Journal of Emerging Trends in Engineering and Applied Sciences: Research Areas

Article Title: Collocation Multistep Methods for Integrating Differential Equations on Manifolds using Lie Group Approach
by J.O. Fatokun and A.B. Gimba

Abstract:
This paper concerns the derivation and analysis of some Collocation Multistep methods that evolve the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. This study is an attempt to examine the linear multi-step collocation methods through Lie group approach in solving the differential equation y=(y)= fi(y)Ei, y = M on manifold M, where Ei are vector fields on M. Collocating the general linear method at X=Xn+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 paper an implicit algorithm was formulated and two approaches presented for its implementation.

Keywords: collocation, multistep methods, homogeneous manifolds, lie group, differential equations on manifolds, geometric integration

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