. Phil. Trans. Roy. Soc. A 357 (1999) 1021–1046. CrossRef
Quispel, G.R.W. and McLaren, D.I., A new class of energy-preserving numerical integration methods. J. Phys. A 41 (2008) 045206. CrossRef
J.E. Scully, A search for improved numerical integration methods using rooted trees and splitting. MSc Thesis, La Trobe University, Australia (2002).
Shampine, L.F., Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. 12B (1986) 1287–1296. CrossRef