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High-Order Symplectic Partitioned Lie Group Methods

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Abstract

In this article, a unified approach to obtain symplectic integrators on \(T^{*}G\) from Lie group integrators on a Lie group \(G\) is presented. The approach is worked out in detail for symplectic integrators based on Runge–Kutta–Munthe-Kaas methods and Crouch–Grossman methods. These methods can be interpreted as symplectic partitioned Runge–Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.

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Notes

  1. This group structure was used by Engø in [12] to construct partitioned Runge–Kutta–Munthe-Kaas methods on \(T^{*}G\), without any special regard to symplecticity.

  2. Variational methods for degenerate Hamiltonian systems using Type II generating functions have been proposed by Leok and Zhang [14].

  3. Patrick and Cuell [19] demonstrate an inaccuracy in the proof in [5]. However, they also show that the relevant result still holds.

  4. Since \(\hat{b}_i=b_i\) and the right-hand side is independent of \(S\), we could instead have grouped \(S\) with \(p\) without any change.

References

  1. Peter E. Crouch and Robert L. Grossman. ‘Numerical integration of ordinary differential equations on manifolds’. In: J. Nonlinear Sci. 3.1 (1993), pp. 1–33, doi:10.1007/BF02429858.

  2. Hans Z.Munthe-Kaas. ‘High order Runge–Kutta methods on manifolds’. In: Proceedings of the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Differential Equations (Golden, CO, 1997). Vol. 29. 1. 1999, pp. 115–127. doi:10.1016/S0168-9274(98)00030-0.

  3. Arieh Iserles, Hans Z. Munthe-Kaas, Syvert Paul Nørsett and Antonella Zanna. ‘Lie-group methods’. In: Acta numerica, 2000. Vol. 9. Acta Numer. Cambridge: Cambridge Univ. Press, 2000, pp. 215–365. doi:10.1017/S0962492900002154.

  4. Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. 2nd ed. Vol. 31. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 2006, pp. xviii+644. doi:10.1007/3-540-30666-8.

  5. Jerrold Eldon Marsden and Matthew West. ‘Discrete mechanics and variational integrators’. In: Acta Numer. 10 (2001), pp. 357–514. doi:10.1017/S096249290100006X.

  6. Melvin Leok. Variational Integrators. In: Encyclopedia of Applied and Computational Mathematics. Ed. by Björn Engquist. Springer-Verlag, 2013.

  7. Nawaf M. Bou-Rabee and Jerrold Eldon Marsden. ‘Hamilton–Pontryagin integrators on Lie groups. I. Introduction and structure-preserving properties’. In: Found. Comput. Math. 9.2 (2009), pp. 197–219. doi:10.1007/s10208-008-9030-4.

  8. Elena Celledoni, Håkon Marthinsen and Brynjulf Owren. ‘An introduction to Lie group integrators – basics, new developments and applications’. In: J. Comput. Phys. 257 (2014), pp. 1040–1061. doi:10.1016/j.jcp.2012.12.031.

  9. Melvin Leok. ‘Foundations of computational geometric mechanics’. PhD thesis. California Institute of Technology, 2004. URL: http://resolver.caltech.edu/CaltechETD:etd-03022004-000251.

  10. James Hall and Melvin Leok. Lie Group Spectral Variational Integrators. 2014. arXiv: 1402.3327 [math.NA]. Submitted.

  11. Vladimir Igorevich Arnold and Boris A. Khesin. Topological methods in hydrodynamics. Vol. 125. Applied Mathematical Sciences. Springer-Verlag, New York, 1998, pp. xvi+374. doi:10.1007/b97593.

  12. Kenth Engø. ‘Partitioned Runge–Kutta methods in Lie-group setting’. In: BIT 43.1 (2003), pp. 21–39. doi:10.1023/A:1023668015087.

  13. Yuri Borisovich Suris. ‘Hamiltonian methods of Runge–Kutta type and their variational interpretation’. In: Mat. Model. 2.4 (1990), pp. 78–87. URL: http://mi.mathnet.ru/eng/mm2356.

  14. Melvin Leok and Jingjing Zhang. ‘Discrete Hamiltonian variational integrators’. In: IMA J. Numer. Anal. 31.4 (2011), pp. 1497–1532. doi:10.1093/imanum/drq027.

  15. Elena Celledoni, Arne Marthinsen and Brynjulf Owren. ‘Commutator-free Lie group methods’. In: Future Generation Computer Systems 19.3 (2003), pp. 341–352. doi:10.1016/S0167-739X(02)00161-9.

  16. Brynjulf Owren. ‘Order conditions for commutator-free Lie group methods’. In: J. Phys. A 39.19 (2006), pp. 5585–5599. doi:10.1088/0305-4470.

  17. Jerrold Eldon Marsden and Tudor S. Ratiu. Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems. 2nd ed. Vol. 17. Texts in Applied Mathematics. Springer-Verlag, New York, 1999, pp. xviii+582. doi:10.1007/978-0-387-21792-5.

  18. Brynjulf Owren and Arne Marthinsen. ‘Runge–Kutta methods adapted to manifolds and based on rigid frames’. In: BIT 39.1 (1999), pp. 116–142. doi:10.1023/A:1022325426017.

  19. George W. Patrick and Charles Cuell. ‘Error analysis of variational integrators of unconstrained Lagrangian systems’. In: Numer. Math. 113.2 (2009), pp. 243–264. doi:10.1007/s00211-009-0245-3.

  20. Ander Murua. ‘On order conditions for partitioned symplectic methods’. In: SIAM J. Numer. Anal. 34.6 (1997), pp. 2204–2211. doi:10.1137/S0036142995285162.

  21. Haruo Yoshida. ‘Construction of higher order symplectic integrators’. In: Phys. Lett. A 150.5–7 (1990), pp. 262–268. doi:10.1016/0375-9601(90)90092-3.

  22. Kenth Engø, Arne Marthinsen and Hans Z. Munthe-Kaas. ‘DiffMan: an object-oriented MATLAB toolbox for solving differential equations on manifolds’. In: Appl. Numer. Math. 39.3–4 (2001). Special issue: Themes in geometric integration, pp. 323–347. doi:10.1016/S0168-9274(00)00042-8.

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Acknowledgments

We would like to thank our supervisor, Brynjulf Owren, for encouragement and many helpful discussions. We would also like to thank Klas Modin for fruitful discussions leading to the writing of this paper. Finally, we would like to thank the two anonymous referees for helpful comments and suggestions. The research was supported by the Research Council of Norway and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

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  1. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Geir Bogfjellmo & Håkon Marthinsen

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  1. Geir Bogfjellmo
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Correspondence to Geir Bogfjellmo.

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Communicated by Arieh Iserles.

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Bogfjellmo, G., Marthinsen, H. High-Order Symplectic Partitioned Lie Group Methods. Found Comput Math 16, 493–530 (2016). https://doi.org/10.1007/s10208-015-9257-9

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