Advertisement

Numerische Mathematik

, Volume 87, Issue 4, pp 763–790 | Cite as

Integration methods based on canonical coordinates of the second kind

  • Brynjulf Owren
  • Arne Marthinsen
Original article

Summary.

We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

Mathematics Subject Classification (1991): 65L05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Brynjulf Owren
    • 1
  • Arne Marthinsen
    • 1
  1. 1.Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway; e-mail: {Brynjulf.Owren,Arne.Marthinsen}@math.ntnu.no NO

Personalised recommendations