A Runge-Kutta for all Seasons

Traditionally, Runge-Kutta integrations proceed one step at a time, with several function evaluations in each step. Upon proceeding to the next step, one abandons all information about the behavior of the solution that became available in any previous step. It is proposed to proceed in blocks of N steps. The self-starting feature of Runge-Kutta methods is still present, only for a block instead of a single step. If there is any need to prevent the last block from running past the end of the region of integration, one can take its step interval to be half as big, or a fourth as big. In each step of a block one can call on information from the other steps of the block. Thus fewer function evaluations per step are needed, and on this point the method becomes competitive with classic predictor-corrector methods. Also, by analyzing the trends over N steps at once, stimates of the accuracy can be derived which compare in reliability with those for classic predictor-corrector methods. One can adjust the step interval for each block so as to maintain essentially the optimum step interval at all times. Stability characteristics resemble those of classical Rurige-Kutta methods. The block method is also applied to improve the classic predictor-corrector methods