Skip to main content
SpringerLink
Log in

Lie Group Integrators

  • Conference paper
  • First Online:
Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 267))

Abstract

In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the activity has since then branched out in several different subareas, focussing both on theoretical and practical issues. From two alternative setups, using either frames or Lie group actions on a manifold, we here introduce the most important classes of schemes used to integrate nonlinear ordinary differential equations on Lie groups and manifolds. We describe a number of different applications where there is a natural action by a Lie group on a manifold such that our integrators can be implemented. An issue which is not well understood is the role of isotropy and how it affects the behaviour of the numerical methods. The order theory of numerical Lie group integrators has become an advanced subtopic in its own right, and here we give a brief introduction on a somewhat elementary level. Finally, we shall discuss Lie group integrators having the property that they preserve a symplectic structure or a first integral.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3), 359–390 (2002)

    Article  MathSciNet  Google Scholar 

  2. Berland, H., Owren, B.: Algebraic structures on ordered rooted trees and their significance to Lie group integrators. In: Group Theory and Numerical Analysis. CRM Proceedings and Lecture Notes, vol. 39, pp. 49–63. American Mathematical Society, Providence, RI (2005)

    Chapter  Google Scholar 

  3. Berland, H., Owren, B., Skaflestad, B.: \(B\)-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43(4), 1715–1727 (2005) (electronic)

    Article  MathSciNet  Google Scholar 

  4. Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470(5–6), 151–238 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bobenko, A.I., Suris, Yu.B.: Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products. Lett. Math. Phys. 49(1), 79–93 (1999)

    Google Scholar 

  6. Bobenko, A.I., Suris, Yu.B.: Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Comm. Math. Phys. 204(1), 147–188 (1999)

    Article  MathSciNet  Google Scholar 

  7. Bogfjellmo, G., Marthinsen, H.: High-Order Symplectic Partitioned Lie Group Methods. Foundations of Computational Mathematics, pp. 1–38 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bou-Rabee, N., Marsden, J.E.: Hamilton-Pontryagin integrators on Lie groups. I. Introduction and structure-preserving properties. Found. Comput. Math. 9(2), 197–219 (2009)

    Article  MathSciNet  Google Scholar 

  9. Bourbaki, N.: Lie Groups and Lie Algebras. Part I, Chapters 1–3, Addison-Wesley (1975)

    Google Scholar 

  10. Bras, S., Cunha, R., Silvestre, C.J., Oliveira, P.J.: Nonlinear attitude observer based on range and inertial measurements. IEEE Trans. Control Syst. Technol. 21(5), 1889–1897 (2013)

    Article  Google Scholar 

  11. Bruls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5(3), 031002 (2010)

    Article  Google Scholar 

  12. Bryant, R.L.: An introduction to Lie groups and symplectic geometry. In: Freed, D.S., Uhlenbeck, K.K. (eds.) Geometry and Quantum Field Theory. IAS/Park City Mathematics Series, vol. 1, 2nd edn. American Mathematical Society (1995)

    Google Scholar 

  13. Calvo, M.P., Iserles, A., Zanna, A.: Runge-Kutta methods for orthogonal and isospectral flows. Appl. Numer. Math. 22, 153–163 (1996)

    Article  MathSciNet  Google Scholar 

  14. Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Math. Comp. 66(220), 1461–1486 (1997)

    Article  MathSciNet  Google Scholar 

  15. Casas, F., Owren, B.: Cost efficient Lie group integrators in the RKMK class. BIT Numer. Math. 43(4), 723–742 (2003)

    Article  MathSciNet  Google Scholar 

  16. Celledoni, E.: A note on the numerical integration of the KdV equation via isospectral deformations. J. Phys. A: Math. Gener. 34(11), 2205–2214 (2001)

    Article  MathSciNet  Google Scholar 

  17. Celledoni, E., Iserles, A.: Approximating the exponential from a Lie algebra to a Lie group. Math. Comp. (2000). Posted on March 15, PII S 0025-5718(00)01223-0 (to appear in print)

    Google Scholar 

  18. Celledoni, E., Iserles, A.: Methods for the approximation of the matrix exponential in a Lie-algebraic setting. IMA J. Numer. Anal. 21(2), 463–488 (2001)

    Article  MathSciNet  Google Scholar 

  19. Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie group methods. Future Gener. Comput. Syst. 19, 341–352 (2003)

    Article  Google Scholar 

  20. Celledoni, E., Marthinsen, H., Owren, B.: An introduction to Lie group integrators—basics, new developments and applications. J. Comput. Phys. 257(part B), 1040–1061 (2014)

    Article  MathSciNet  Google Scholar 

  21. Celledoni, E., Owren, B.: A class of intrinsic schemes for orthogonal integration. SIAM J. Numer. Anal. 40(6), 2069–2084 (2002, 2003) (electronic)

    Article  MathSciNet  Google Scholar 

  22. Celledoni, E., Owren, B.: On the implementation of Lie group methods on the Stiefel manifold. Numer. Algorithms 32(2–4), 163–183 (2003)

    Article  MathSciNet  Google Scholar 

  23. Celledoni, E., Owren, B.: Preserving first integrals with symmetric Lie group methods. Discrete Contin. Dyn. Syst. 34(3), 977–990 (2014)

    Article  MathSciNet  Google Scholar 

  24. Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge Tracts in Mathematics, vol. 108. Cambridge University Press (1993)

    Google Scholar 

  25. Crouch, P.E., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3, 1–33 (1993)

    Article  MathSciNet  Google Scholar 

  26. Dahlby, M., Owren, B., Yaguchi, T.: Preserving multiple first integrals by discrete gradients. J. Phys. A, 44(30), 305205, 14 (2011)

    Article  MathSciNet  Google Scholar 

  27. Ebrahimi-Fard, K., Lundervold, A., Munthe-Kaas, H.: On the Lie enveloping algebra of a post-Lie algebra. J. Lie Theory 25(4), 1139–1165 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Engø, K.: Partitioned Runge-Kutta methods in Lie-group setting. BIT 43(1), 21–39 (2003)

    Article  MathSciNet  Google Scholar 

  29. Engø, K., Faltinsen, S.: Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy. SIAM J. Numer. Anal. 39(1), 128–145 (2001)

    Article  MathSciNet  Google Scholar 

  30. Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)

    Article  MathSciNet  Google Scholar 

  31. Grossman, R.L., Larson, R.G.: Differential algebra structures on families of trees. Adv. Appl. Math. 35(1), 97–119 (2005)

    Article  MathSciNet  Google Scholar 

  32. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  33. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Springer-Verlag (1972)

    Google Scholar 

  34. Iserles, A.: Magnus expansions and beyond. In: Combinatorics and Physics. Contemporary Mathematics, vol. 539, pp. 171–186. American Mathematical Society, Providence, RI (2011)

    Google Scholar 

  35. Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numerica 9, 215–365 (2000)

    Article  MathSciNet  Google Scholar 

  36. Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Trans. Graph. 28(2) (2009)

    Article  Google Scholar 

  37. Krogstad, S.: A low complexity Lie group method on the Stiefel manifold. BIT 43(1), 107–122 (2003)

    Article  MathSciNet  Google Scholar 

  38. Krogstad, S., Munthe-Kaas, H., Zanna, A.: Generalized polar coordinates on Lie groups and numerical integrators. Numer. Math. 114(1), 161–187 (2009)

    Article  MathSciNet  Google Scholar 

  39. Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods Appl. Mech. Eng. 196(29–30), 2907–2924 (2007)

    Article  MathSciNet  Google Scholar 

  40. Lewis, D., Olver, P.J.: Geometric integration algorithms on homogeneous manifolds. Found. Comput. Math. 2(4), 363–392 (2002)

    Article  MathSciNet  Google Scholar 

  41. Lewis, D., Simo, J.C.: Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups. J. Nonlinear Sci. 4, 253–299 (1994)

    Article  MathSciNet  Google Scholar 

  42. Lundervold, A., Munthe-Kaas, H.: Hopf algebras of formal diffeomorphisms and numerical integration on manifolds. In: Combinatorics and Physics. Contemporary Mathematics, vol. 539, pp. 295–324. American Mathematical Society, Providence, RI (2011)

    Google Scholar 

  43. Malham, S.J.A., Wiese, A.: Stochastic Lie group integrators. SIAM J. Sci. Comput. 30(2), 597–617 (2008)

    Article  MathSciNet  Google Scholar 

  44. Marsden, J.E., Pekarsky, S., Shkoller, S.: Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12(6), 1647–1662 (1999)

    Article  MathSciNet  Google Scholar 

  45. Marsden, J.E., Pekarsky, S., Shkoller, S.: Symmetry reduction of discrete Lagrangian mechanics on Lie groups. J. Geom. Phys. 36(1–2), 140–151 (2000)

    Article  MathSciNet  Google Scholar 

  46. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol. 17, 2nd edn. Springer-Verlag (1999)

    Google Scholar 

  47. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)

    Article  MathSciNet  Google Scholar 

  48. McLachlan, R., Modin, K., Verdier, O.: A minimal-variable symplectic integrator on spheres. Math. Comp. 86(307), 2325–2344 (2017)

    Article  MathSciNet  Google Scholar 

  49. McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Phil. Trans. Royal Soc. A 357, 1021–1046 (1999)

    Article  MathSciNet  Google Scholar 

  50. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20(4), 801–836 (1978)

    Article  MathSciNet  Google Scholar 

  51. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003) (electronic)

    Article  MathSciNet  Google Scholar 

  52. Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys. 139(2), 217–243 (1991)

    Article  MathSciNet  Google Scholar 

  53. Munthe-Kaas, H.: Lie-Butcher theory for Runge-Kutta methods. BIT 35(4), 572–587 (1995)

    Article  MathSciNet  Google Scholar 

  54. Munthe-Kaas, H.: Runge-Kutta methods on Lie groups. BIT 38(1), 92–111 (1998)

    Article  MathSciNet  Google Scholar 

  55. Munthe-Kaas, H.: High order Runge-Kutta methods on manifolds. Appl. Numer. Math. 29, 115–127 (1999)

    Article  MathSciNet  Google Scholar 

  56. Munthe-Kaas, H., Lundervold, A.: On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013)

    Article  MathSciNet  Google Scholar 

  57. Munthe-Kaas, H., Owren, B.: Computations in a free Lie algebra. Phil. Trans. Royal Soc. A 357, 957–981 (1999)

    Article  MathSciNet  Google Scholar 

  58. Munthe-Kaas, H., Verdier, O.: Integrators on homogeneous spaces: isotropy choice and connections. Found. Comput. Math. 16(4), 899–939 (2016)

    Article  MathSciNet  Google Scholar 

  59. Munthe-Kaas, H., Wright, W.M.: On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8(2), 227–257 (2008)

    Article  MathSciNet  Google Scholar 

  60. Munthe-Kaas, H., Zanna, A.: Numerical integration of differential equations on homogeneous manifolds. In: Cucker, F., Shub, M. (eds.) Foundations of Computational Mathematics, pp. 305–315. Springer-Verlag (1997)

    Google Scholar 

  61. Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT 54(1), 257–282 (2014)

    Article  MathSciNet  Google Scholar 

  62. Murua, A.: Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators. Appl. Numer. Math. 29(2), 221–251 (1999)

    Article  MathSciNet  Google Scholar 

  63. Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6(4), 387–426 (2006)

    Article  MathSciNet  Google Scholar 

  64. Owren, B.: Order conditions for commutator-free Lie group methods. J. Phys. A 39(19), 5585–5599 (2006)

    Article  MathSciNet  Google Scholar 

  65. Owren, B., Marthinsen, A.: Runge-Kutta methods adapted to manifolds and based on rigid frames. BIT 39(1), 116–142 (1999)

    Article  MathSciNet  Google Scholar 

  66. Owren, B., Marthinsen, A.: Integration methods based on canonical coordinates of the second kind. Numer. Math. 87(4), 763–790 (2001)

    Article  MathSciNet  Google Scholar 

  67. Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A, 41(4), 045206, 7 (2008)

    Article  MathSciNet  Google Scholar 

  68. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol. 7. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1993)

    Google Scholar 

  69. Rodrigues, O.: Des lois géometriques qui regissent les déplacements d’ un systéme solide dans l’ espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendent des causes qui peuvent les produire. J. Math. Pures Appl. 5, 380–440 (1840)

    Google Scholar 

  70. Saccon, A.: Midpoint rule for variational integrators on Lie groups. Int. J. Numer. Methods Eng. 78(11), 1345–1364 (2009)

    Article  MathSciNet  Google Scholar 

  71. Terze, Z., Muller, A., Zlatar, D.: Lie-group integration method for constrained multibody systems in state space. Multibody Syst. Dyn. 34(3), 275–305 (2015)

    Article  MathSciNet  Google Scholar 

  72. Veselov, A.P.: Integrable discrete-time systems and difference operators. Funct. Anal. Appl. 22, 83–93 (1988)

    Article  MathSciNet  Google Scholar 

  73. Wensch, J., Knoth, O., Galant, A.: Multirate infinitesimal step methods for atmospheric flow simulation. BIT 49(2), 449–473 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Norwegian University of Science and Technology, Alfred Getz vei 1, 7491, Trondheim, Norway

    Brynjulf Owren

Authors
  1. Brynjulf Owren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brynjulf Owren .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology—NTNU, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  2. Institute for the Mathematical Sciences (ICMAT), Madrid, Spain

    María Barbero Liñán

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Owren, B. (2018). Lie Group Integrators. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_2

Download citation

Publish with us

Policies and ethics

Search

Navigation