Variable-order, variable-step methods for second-order initial-value problems

We develop a variable-order, variable-step algorithm for solving second-order initial-value problems y″ = f(t,y), y(0), y′(0) given. This algorithm employs a family of implicit two-step methods, in which the differential equation is evaluated at a number of off-step points. We have derived the order conditions for this family of methods in previously published work and we have also shown that the remaining free parameters of the method may be chosen to given P-stability. Here, the emphasis is on the combination of second-, fourth-, sixth- and eighth-order methods from this family in a variable-order, variable-step algorithm. The aim is to choose particular P-stable methods from the family in such a way that a saving can be made in the amount of computational effort involved when they are combined together.