Modified Runge-Kutta method. II

Modern Runge-Kutta method of solving ODE bears a slight resemblance with the classical (explicit) method and is based on the transformation of the differential equation to the integral one. The contents of the mathematical theory was formulated by J.C. Butcher et all. Nevertheless, the technique of constructing fundamental equations of RK-method remained unchanged. In this paper new concepts are lying in the basis of constructing fundamental equations. Some new ideas are used for solving the fundamental equations, in particular, the principle of nilpotency for explicit, diagonal and singly-implicit RK-methods is successively performed. In the second part of this paper the investigation of the nilpotency method is continued. The main attention is paid to the singly implicit RK-method. The new set of the singly implicit RK-schemes of high accuracy was proposed.