Phase properties of high order,almostP-stable formulae

Cash 3] and Chawla 4] derive families of two-step, symmetric,P-stable (hybrid) methods for solving periodic initial value problems numerically. Chawla demonstrates the existence of a family of fourth order methods while Cash derives both fourth order and sixth order methods. In this paper, we demonstrate that these methods, which are dependent on certain free parameters, havein phase particular solutions. We consider more general families of 2-step symmetric methods, including those derived by Cash and Chawla, and show that some members of these families have higher order phase lag and are almostP-stable. We also consider how the free parameters can be chosen so as to lead to an efficient implementation of the fourth order methods for large periodic systems.Most of this work was carried out while the author was at the Department of Computer Studies, University of Leeds, Leeds, England.