On global error estimation and control for initial value problems

Abstract.
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.

Key words.
Numerical integration for ODEs, global error estimation, global error control, defects
and local errors, tolerance proportionality, adjoint method, small sample statistical initialization