NUMERICAL-INTEGRATION OF ORDINARY DIFFERENTIAL-EQUATIONS ON MANIFOLDS

This paper is concerned with the problem of developing numerical integ ration algorithms for differential equations that, when viewed as equa tions in some Euclidean space, naturally evolve on some embedded subma nifold. It is desired to construct algorithms whose iterates also evol ve on the same manifold. These algorithms can therefore be viewed as i ntegrating ordinary differential equations on manifolds. The basic met hod ''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 o perators 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.