Families of Direct Fourth‐Order Methods for the Numerical Solution of General Second‐Order Initial‐Value‐Problems

We consider the construction of certain families of direct fourth‐order methods based on four function‐evaluations for the general second‐order initial‐value problem: y″ = f(x, y, y′), y(x 0 ) = A, y′(x 0 ) = B. We obtain these families of methods from the set of conditions sufficient for order four (see Hairer and Wanner [3], Table 2). We obtain a four‐parameter family of methods in the general case, whereas in each of the four special cases we obtain a three‐parameter family of methods. The classical Runge‐Kutta‐Nyström method is a member of the family of methods in one of the special cases. While determining particular methods from these families we require that when these methods are applied to the linear differential equation y″ ‐(α + β)y′ + αβy = 0, the measure of the relative error F, introduced by Rutishauser [5], should have the properties that (i) when α = −β, F should not deteriorate, (ii) when α = –β, F should be least. It was pointed out by Rutishauser [5] that the classical Runge‐Kutta‐Nyström method lacks these properties; it is interesting to note that by a suitable adjustment of some of the parameters, we obtain a new Runge‐Kutta‐Nyström type method which possesses the above two desirable properties. Some particular methods are listed and illustrated by numerical examples; it will be observed from the numerical examples that all these methods perform far better than the classical Runge‐Kutta‐Nyström method, particularly for large x.