Local and global error estimation in Nordsieck methods

Russian Journal of Numerical Analysis and Mathematical Modelling, 23(6) (2008), 567-595
http://dx.doi.org/10.1515/RJNAMM.2008.033

This paper deals with asymptotically correct methods to evaluate the local and global errors of Nordsieck formulas applied to ordinary differential equations. It extends naturally the results developed by Kulikov and Shindin [Comp. Math. Math. Phys. (2000) 40, 1255–1275] in local and global error computation of multistep methods, but shows that Kulikov and Shindin's technique becomes more complicated when implemented in numerical methods, for which the concepts of consistency and quasi-consistency are not equivalent (see Skeel [SIAM J. Numer. Anal. (1976) 13, 664–685]). A new property termed super quasi-consistency is introduced and special cases of Nordsieck formulas with cheaper error estimation are found. Numerical examples are included to confirm practically the theory presented in this paper.