Journal of Nonlinear Science

, Volume 3, Issue 1, pp 1–33 | Cite as

Numerical integration of ordinary differential equations on manifolds

  • P. E. Crouch
  • R. Grossman


This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.

Key words

numerical integration manifold differential equation flow lie algebra algorithm symbolic computation frozen coefficients 

AMS Subject Classifications

34A50 34A34 65L06 93C15 58F99 


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Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • P. E. Crouch
    • 1
  • R. Grossman
    • 2
  1. 1.Center for Systems Science and EngineeringArizona State UniversityTempeUSA
  2. 2.Laboratory for Advanced ComputingUniversity of Illinois at ChicagoChicagoUSA

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