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The optimum Runge-Kutta method of a particular order is the one whose truncation error is a minimum. Various measures of the size of the truncation error are considered. The optimum method is practically independent of the measure being used. Moreover, among methods of the same order which one might consider using the difference in size of the estimated error is not more than a factor of 2 or 3. These results are confirmed in practice insofar as the choice of optimum method is concerned, but they underestimate the variation in error between different methods.
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Hull et al. (1964) studied this question.
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