Variable Step Runge-Kutta-Nyström Methods for the Numerical Solution of Reversible Systems

Fixed step, symmetric Runge-Kutta-Nyström formulae have proved to be very efficient for the numerical integration of a large class of reversible second order systems of ordinary differential equations of the special form d 2y dt2 = f(t, y). However for some important classes of problems it is necessary, for the sake of efficiency, to allow a variable steplength of integration to be used and in this case existing numerical methods tend to be much less satisfactory. In the present paper we examine in detail the problem of implementing reversible Runge-Kutta-Nyström integration formulae with varying time steps. We show that, even though enormous gains in efficiency can be made if the methods are implemented in an appropriate way, there are still some important practical problems to be overcome.