A collocation multistep methods for integrating ordinary differential equations on manifolds.

Abstract

This paper concerns a family of generalized collocation multistep methods that evolves the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. Collocating the general linear method at x = xx+k for k = 0,1,….s, , we obtain the discrete scheme which can be adapted to homogeneous spaces. Varying the values ofk in the collocation process, the standard Munthe-Kass (k = 1) and the linear multistep methods (k = s) are recovered. Any classical multistep methods may be employed as an invariant method and the order of the invariant method is as high as in the classical setting. In this paper an implicit algorithm was formulated and two approaches presented for its implementation. Key words: Collocation, multistep methods, homogeneous manifolds, implicit methods, invariant methods, differential equations on manifolds, geometric integration.