Two-step processes by one-step methods of order 3 and of order 4

Local error estimation by doubling

“…Doubling allows us to advance two half steps with any four stage, fourth order formula and then to estimate the local error by a whole step at a cost of three extra evaluations of f. It is reasonable to ask if an error estimating companion can be found which requires fewer extra evaluations.Shintani [,16] showed that if a particular basic formula is used, then the error can be estimated with only one additional evaluation of f. England[-3] has given a family of such procedures. A procedure of some popularity is based on the formula…”

Local extrapolation in the solution of ordinary differential equations

“…Of course, the parameters are chosen so that most function evaluations are used in both formulas to keep the total cost down to six evaluations per step. Shintani [5] has given an effective scheme using seven evaluations per double step with a similar derivation. Along with England's scheme, and perhaps Zonneveld's, we feel these are the principal possibilities for an efficient Runge-Kutta code.…”

A block 6(4) Runge-Kutta formula for nonstiff initial value problems

“…Fourth-order formulas of this class have been popular for some time, starting with the well-known "classical" fourth-order formula [12] with four stages (which was often used with a constant step size), developing through the embedded formulas of Merson [13], England [8], and Shintani [17] (which were inspired by the ideas of Fehlberg [lo]), and culminating in the widely used code RKF45 of Shampine and Watts [16]. Although this is still widely regarded as a "state-ofthe-art" code, recent work by Dormand and Prince [6,71 has suggested other fourth-order formulas that may be competitive.…”