Numerical approaches for solution of differential equations on manifolds

摘要

Numerical approaches for the solution of vector fields (differential equations defined on a manifold) have attracted wide attention over the past few years. This paper first reviews the various numerical approaches available in the literature for the solution of vector fields namely, Parameterization approach, Constraint Stabilization approach, and Perturbation approach (PA). In the process, the paper also makes the following useful contributions: an expanded analysis and a new perturbation scheme for the PA; and a new way of choosing integration error tolerances for the parameterization approach. A comparison of all the approaches is carried out, both by means of a crude cost analysis as well as by studying their numerical performance on examples of vector fields arising from constrained mechanical systems (CMS). Based on this comparison, recommendations are made for a proper choice of a suitable approach. Overall, the PA performs ‘better’ than the other approaches.