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.