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 onthe 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.

Numerical integration for ODEs, global error estimation, global error control, defects and local errors, tolerance proportionality, adjoint method, small sample statistical initialization
S.I.A.M.
SIAM Journal on Scientific Computing
Computational Dynamics

Lang, J, & Verwer, J.G. (2007). On global error estimation and control for initial value problems. SIAM Journal on Scientific Computing, 29(4), 1460–1475.