A note on a diagonally implicit Runge-Kutta-Nyström method (Q1098578)

For the numerical integration of the special second-order initial value problem \(y''=f(t,y),\quad y(t_ 0)=y_ 0,\quad y'(t_ 0)=y_ 0'\) it is often advantageous applying a direct method for this type of differential equations, rather than rewriting that to its first-order form. Fourth-order accurate diagonally implicit (or semi-explicit) Runge- Kutta-Nyström methods with only 2 stages are shown can be obtained. The scheme is given with the largest interval (0,12) of periodicity, and the requirement of P-stability decreases the order to 2.