A new fourth order Runge-Kutta formula for \(y'=Ay\) with stepsize control (Q1106628)

For the single differential equation \(y'(x)=f(x,y(x))\), the authors propose the derivative evaluations at four points be used to obtain a conventional Runge-Kutta estimate and a ``geometric-mean'' estimate of the solution at the end of each step. The solution is to be propagated by the former, and the difference between the two forms an error estimate. In the proposed method, the first formula has order 3 for non-linear problems, and order 4 for problems linear in x and y, the second formula has order 4 for all problems, and the error estimator is appropriate for \(y'=Ay\). A particular method is applied to the problem \(y'=-y\). Although the local error is not recorded, the error estimate appears to accurately estimate the actual error on each step for this problem. Perhaps a growing exponential solution would provide further evidence of its utility. Because the method has a very limited application, it will not be of general interest.