“Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications”Authors: Arne Marthinsen, Hans Munthe-Kaas and Brynjulf OwrenAffiliation: NTNU (Department of Mathematical Sciences) Reference: 1997, Vol. 18, No. 1, pp. 75-88 |
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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.
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DOI: 10.4173/mic.1997.1.4
DOI forward links to this article:
[1] P. Krysl 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 William Shelton Jr, (2007), doi:10.1002/nme.1874
[6] Stig Faltinsen Arne Marthinsen Hans Z. Munthe-Kaas, (2001), doi:10.1016/S0168-9274(01)00103-9
[7] Hans Munthe-Kaas, (1998), doi:10.1007/BF02510919
[8] L.O. Jay, (2004), doi:10.1016/j.camwa.2003.02.013
References:
[1] ARNOLD, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Second edition.
[2] CALVO, M. P., ISERLES, A. and ZANNA, A. (1995). Numerical Solution ofIsospectral Flows, Tech. Rep. DAMTP 1995/NA03, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
[3] CALVO, M. P., ISERLES, A. and ZANNA, A. (1995). Runge-Kutta Methods on Manifolds, Preprint.
[4] CROUCH, P. E. and 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. and 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. and WANNER, G. (1993). Solving Ordinary Differential Equations 1, SCM 8, Springer-Verlag, Second Edition.
[7] HILBER, H. M., HUGHES, T. J. R. and 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. and 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. and 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. and 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. and CALVO, M. P. (1994). Numerical Hamiltonian Problems, Chapman and Hall.
[14] SIMO, J. C. and 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. and 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}
};
