Fixed versus variable order Runge-Kutta

Popular codes for the numerical solution of nonstiff ordinary differential equations (ODEs) are based on a (fixed order) Runge-Kutta method, a variable order Adams method, or an extrapolation method. Extrapolation can be viewed as a variable order Runge-Kutta method. It is plausible that variation of order could lead to a much more efficient Runge-Kutta code, but numerical comparisons have been contradictory. We reconcile previous comparisons by exposing differences in testing methodology and incompatibilities of the implementations tested. An experimental Runge-Kutta code is compared to a state-of-the-art extrapolation code. With some qualifications, the extrapolation code shows no advantage. Extrapolation does not appear to be a particularly effective way to vary the order of Runge-Kutta methods. Although an acceptable way to solve nonstiff problems, our tests raise the question as to whether there is any point in pursuing it as a separate method.

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