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BIT Numerical Mathematics

, Volume 39, Issue 1, pp 116–142 | Cite as

Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

  • B. Owren
  • A. Marthinsen
Article

Abstract

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E1, ... , En on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E1, ... , En that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ℝ n . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.

Geometric integration numerical integration of ordinary differential equations on manifolds Runge-Kutta methods 

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Copyright information

© Swets & Zeitlinger 1999

Authors and Affiliations

  • B. Owren
    • 1
  • A. Marthinsen
    • 1
  1. 1.Department of Mathematical SciencesTrondheimNorway

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