Comparing Numerical Methods for Ordinary Differential Equations

Numerical methods for systems of first order ordinary differential equations are tested on a variety of initial value problems. The methods are compared primarily as to how well they can handle relatively routine integration steps under a variety of accuracy requirements, rather than how well they handle difficulties caused by discontinuities, stiffness, roundoff or getting started. According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer. However, when function evaluations are relatively expensive, variable-order methods based on Adams formulas are best. The overhead costs are lower for the method of Bulirsch and Stoer, but the Adams methods require considerably fewer function evaluations. Krogh's implementation of a variable-order Adams method is the best of those tested, but one due to Gear is also very good. In general, Runge–Kutta methods are not competitive, but fourth or fifth order methods of this type are best for restricted classes of problems in which function evaluations are not very expensive and accuracy requirements are not very stringent. The problems, methods and comparison criteria are specified very carefully. One objective in doing so is to provide a rigorous conceptual basis for comparing methods. Another is to provide a useful standard for such comparisons. A program called DETEST is used to obtain the relevant statistics for any particular method.