A mono-implicit Runge-Kutta-Nystrِm modification of the Numerov method

We present two two-parameter families of fourth-order mono-implicit Runge-Kutta-Nystrِm methods. Each member of these families can be considered as a modification of the Numerov method. We analyze the stability and periodicity properties of these methods. It is shown that (i) within one of these families there exist A-stable (even L-stable) and P-stable methods, and (ii) in both families there exist methods with a phase lag of order six.