Adaptive Nordsieck Formulas with Advanced Global Error Control Mechanisms

Russian Journal of Numerical Analysis and Mathematical Modelling, 28(4) (2013), 321-352
http://dx.doi.org/10.1515/rnam-2013-0020

In this paper we develop efficient numerical schemes to solve ordinary differential equations. Our methods are of the Nordsieck type, adaptive and capable of automatically controlling the global error of a numerical solution. A special feature of the new stepsize selection algorithms introduced here is the global error estimation quality control. Two different ways of attaining the preassigned accuracy of computation are examined in the paper. Namely, we implement the global error control mechanism based on reducing the maximum stepsize bound and the other one is based on reducing the local error tolerance. An accurate starting procedure for the adaptive Nordsieck methods is presented in full detail. Our intention here is to find the most effective strategy of stepsize selection. Theoretical investigation is supplied with numerical tests.