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Abstract.This paper describes a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations. The basis of a fair comparison is discussed in detail. Measurements of cost and reliability are made over a collection of 25 carefully selected problems. The problems have been designed to show how certain major factors affect the performance of a method.The technique is applied to five methods, of which three turn out to be quite good, including one based on backward differentiation formulas, another on second derivative formulas, aaud a third on extrapolation. However, each of the three has a weakness of its own, which can be identified with particular problem characteristics.

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with O ( h 5 ) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with O ( h 6 ) local truncation error for this RK formula.

We present a discussion and description of a collection of FORTRAN routines designed to aid in the assessment of initial value methods for ordinary differential equations. Although the overall design characteristics are similar to those of earlier testing packages [2,6] that were used for the comparison of methods [5,7], the details and objectives of the current collection are quite different. Our principal objective is the development of testing tools that can be used to assess the efficiency and reliability of a standard numerical method without requiring significant modifications to the method and without the tools themselves affecting the performance of the method.