The Relative Efficiency of Alternative Defect Control Schemes for High-Order Continuous Runge–Kutta Formulas

The development of continuous Runge-Kutta formulas for nonstiff problems has been the subject of several investigations in recent years. In Enright [SIAM J. Numer. Anal., 26 (1989), pp. 588–599] the author proposed a new error and stepsize control strategy (based on controlling the size of the defect of the underlying continuous solution) that is particularly appropriate for numerical methods based on this class of formulas. In this investigation the author presents an analysis of the relative efficiency of existing continuous Runge-Kutta methods. The author introduces theoretical measures useful in evaluating the potential of such methods and apply these measures to several methods of orders 4 through 8. The author also presents numerical results to illustrate that the theoretical measures of potential are consistent with the observed performance of nonstiff problems over a wide range of error tolerances.