BIT Numerical Mathematics

, Volume 41, Issue 5, pp 996–1007 | Cite as

Geometric Integration of Ordinary Differential Equations on Manifolds

  • E. Hairer


This article illustrates how classical integration methods for differential equations on manifolds can be modified in order to preserve certain geometric properties of the exact flow. Projection methods as well as integrators based on local coordinates are considered. The central ideas of this article have been presented at the 40th anniversary meeting of the journal BIT.

Geometric integration differential equations on manifolds symmetric methods projection methods methods based on local coordinates 


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  1. H. C. Andersen, Rattle: A “velocity” version of the Shake algorithm for molecular dynamics calculations, J. Comput. Phys., 52 (1983), pp. 24–34.Google Scholar
  2. U. M. Ascher, H. Chin, and S. Reich, Stabilization of DAEs and invariant manifolds, Numer. Math., 67 (1994) pp. 131–149.Google Scholar
  3. U. M. Ascher and S. Reich, On some difficulties in integrating highly oscillatory Hamiltonian systems, Lecture Notes in Computational Science and Engineering, 4 (1998), pp. 281–296.Google Scholar
  4. J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comput. Methods Appl. Mech. Engrg., 1 (1972), pp. 11–16.Google Scholar
  5. K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland, New York, 1989.Google Scholar
  6. P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3 (1993), pp. 1–33.Google Scholar
  7. F. Diele, L. Lopez, and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math., 8 (1998), pp. 317–334.Google Scholar
  8. E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics, Teubner, Stuttgart, 1998.Google Scholar
  9. C. W. Gear, Simultaneous numerical solution of differential-algebraic equations, IEEE Trans. Circuit Theory, CT-18 (1971), pp. 89–95.Google Scholar
  10. E. Griepentrog and R. März: Differential-algebraic equations and their numerical treatment, Teubner Texte zur Math., Vol. 88, Teubner, Stuttgart, 1986.Google Scholar
  11. E. Hairer, Numerical Geometric Integration, Unpublished Lecture Notes, March 1999, available on Scholar
  12. E. Hairer, Symmetric projection methods for differential equations on manifolds, BIT, 40:4 (2000), pp. 726–734.Google Scholar
  13. E. Hairer, Ch. Lubich, and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Vol. 1409, Springer-Verlag, Berlin, 1989.Google Scholar
  14. E. Hairer, Ch. Lubich, and G. Wanner, Geometric Numerical Integration, monograph in preparation.Google Scholar
  15. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer Series in Comput. Math., Vol. 14, Springer-Verlag, Berlin, 1996.Google Scholar
  16. A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), pp. 215–365.Google Scholar
  17. L. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal., 33 (1996), pp. 368–387.Google Scholar
  18. B. J. Leimkuhler and R. D. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comput. Phys., 112 (1994), pp. 117–125.Google Scholar
  19. H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Appl. Numer. Math., 29 (1999), pp. 115–127.Google Scholar
  20. F. A. Potra and W. C. Rheinboldt, On the numerical solution of Euler-Lagrange equations, Mech. Struct. Mech., 19 (1991), pp. 1–18.Google Scholar
  21. S. Reich, Symplectic integration of constrained Hamiltonian systems by composition methods, SIAM J. Numer. Anal., 33 (1996), pp. 475–491.Google Scholar
  22. W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds, Math. Comp., 43 (1984), pp. 473–482.Google Scholar
  23. J.-P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys., 23 (1977), pp. 327–341.Google Scholar
  24. L. F. Shampine, Conservation laws and the numerical solution of ODEs, Comput. Maths. Appls., 12B (1986) pp. 1287–1296.Google Scholar
  25. R. A. Wehage and E. J. Haug, Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems, J. Mechanical Design, 104 (1982), pp. 247–255.Google Scholar
  26. A. Zanna, K. Engø, and H. Munthe-Kaas, Adjoint and selfadjoint Lie-group methods, BIT, 41:2 (2001), pp. 395–421.Google Scholar

Copyright information

© Swets & Zeitlinger 2001

Authors and Affiliations

  • E. Hairer
    • 1
  1. 1.Section de mathématiquesUniversité de GenèveGenève 24Switzerland

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