Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

Summary:For initial value problem (IVPs) in ordinary second order differential equations of the special form $y^{\prime \prime }=f\left(x,y\right)$ possessing oscillating solutions, diagonally implicit Runge–Kutta–Nystrom (DIRKN) formula-pairs of orders 5(4) in 5-stages are derived in this paper. The method is zero dissipative, thus it possesses a non-empty interval of periodicity. Some numerical results are presented to show the applicability of the new method compared with existing Runge–Kutta (RK) method applied to the problem reduced to first-order system