Runge-Kutta pairs for periodic initial value problems (Q1310616)

Pairs of different order (namely \(p\) and \(p-1\)) explicit \(s\)-stage Runge- Kutta formulae may be used to solve the initial value problem, the difference between the two solutions being used for local error approximation. In this paper, the authors discuss the phase-lag property of such methods when applied to problems with periodic solutions. They analyze two families, one of 5(4) pairs and the other of 6(5) pairs and from each select a pair with the highest possible order of phase-lag. New 5(4) pairs, based on pairs proposed by \textit{E. Fehlberg} [Computing 6, 61-71 (1970; Zbl 0217.530)] and by \textit{J. R. Dormand} and \textit{P. J. Prince} [J. Comput. Appl. Math. 6, 19-26 (1980; Zbl 0448.65045)], respectively, are of phase-lag order 8(4) and 8(6) and are not dissipative. The 6(5) pairs are of phase-lag order 10(6) and are dissipative (a dissipative scheme has non-vanishing imaginary stability interval). The paper concludes by demonstrating the generally improved performance of the new methods over other well known methods on some commonly utilized test problems whose solutions exhibit periodic behaviour.