Numerical integration of ordinary differential equations on manifolds |
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Authors: | P. E. Crouch R. Grossman |
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Affiliation: | (1) Center for Systems Science and Engineering, Arizona State University, 85287-7606 Tempe, AZ, USA;(2) Laboratory for Advanced Computing, University of Illinois at Chicago, 60680 Chicago, IL, USA |
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Abstract: | Summary 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. |
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Keywords: | numerical integration manifold differential equation flow lie algebra algorithm symbolic computation frozen coefficients |
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