Abstract
A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special formy″(t) = f(t,y(t)). The method is based on the exact integration of some given functions. If the second derivative of the solution is a linear combination of these functions, then the method is exact, and if this is not the case, the algebraic order (order of accuracy) of the method is very important. The algebraic order of the method is investigated by using the power series expansions of the coefficients, which are functions of the stepsize and independent variablet. It is shown that the method has the same algebraic order as those of the direct collocation Runge-Kutta-Nyström method by Van der Houwenet al., when the collocation points are identical with that method. Experimental results which demonstrate the validity of the theoretical analysis are presented.
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References
J.R. Cash, A note on the exponential fitting of blended, extended linear multistep methods. BIT,21 (1981), 450–453.
J.P. Coleman, Mixed interpolation methods with arbitrary nodes. J. Comput. Appl. Math.,92 (1998), 69–83.
J.P. Coleman and S.C. Duxbury, Mixed collocation methods fory″ =f(x,y). J. Comput. Appl. Math.,126 (2000), 47–75.
W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math.,3 (1961), 381–397.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I (Second Revised Edition). Springer-Verlag, 1992.
T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal.,9 (1972), 603–637.
K. Ozawa, A Four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems. Japan J. Indust. Appl. Math.,16 (1999), 25–46.
K. Ozawa, Functional fitting Runge-Kutta method with variable coefficients. Japan J. Indust. Appl. Math.,18 (2001), 105–128.
B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math.,28 (1998), 401–412.
A.D. Raptis and T.E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problems. BIT,31 (1991), 160–168.
T.E. Simos, Some new four-step exponential-fitting methods for the numerical solution of the radial Schrödinger equation. IMA J. Numer. Anal.,11 (1991), 347–356.
E. Stiefel and D.G. Bettis, Stabilization of Cowell’s method. Numer. Math.,13 (1969), 154–175.
P. J. Van der Houwen, B.P. Sommeijer and N.H. Cong, Stability of collocation-based Runge-Kutta-Nyström methods. BIT,31 (1991), 469–481.
G. Vanden Berghe, H. De Meyer, M. Van Daele and T. Van Hecke, Exponentially-fitted Runge-Kutta methods. Preprint, 1999.
J. Vanthournout, H. De Meyer and G. Vanden Berghe, Multistep methods for ordinary differential equations based on algebraic and first order trigonometric polynomials. Computational Ordinary Differential Equations (eds. J.R.Cash and I.Gladwell), 1992, 61–71.
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Ozawa, K. A functional fitting Runge-Kutta-Nyström method with variable coefficients. Japan J. Indust. Appl. Math. 19, 55–85 (2002). https://doi.org/10.1007/BF03167448
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DOI: https://doi.org/10.1007/BF03167448