CLASSICAL FIFTH-ORDER AND SEVENTH-ORDER RUNGE-KUTTA FORMULAS WITH STEPSIZE CONTROL

New explicit fifth- and seventh-order Runge-Kutta formulas are derived. They include a stepsize control procedure based on a complete coverage of the leading term of the local truncation error. These formulas require fewer evaluations per step than other Runge-Kutta formulas of corresponding order if the latter ones are also used with stepsize control (richardson's extrapolation to the limit). By a proper choice of some parameters the leading truncation error term of our formulas can be reduced substantially, thereby allowing an increase in the stepsize without loss of accuracy. A numerical example is presented. Our results being of the same accuracy, we save in this example 40% to 60% computer time compared with the known Runge-Kutta formulas of corresponding order. © 1969 Springer-Verlag.