Conservative and Approximately Conservative Algorithms on Manifolds

Algorithms for the numerical integration of initial value problems have traditionally been either very general or very specialized. That is, the algorithms have either been intended to perform well for most vector fields or optimized to perform very well for a specific differential equation. While some classes of general purpose algorithms, e.g. implicit methods, possess combinations of advantages and disadvantages that make them particularly appropriate for some applications and inappropriate for others, the methods are still applicable to a very broad range of initial value problems. In recent years, there has been increasing interest in algorithms for specific families of dynamical systems with some common features; algorithms have been developed to accurately capture these features even when the overall accuracy of the method is relatively low.