Variable step size commutator free Lie group integrators

Abstract

We introduce variable step size commutator free Lie group integrators, where the error control is achieved using embedded Runge–Kutta pairs. These are schemes for the integration of initial value problems posed on homogeneous spaces by means of Lie group actions. The focus is on commutator free methods, in which the approximation evolves by composing flows generated by Lie group exponentials. Such methods are encoded by a generalization of Butcher’s Runge–Kutta tableaux, but it is known that more order conditions must be satisfied to obtain a scheme of a given order than are required for classical RK schemes. These extra considerations complicate the task of designing embedded pairs. Moreover, whilst the computational cost of RK schemes is typically dominated by function evaluations, in most situations, the dominant cost of commutator free Lie group integrators comes from computing Lie group exponentials. We therefore give Butcher tableaux for several families of methods of order 3(2) and 4(3), designed with a view to minimizing the number of Lie group exponentials required at each time step, and briefly discuss practical error control mechanisms. The methods are then applied to a selection of examples illustrating the expected behaviour.

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

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie group methods. Futur. Gener. Comput. Syst. 19(3), 341–352 (2003)

    Article  Google Scholar 

  3. 3.

    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)

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

    Curry, C., Schmeding, S.: Convergence of Lie group integrators. arXiv:1807.11829 (2018)

  6. 6.

    Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Faltinsen, S.: Backward error analysis for Lie-group methods. BIT 40(4), 652–670 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010). Structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition

    Google Scholar 

  9. 9.

    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, Volume 8 of Springer Series in Computational Mathematics, 2nd edn. Springer-Verlag, Berlin (1993). Nonstiff problems

    Google Scholar 

  10. 10.

    Holm, D.D.: Geometric Mechanics. Part II. Rotating, translating and rolling, 2nd edn. Imperial College Press, London (2011)

    Google Scholar 

  11. 11.

    Iserles, A., Munthe-Kaas, H.Z., Nrsett, S.P., Zanna, A.: Lie-group methods. In: Acta Numerica, 2000, Volume 9 of Acta Numer., pp. 215–365. Cambridge University Press, Cambridge (2000)

  12. 12.

    Lundervold, A., Munthe-Kaas, H.Z.: On algebraic structures of numerical integration on vector spaces and manifolds. In: Faà Di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series, volume 21 of IRMA Lect. Math. Theor. Phys., pp. 219–263. Eur. Math. Soc., Zürich (2015)

  13. 13.

    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, volume 17 of Texts in Applied Mathematics, 2nd edn. Springer-Verlag, New York (1999). A basic exposition of classical mechanical systems

    Google Scholar 

  14. 14.

    Marthinsen, A., Munthe-Kaas, H.Z., Owren, B.: Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications. Model. Identif. Control 18(1), 75–88 (1997)

    Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

    Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Dedicated to C. William Gear on the occasion of his 60th birthday

    MathSciNet  Article  Google Scholar 

  18. 18.

    Wensch, J.: Extrapolation methods in Lie groups. Numer. Math. 89(3), 591–604 (2001)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 691070, and from The Research Council of Norway (project 231632).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Charles Curry.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Curry, C., Owren, B. Variable step size commutator free Lie group integrators. Numer Algor 82, 1359–1376 (2019). https://doi.org/10.1007/s11075-019-00659-0

Download citation

Keywords

  • Lie group integrators
  • Adaptive error control
  • Geometric integration
  • Commutator-free methods