Absolute stability of explicit Runge-Kutta-Nyström methods for \(y''=f(x,y,y')\) (Q1060543)

We examine absolute stability of s-stage explicit Runge-Kutta-Nyström (R-K-N) methods of order s for \(s=2,3,4\) for \(y''=f(x,y,y')\) by applying these methods to the test equation: \(y''+2\lambda y'+\lambda^ 2y=0\), \(\lambda >0\). We show the existence of R-K-N methods of orders two, three and four possessing intervals of absolute stability as large as that of explicit Runge-Kutta methods of respective orders.