The method of Verner

An example of a 7-stage method with an 8-stage error estimation was developed by Verner (Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error, SIAM Journal on Numerical Analysis, Vol. 15, No. 4 (Aug., 1978), pp. 772-79). The Butcher tableau is shown below.



Once again, the generalized RK methodology can be used quite easily with the above table for the Verner’s method. The error estimate is obtained using the ycap coefficients from the last row of coefficients, and can be used to develop a variable step algorithm.