Test Results on Initial Value Methods for Non-Stiff Ordinary Differential Equations

Numerical methods for the solution of the initial value problem in ordinary differential equations are evaluated and compared. Earlier comparisons (Hull et al. (1972)) are extended and twenty numerical methods are assessed on the basis of how well they solve a collection of routine non-stiff differential equations under a variety of accuracy requirements. The methods tested include extrapolation methods, variable-order Adams methods, Runge–Kutta methods based on the formulas of Fehlberg, and appropriate methods from the SSP and IMSL subroutine libraries. (In some cases the methods had to be modified before being tested, in order to make them conform to the standardized requirements of the test program.) The results indicate that, to be efficient over a range of error tolerances, a numerical method must have the ability to vary its order. It is found that when derivative evaluations are relatively expensive (in which case the efficiency will depend primarily on the number of function evaluations required to solve the problems) the variable-order Adams methods of Sedgwick and Krogh are best. On the other hand, when derivative evaluations are relatively inexpensive (in which case efficiency will depend primarily on the overhead of the method), the Runge–Kutta–Fehlberg methods are best (with the reservation that the higher order ones cannot be used for quadrature). The extrapolation methods are also good in this case.