On global error estimation and control for initial value problems

This paper addresses global error estimation and control for initial value problems for ordinary differential equations. The focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation. Control is achieved through tolerance proportionality. Both approaches are found to work well and to enable estimation and control in a reliable manner. However, the novel approach is not found to be competitive with the classical approach, mainly because of its huge storage demand for large problems.