Comment on the Runge-Kutta-Merson Algorithm Dear John:

It seems to me that some people are using the Runge-Kutta-Merson algorithm for estimating the truncation errors of the order of h5 without being fully aware of its limitations. R. E. Scraton1 has pointed out that Merson’s esti-mate * of truncation error is for the equation, or the equation close to, where a, b, and c are constants. For a general equation the value of Merson’s estimate is of the order of h4; hence it cannot in general estimate the truncation error of the order of h5. Scraton also gave an example in his paperl to show that the result from Merson’s estimate was ten times the actual error where the step size was 0.1. If the term (h4/24)Yn&dquo;&dquo; in yn+1 of an equation is expanded to where and then the value of Merson’s estimate is which is of the order of h4. If, however, the equation is ( 1), then, for this case, and Merson’s estimate will be close to the actual error: Take a simple example which is a first-order linear system with input sin x. The following table shows the numerical results for the errors of y(h): Hence Merson’s estimates are not even close to the actual errors. I suggest that the better methods for estimating the truncation errors are D. Sarafyan’s formula2 for single step and J. Kuntzmann’s formula3 for multisteps. They have been quoted in my paper4. *Merson original paper5 does not imply this form of the esti-mate in spite of the fact it is commonly used