Variable order and step integrating algorithm based on the explicit two-stage Runge–Kutta method

The inequality for a stability control of the explicit two-stage Runge–Kutta like method is obtained.With the usage of stages of this scheme, the methods of first and second order are developed.The method of first order has a maximal length of the stability interval equal to 8. The algorithm of variable order and step is created, for which the most efficient computational scheme is chosen from the stability criterion. Numerical results with an additional stability control and variable order demonstrate an increase in efficiency.