A Theoretical Criterion for Comparing Runge–Kutta Formulas

A criterion is proposed for determining which explicit Runge–Kutta formulas are the most promising as a basis for developing good library subroutines for solving nonstiff initial-value problems associated with ordinary differential equations. The criterion is based on a theoretical measure of the cost of solving classes of linear homogeneous problems with constant coefficients. It takes into account the trade-off between reliability and efficiency, as well as the user’s accuracy requirement. Results are presented for twenty-one formulas; one of the recently developed formula pairs due to Verner appears to be one of the most promising currently available. Moreover, our results support the thesis that the use of local extrapolation will usually improve the performance of a method. These conclusions are also supported by some empirical results collected over a relatively wide class of nonstiff problems. Finally, some suggestions are given on how to exploit these results, particularly if local extrapolation i...

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