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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References
C. Bischof, A. Carle, G. Corliss, A. Griewank and P. Hovland, ADIFOR—Generating derivative codes from Fortran programs. Scientific Programming,1 (1992), 11–29.
R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math.,8 (1966), 1–13.
J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations. John Wiley & Sons, New York, 1987.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Non stiff Problems. Springer-Verlag, Berlin, 1993.
M. Iri, Simultaneous Computation of Functions, Partial Derivatives and Estimates of Rounding Errors — Complexity and Practicality —. Japan J. Appl. Math.,1 (1984), 223–252.
K. Kubota, PADRE2, a FORTRAN precompiler yielding error estimates and second derivatives. Proc. SIAM Workshop on Automatic Differentiation of Algorithms — Theory, Implementation and Application, 1991.
NUMPAC vol.3, User’s guide for library programs. Computer Center, Nagoya Univ., 1991 (in Japanese).
H. Ono, Five and six stage Runge-Kutta type formulas of orders numerically five and six. J. Inform. Process.,12 (1989), 251–260.
H. Ono, Limiting formulas of nine-stage explicit Runge-Kutta methods of order eight. Trans. IPS Japan,38 (1997), 1886–1893.
H. Ono and H. Toda, Explicit Runge-Kutta methods using second derivatives. Ann. Numer. Math.,1 (1994), 171–182.
L.B. Rall, Automatic Differentiation: Techniques and Applications. Lecture Notes in Computer Science120, Springer, 1981.
H. Toda, On the truncation error of a limiting formula of Runge-Kutta methods. Trans. IPS Japan,21 (1980), 285–296 (in Japanese).
T. Yoshida, Automatic derivative derivation system. Trans. IPS Japan,30 (1989), 799–806 (in Japanese).
T. Yoshida and H. Ono, Two stage explicit Runge-Kutta Type method using second and third derivatives. Trans. IPS Japan,44, No. 1 (2003), 82–87.
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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