Two-stage explicit Runge-Kutta type methods using derivatives

Two-stage explicit Runge-Kutta type methods using derivatives for the systemy′(t) =f(y(t)),y(t0) =y0 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 κth-order Taylor series method uses derivativesf∼’,f∼",…,f(κ−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 tof(κ−3) in the first stage andf,f∼’ in the second stage. They can achieve κth-order accuracy and construct embedded formula for the error estimation.