Two-stage explicit Runge-Kutta type methods using derivatives

Two-stage explicit Runge-Kutta type methods using derivatives for the system y'(t) = f(y(t)), y(t(0)) = y(0) are considered. Derivatives in the first stage have the standard form, but in the second stage, they have the form included in the limiting formula. The kthorder Taylor series method uses derivatives f', f'' ,..., f((k-1)). Though the values of derivatives can be easily obtained by using automatic differentiation, the cost increases proportional to square of the order of differentiation. Two-stage methods considered here use the derivatives up to f((k-3)) in the first stage and f, f' in the second stage. They can achieve kth-order accuracy and construct embedded formula for the error estimation.