Families of Imbedded Runge–Kutta Methods

Using the s stages of an explicit Runge–Kutta method of order p, approximations of various lower orders may be obtained. A linear system of equations classifies and characterizes parameters of all properly imbedded methods. For all methods with $p = 2,3$, and some with $p = 5$, and s minimal, approximations of all lower orders may be obtained. Otherwise, it appears that the imbedding of a method of order $p - 1$ in one of order p requires at least one additional stage. For $p = 6,7,8,9$ matched pairs of methods of orders $p - 1$ and p requiring s stages (where s is larger by 1 than the minimum number of stages required for the method of order p) are known. For some, if not all of these, imbedded methods of all lower orders may be obtained. Some examples and suggestions for their utilization are given.