A fifth order implicit method for the numerical solution of differential-algebraic equations

An implicit two-step Runge–Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.