Skip to main content
SpringerLink
Log in

Cost Efficient Lie Group Integrators in the RKMK Class

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge—Kutta—Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to order seven based on some of the most popular Runge—Kutta schemes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. S. Blanes and F. Casas, Optimization of Lie group methods for differential equations, Future Generation Computer Systems, 19 (2003), pp. 331-339.

    Article  Google Scholar 

  2. S. Blanes, F. Casas, and J. Ros, High order optimized geometric integrators for linear differential equations, BIT, 42(2) (2002), pp. 262-284.

    Article  Google Scholar 

  3. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Wiley, Chichester, 1987.

    Google Scholar 

  4. F. Casas and B. Owren, Cost efficient Lie group integrators in the RKMK class, Technical Report Numerics no. 5/2002, Norwegian University of Science and Technology, Trondheim, Norway, 2002.

    Google Scholar 

  5. E. Celledoni, A. Iserles, S. P. Nørsett, and B. Orel, Complexity theory for Lie-group solvers, J. Complexity, 18(1) (2002), pp. 242-286.

    Article  Google Scholar 

  6. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, 2nd revised ed., 1993.

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

    Article  Google Scholar 

  8. A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, Phil. Trans. Royal Soc. A, 357 (1999), pp. 983-1020.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Jaume I, 12071, Castellón, Spain

    F. Casas

  2. Department of Mathematical Sciences, NTNU, N-7491, Trondheim, Norway

    B. Owren

  3. Center for Advanced Study, Drammensveien 78, 0271, Oslo, Norway

    B. Owren

Authors
  1. F. Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Owren
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Casas, F., Owren, B. Cost Efficient Lie Group Integrators in the RKMK Class. BIT Numerical Mathematics 43, 723–742 (2003). https://doi.org/10.1023/B:BITN.0000009959.29287.d4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:BITN.0000009959.29287.d4

Search

Navigation