A Collocation Multistep Method for Integrating Ordinary Differential Equations on Manifolds

This study 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-x n + k, for k - 0,1...s, we obtain the discrete scheme which can be adapted to homogeneous spaces. Varying the values of k 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 study an implicit algorithm was formulated and 2 approaches presented for its implementation.