Lie Group Integrators

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


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.


Lie group integrators Geometric integration Symplectic integrators Geometric mechanics Structure preserving integrators Energy preservation Numerical methods on manifolds Lie groups Homogeneous spaces 


65L05 34C40 34G20 37M15 


  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 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. 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bogfjellmo, G., Marthinsen, H.: High-Order Symplectic Partitioned Lie Group Methods. Foundations of Computational Mathematics, pp. 1–38 (2015)MathSciNetCrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourbaki, N.: Lie Groups and Lie Algebras. Part I, Chapters 1–3, Addison-Wesley (1975)Google Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Bruls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5(3), 031002 (2010)CrossRefGoogle Scholar
  12. 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. 13.
    Calvo, M.P., Iserles, A., Zanna, A.: Runge-Kutta methods for orthogonal and isospectral flows. Appl. Numer. Math. 22, 153–163 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Math. Comp. 66(220), 1461–1486 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Casas, F., Owren, B.: Cost efficient Lie group integrators in the RKMK class. BIT Numer. Math. 43(4), 723–742 (2003)MathSciNetCrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 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. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie group methods. Future Gener. Comput. Syst. 19, 341–352 (2003)CrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Celledoni, E., Owren, B.: A class of intrinsic schemes for orthogonal integration. SIAM J. Numer. Anal. 40(6), 2069–2084 (2002, 2003) (electronic)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Celledoni, E., Owren, B.: On the implementation of Lie group methods on the Stiefel manifold. Numer. Algorithms 32(2–4), 163–183 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Celledoni, E., Owren, B.: Preserving first integrals with symmetric Lie group methods. Discrete Contin. Dyn. Syst. 34(3), 977–990 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge Tracts in Mathematics, vol. 108. Cambridge University Press (1993)Google Scholar
  25. 25.
    Crouch, P.E., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3, 1–33 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Dahlby, M., Owren, B., Yaguchi, T.: Preserving multiple first integrals by discrete gradients. J. Phys. A, 44(30), 305205, 14 (2011)MathSciNetCrossRefGoogle Scholar
  27. 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)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Engø, K.: Partitioned Runge-Kutta methods in Lie-group setting. BIT 43(1), 21–39 (2003)MathSciNetCrossRefGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Grossman, R.L., Larson, R.G.: Differential algebra structures on families of trees. Adv. Appl. Math. 35(1), 97–119 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Springer-Verlag (1972)Google Scholar
  34. 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. 35.
    Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numerica 9, 215–365 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Trans. Graph. 28(2) (2009)CrossRefGoogle Scholar
  37. 37.
    Krogstad, S.: A low complexity Lie group method on the Stiefel manifold. BIT 43(1), 107–122 (2003)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Krogstad, S., Munthe-Kaas, H., Zanna, A.: Generalized polar coordinates on Lie groups and numerical integrators. Numer. Math. 114(1), 161–187 (2009)MathSciNetCrossRefGoogle Scholar
  39. 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)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Lewis, D., Olver, P.J.: Geometric integration algorithms on homogeneous manifolds. Found. Comput. Math. 2(4), 363–392 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Lewis, D., Simo, J.C.: Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups. J. Nonlinear Sci. 4, 253–299 (1994)MathSciNetCrossRefGoogle Scholar
  42. 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. 43.
    Malham, S.J.A., Wiese, A.: Stochastic Lie group integrators. SIAM J. Sci. Comput. 30(2), 597–617 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Marsden, J.E., Pekarsky, S., Shkoller, S.: Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12(6), 1647–1662 (1999)MathSciNetCrossRefGoogle Scholar
  45. 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)MathSciNetCrossRefGoogle Scholar
  46. 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. 47.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)MathSciNetCrossRefGoogle Scholar
  48. 48.
    McLachlan, R., Modin, K., Verdier, O.: A minimal-variable symplectic integrator on spheres. Math. Comp. 86(307), 2325–2344 (2017)MathSciNetCrossRefGoogle Scholar
  49. 49.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Phil. Trans. Royal Soc. A 357, 1021–1046 (1999)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20(4), 801–836 (1978)MathSciNetCrossRefGoogle Scholar
  51. 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)MathSciNetCrossRefGoogle Scholar
  52. 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)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Munthe-Kaas, H.: Lie-Butcher theory for Runge-Kutta methods. BIT 35(4), 572–587 (1995)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Munthe-Kaas, H.: Runge-Kutta methods on Lie groups. BIT 38(1), 92–111 (1998)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Munthe-Kaas, H.: High order Runge-Kutta methods on manifolds. Appl. Numer. Math. 29, 115–127 (1999)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Munthe-Kaas, H., Lundervold, A.: On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Munthe-Kaas, H., Owren, B.: Computations in a free Lie algebra. Phil. Trans. Royal Soc. A 357, 957–981 (1999)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Munthe-Kaas, H., Verdier, O.: Integrators on homogeneous spaces: isotropy choice and connections. Found. Comput. Math. 16(4), 899–939 (2016)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Munthe-Kaas, H., Wright, W.M.: On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8(2), 227–257 (2008)MathSciNetCrossRefGoogle Scholar
  60. 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. 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)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Murua, A.: Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators. Appl. Numer. Math. 29(2), 221–251 (1999)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6(4), 387–426 (2006)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Owren, B.: Order conditions for commutator-free Lie group methods. J. Phys. A 39(19), 5585–5599 (2006)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Owren, B., Marthinsen, A.: Runge-Kutta methods adapted to manifolds and based on rigid frames. BIT 39(1), 116–142 (1999)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Owren, B., Marthinsen, A.: Integration methods based on canonical coordinates of the second kind. Numer. Math. 87(4), 763–790 (2001)MathSciNetCrossRefGoogle Scholar
  67. 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)MathSciNetCrossRefGoogle Scholar
  68. 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. 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. 70.
    Saccon, A.: Midpoint rule for variational integrators on Lie groups. Int. J. Numer. Methods Eng. 78(11), 1345–1364 (2009)MathSciNetCrossRefGoogle Scholar
  71. 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)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Veselov, A.P.: Integrable discrete-time systems and difference operators. Funct. Anal. Appl. 22, 83–93 (1988)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Wensch, J., Knoth, O., Galant, A.: Multirate infinitesimal step methods for atmospheric flow simulation. BIT 49(2), 449–473 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Norwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations