Abstract.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, have in 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 almost P-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.

SUMMARYWe develop a new class of multistage algorithms for second-order systems of ordinary differential equations (ODES) based on methods by Zienkiewicz etOur new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques. In a companion paper, we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes.

We consider numerical methods for initial value problems for second-order systems of ordinary differential equations, analysing them by applying them to the test equation x + c2x = 0. We discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property. We also use a

No abstract

SUMMARYIn a companion paper, Thomas and Gladwell: the authors designed a class of multistage methods for secondorder systems of ordinary differential equations (ODES), together with local error estimators based on embedding techniques. Here we provide a set of subroutines implementing the algorithms and discuss our numerical experience with the resulting codes.

No abstract

Abstract.A class of two-step (hybrid) methods is considered for solving pure oscillation second order initial value problems. The nonlinear system, which results on applying methods of this type to a nonlinear differential system, may be solved using a modified Newton iteration scheme. From this class the author has derived methods which are fourth order accurate, P-stable, require only two (new) function evaluations per iteration and have a true real perfect square iteration matrix. Now, we propose an extension to sixth order, P-stable methods which require only three (new) function evaluations per iteration and for which the iteration matrix is a true real perfect cube.This implies that at most one real matrix "must be factorised at each step. These methods have been implemented in a new variable step, local error controlling code.

No abstract