Optimum Runge-Kutta methods

Hull, T. E.; Johnston, R. L.

doi:10.1090/S0025-5718-1964-0165700-6

Optimum Runge-Kutta methods
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by T. E. Hull and R. L. Johnston PDF

Math. Comp. 18 (1964), 306-310 Request permission

Abstract:

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.

References

S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Soc. 47 (1951), 96–108. MR 39374
J. Kuntzmann, Deux formules optimales du type de Runge-Kutta, Chiffres 2 (1959), 21–26 (French, with English, German and Russian summaries). MR 136080
Max Lotkin, On the accuracy of Runge-Kutta’s method, Math. Tables Aids Comput. 5 (1951), 128–133. MR 43566, DOI 10.1090/S0025-5718-1951-0043566-3
Anthony Ralston, Runge-Kutta methods with minimum error bounds, Math. Comp. 16 (1962), 431–437. MR 150954, DOI 10.1090/S0025-5718-1962-0150954-0
Michael J. Romanelli, Runge-Kutta methods for the solution of ordinary differential equations, Mathematical methods for digital computers, Wiley, New York, 1960, pp. 110–120. MR 0117915