Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error

Efficient algorithms for the approximate solution of ordinary differential equations rely on controlling estimates of the error through adjustment of stepsize (and possibly, of order). For explicit Runge-Kutta methods estimates of the local truncation error are normally used, and currently the most efficient estimates may be obtained as differences of certain pairs of methods of successive orders. Because of the difficulty of deriving methods of intermediate orders of accuracy, and because such methods are efficient for many problems with moderate tolerance requirements, the methods derived by Fehlberg have become particularly well-known. Unfortunately, for problems that reduce to the evaluation of quadratures, Fehlberg’s methods give error estimates which are identically zero, and hence the estimates are unreliable for problems that are at least partially of this type. The methods developed here are designed to overcome this difficulty. The approach yields new methods of arbitrarily high orders of accuracy, and the parameters of four particular pairs of methods of orders 5(6), 6(7), 7(8), 8(9) are included.