On Pseudo-Runge-Kutta Methods with 2 and 3 Stages

Pseudo-Runge-Kutta methods have been proposed by Byrne, Lambert and Costabile. We have seen in [1], [4] and [15] that Pseudo-Runge-Kutta methods have order p*(r) = r+l (r = 2, 3,4). Byrne, Lambert and many other authors have shown that Pseudo-RungeKutta methods are less accurate than Runge-Kutta methods in the same order. In this paper, we shall present new Pseudo-Runge-Kutta methods which have order p*(r) = r + 2 (r = 2, 3, 4). In comparing our methods and other methods in the same order, our methods have almost the same accuracy as the Runge-Kutta type methods in order 5 and 6. The outline of this paper is as follows: In Section 2, we present our new