Symp. Data Processing, Weapons Research Establishment, Salisbury, Australia, 1957, pp 110-125) came up with an RK formulation of order 4. The Butcher tableau for the Merson’s method (ref: Hairer, Nørsett, Wanner, Solving ordinary differential equations, vol.1. Nonstiff problems, 2nd ed) is shown below:
Notice a second row (ycap1) at the end in the tableau. By using a 4th order formulation, followed by a 5th order estimate for the new coefficients in the last row, an error estimate can be calculated.
The generalized Butcher tableau for with error estimation then becomes
The calculations for ki’s and y(n+1) remain the same from the Generalized RK procedure. The error estimation equation is defined by
The difference between the two values of y then serves as an estimate of the error. A variable step algorithm can then be developed based on the error estimate, similar to the one presented for the Euler’s method.
