A Variable Step Runge--Kutta--Nyström Integrator for Reversible Systems of Second Order Initial Value Problems

A class of implicit Runge--Kutta--Nyström (RKN) integration formulae is developed for the numerical integration of reversible systems of second order ordinary differential equations of the form y' = f(x,y). The formulae derived have high stage order, provide a continuous solution, are symmetric, and have asymptotically correct local error estimates available at each step point. The availability of these error estimates, coupled with the symmetry of the algorithm, allows the formulae to be implemented in a variable stepsize mode using the ideas of Stoffer [Computing, 55 (1995), pp. 1--22]. Numerical results presented for both fixed and variable stepsizes of integration show the power of the new methods.