Explicit almost P-stable Runge–Kutta–Nyström methods for the numerical solution of the two-body problem

In this paper, three families of explicit Runge–Kutta–Nyström methods with three stages and third algebraic order are presented. Each family consists of one method with constant coefficients and one corresponding optimized “almost” P-stable method with variable coefficients, zero phase-lag and zero amplification error. The first method with constant coefficients is new, while the second and third have been constructed by Chawla and Sharma. The new method with constant coefficients, constructed in this paper has larger interval of stability than the two methods of Chawla and Sharma. Furthermore, the optimized methods possess an infinite interval of periodicity, excluding some discrete values, while being explicit, which is a very desired combination. The preservation of the algebraic order is examined, local truncation error and stability/periodicity analysis are performed and the efficiency of the new methods is measured via the integration of the two-body problem.