ABSTRACT
This chapter is concerned with the development of modern methods dealing with global error. Many methods have been constructed to provide global error estimation. A typical procedure, often employed when local error control is practised, is called tolerance reduction. A more systematic approach is the classical Richardson extrapolation scheme, which requires a second solution with halved or with doubled step-sizes. This method is fairly reliable but costly. The estimation of local error in Runge-Kutta and in multistep formulae was usually followed by local extrapolation. This procedure, although popular with practitioners, does not find favour with all numerical analysts. With such a large difference in orders the usual assumption of tolerance proportionality is unjustified. An alternative scheme would be based on an estimator embedding. Only a few modifications are required to incorporate global embedding in the RKpair program.
