Regions of absolute stability of explicit Runge-Kutta-Nyström methods for y″ = f(x, y, y′)

We examine regions of 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″ + 2αy′ + β²y = O, α, β ? 0, α + β > 0. We also examine the regions of absolute stability of Runge-Kutta (R-K) methods for first order differential equations of respective orders. Interestingly, it turns out that regions of absolute stability of R-K methods and R-K-N methods of respective orders for which the asymptotic relative error does not deteriorate are identical. Our present investigations are in continuation of the recent results of Chawla and Sharma [1].