On one-step collocation methods with higher derivatives for solving ordinary differential equations

Computational Mathematics and Mathematical Physics, 44(10) (2004), 1696-1720


A theory is developed for one-step collocation methods using higher derivatives for the approximation of ordinary differential equations. These methods are shown to be implicit and A-stable, which makes it possible to propose rigorously justified ways for their practical implementation. An evaluation and control of the local and global errors of the numerical solution are also examined with the aim to automatically attain the prescribed accuracy (assuming that no round-off errors are made). All the theoretical results of this paper are substantiated by numerical results obtained for a test problem.