“Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications”

Authors: Arne Marthinsen, Hans Munthe-Kaas and Brynjulf Owren,
Affiliation: NTNU
Reference: 1997, Vol 18, No 1, pp. 75-88.

Keywords: Ordinary differential equations, manifolds, numerical analysis, initial value problems

Abstract: During the last few years, different approaches for integrating ordinary differential equations on manifolds have been published. In this work, we consider two of these approaches. We present some numerical experiments showing benefits and some pitfalls when using the new methods. To demonstrate how they work, we compare with well known classical methods, e.g. Newmark and Runge-Kutta methods.

PDF PDF (1557 Kb)        DOI: 10.4173/mic.1997.1.4

DOI forward links to this article:
[1] P. Krysl and L. Endres (2005), doi:10.1002/nme.1272
[2] Arne Marthinsen (1999), doi:10.1137/S0036142998338861
[3] A. Zanna (1999), doi:10.1137/S0036142997326616
[4] Hans Munthe-Kaas (1999), doi:10.1016/S0168-9274(98)00030-0
[5] Phani Kumar V. V. Nukala and William Shelton Jr (2007), doi:10.1002/nme.1874
[6] A. Müller and Z. Terze (2013), doi:10.1016/j.cam.2013.10.039
[7] Stig Faltinsen, Arne Marthinsen and Hans Z. Munthe-Kaas (2001), doi:10.1016/S0168-9274(01)00103-9
[8] Hans Munthe-Kaas (1998), doi:10.1007/BF02510919
[9] L.O. Jay (2004), doi:10.1016/j.camwa.2003.02.013
[10] Andreas Müller and Zdravko Terze (2014), doi:10.1016/j.mechmachtheory.2014.06.014
[11] Zdravko Terze, Andreas Müller and Dario Zlatar (2016), doi:10.1007/s11044-016-9518-7
[12] Andreas Müller and Zdravko Terze (2016), doi:10.1007/s00707-016-1760-9
[13] Charles Curry and Brynjulf Owren (2019), doi:10.1007/s11075-019-00659-0
[14] Andreas Müller and Zdravko Terze (2014), doi:10.1007/978-3-319-07260-9_6
References:
[1] ARNOLD, V. I. (1989). Mathematical Methods of Classical Mechanics, Springer-Verlag, GTM 60, Second edition.
[2] CALVO, M. P., ISERLES, A. ZANNA, A. (1995). Numerical Solution of Isospectral Flows, Tech. Rep. DAMTP 1995/NA03, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
[3] CALVO, M. P., ISERLES, A. ZANNA, A. (1995). Runge-Kutta Methods on Manifolds, Preprint.
[4] CROUCH, P. E. GROSSMAN, R. (1993). Numerical Integration of Ordinary Differential Equations on Manifolds, J. Nonlinear Sci., Vol. 3, pp. 1-33 doi:10.1007/BF02429858
[5] DIECI, L., RUSSEL, R. D. VAN VLECK, E. S. (1994). Unitary Integrators and Applications to Continuous Orthonormalization Techniques, SIAM J. Numer. Anal., Vol. 31, No. 1, pp. 261-281 doi:10.1137/0731014
[6] HAIRER, E., NØRSETT, S. P. WANNER, G. (1993). Solving Ordinary Differential Equations 1, SCM 8, Springer-Verlag, Second Edition.
[7] HILBER, H. M., HUGHES, T. J. R. TAYLOR, R. L. (1977). Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics, Earthquake Eng. and Struct. Dynamics, Vol. 5, pp. 283-292 doi:10.1002/eqe.4290050306
[8] ISERLES, A. ZANNA, A. (1995). Qualitative Numerical Analysis of Ordinary Differential Equations, Tech. Rep. DAMTP 1995/NA05, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
[9] MARTHINSEN, A. OWREN, B. (1996). Order Conditions for Integration Methods Based on Rigid Frames, manuscript.
[10] MUNTHE-KAAS, (1995). Lie-Butcher Theory for Runge-Kutta Methods, BIT 35, pp. 572-587 doi:10.1007/BF01739828
[11] MUNTHE-KAAS, H. (1996). Runge-Kutta Methods on Lie Groups, submitted to BIT.
[12] MUNTHE-KAAS, H. ZANNA, A. (1997). Numerical Integration of Differential Equations on Homogeneous Manifolds, in Foundation of Computational Mathematics, CUCKER, F..ed., Springer Verlag, to appear.
[13] SANZ-SERNA, J. M. CALVO, M. P. (1994). Numerical Hamiltonian Problems, Chapman and Hall.
[14] SIMO, J. C. WONG, K. K. (1991). Unconditionally Stable Algorithms for Rigid Body Dynamics that Exactly Preserve Energy and Momentum, Intl. J. for Num. Methods in Engineering, Vol. 31.
[15] ZANNA, A. MUNTHE-KAAS, H. (1997). Iterated Commutators, Lie´s Reduction Method and Ordinary Differential Equations on Matrix Lie Groups, In Foundation of Computational Mathematics, CUCKER. F..ed., Springer Verlag, to appear.


BibTeX:
@article{MIC-1997-1-4,
  title={{Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications}},
  author={Marthinsen, Arne and Munthe-Kaas, Hans and Owren, Brynjulf},
  journal={Modeling, Identification and Control},
  volume={18},
  number={1},
  pages={75--88},
  year={1997},
  doi={10.4173/mic.1997.1.4},
  publisher={Norwegian Society of Automatic Control}
};