On quasi-consistent integration by Nordsieck methods

Journal of Computational and Applied Mathematics, 225 (2009), 268-287
http://dx.doi.org/10.1016/j.cam.2008.07.038

In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. 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.

Here, we apply the above-mentioned 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 methods. This facilitates the local error control in quasi-consistent Nordsieck schemes. Numerical examples presented in this paper confirm clearly the power of quasi-consistent integration in practice.