Abstract
Two-stage explicit Runge-Kutta type methods using derivatives for the systemy′(t) =f(y(t)),y(t 0) =y 0 are considered. Derivatives in the first stage have the standard form, but in the second stage, they have the form included in the limiting formula. The κth-order Taylor series method uses derivativesf∼’,f∼",…,f (κ−1) Though the values of derivatives can be easily obtained by using automatic differentiation, the cost increases proportional to square of the order of differentiation. Two-stage methods considered here use the derivatives up tof (κ−3) in the first stage andf,f∼’ in the second stage. They can achieve κth-order accuracy and construct embedded formula for the error estimation.
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Ono, H., Yoshida, T. Two-stage explicit Runge-Kutta type methods using derivatives. Japan J. Indust. Appl. Math. 21, 361–374 (2004). https://doi.org/10.1007/BF03167588
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DOI: https://doi.org/10.1007/BF03167588