Global Error Control in Consistent and Quasi-Consistent Numerical
Schemes
In this paper we discuss Nordsieck formulas applied to ordinary
differential equations. We focus on local and global error
evaluation techniques in the mentioned numerical schemes. The error
estimators are derived for both consistent Nordsieck methods and
quasi-consistent ones. It is also shown how quasi-consistent
Nordsieck formulas, which suffer, on variable grids, from the order
reduction phenomenon, can be modified in an optimal way to avoid
the order reduction. A special task here is to study advantages of
numerical integration by quasi-consistent Nordsieck formulas. All
quasi-consistent numerical methods possess at least one attractive
property for practical use. The global error of a quasi-consistent
method has the same order as its local error. This means that the
usual local error control will produce a numerical solution for the
prescribed accuracy requirement if the principal term of the local
error dominates strongly over remaining terms. In other words, the
global error control can be as cheap as the local error control in
the methods under discussion. We apply this idea to Nordsieck
Adams-Moulton methods, which are known to be quasi-consistent.
Moreover, some Nordsieck Adams-Moulton methods are even
super-quasi-consistent. The latter property means that their
propagation matrices annihilate two leading terms in the defect
expansion of such methods. In turn, this can impose a strong
relation between the local and global errors of the numerical
solution and allow the global error to be controlled effectively by
a local error control. We also introduce Implicitly Extended
Nordsieck methods such that in some sense they form pairs of
embedded formulas with their source Nordsieck counterparts. This
facilitates the local error control in quasi-consistent Nordsieck
schemes. As the most promising result, we study double
quasi-consistency of numerical schemes. This property means that
the principal terms of the global and local errors of a doubly
quasi-consistent numerical technique coincide. This benefits the
global error control, significantly. We show that doubly
quasi-consistent schemes do exist and belong to the class of
general linear methods. The notion of double quasi-consistency
creates potentially a new area of research in numerical methods for
differential equations. Numerical examples presented in this paper
confirm clearly the power of the above-mentioned global error
estimators in practice.
