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Some Modified Runge-Kutta Methods for the Numerical Solution of Initial-Value Problems with Oscillating Solutions

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Abstract

Two new modified Runge-Kutta methods with minimal phase-lag are developed for the numerical solution of initial-value problems with oscillating solutions which can be analyzed to a system of first order ordinary differential equations. These methods are based on the well known Runge-Kutta RK5(4)7FEq1 method of Higham and Hall (1990) of order five. Also, based on the property of the phase-lag a new error control procedure is introduced. Numerical and theoretical results show that this new approach is more efficient compared with the well known Runge-Kutta Dormand-Prince RK5(4)7S method [see Dormand and Prince (1980)] and the well known Runge-Kutta RK5(4)7FEq1 method of Higham and Hall (1990).

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REFERENCES

  • Brusa, L., and Nigro, L. (1980). A one-step method for direct integration of structural dynamics equations, Int'l. J. Number. Methods Engrg. 15, 685-699.

    Google Scholar 

  • Chawla, M. M., and Rao, P. S. (1984). A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, J. Comput. Appl. Math. 11, 277-281.

    Google Scholar 

  • Chawla, M. M., and Rao, P. S. (1986). A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems II. Explicit method, J. Comput. Appl. Math. 15, 329-337.

    Google Scholar 

  • Chawla, M. M., and Rao, P. S. (1987). An explicit sixth-order method with phase-lag of order eight for y″ = f(t, y), J. Comput. Appl. Math. 17, 365-368.

    Google Scholar 

  • Chawla, M. M., and Rao, P. S., and Neta, B. (1986). Two-step fourth order P-stable methods with phase-lag of order six for y″ = f(t, y), J. Comput. Appl. Math. 16, 233-236.

    Google Scholar 

  • Coleman, J. P. (1989). Numerical methods for y″ = f(x, y) via rational approximation for the cosine, IMA J. Number. Anal. 9, 145-165.

    Google Scholar 

  • Dormand, J. R., and Prince, P. J. (1980). A family of embedded Runge-Kutta formula, J. Comput. Appl. Math. 6, 19-26.

    Google Scholar 

  • Higham, D. J., and Hall, G. (1990). Embedded Runge-Kutta formula with stable equilibrium states, J. Comput. Appl. Math. 29, 25-33.

    Google Scholar 

  • Fehlberg, E. (undated). Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems, NASA Technical Report 315.

  • Raptis, A. D., and Simos, T. E. (1990). A four-step phase fitted method for the numerical integration of second order initial-value problems, BIT 31, 89-121.

    Google Scholar 

  • Sideridis, A. B., and Simos, T. E. (1992). A low order embedded Runge-Kutta method for periodic initial-value problems, J. Comput. Appl. Math. 44, 235-244.

    Google Scholar 

  • Simos, T. E. (1993). Runge-Kutta interpolants with minimal phase-lag, Comput. Math. Appl. 26, 43-49.

    Google Scholar 

  • Simos, T. E. (1991). A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems, Int'l. J. Comput. Math. 39, 135-140.

    Google Scholar 

  • Simos, T. E., and Raptis, A. D. (1990). Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation, Computing 45, 175-181.

    Google Scholar 

  • Simos, T. E. (1995). Modified Runge-Kutta Fehlberg methods for periodic initial-value problems, Japan J. Indust. Appl. Math. 12, 109-122.

    Google Scholar 

  • Thomas, R. M. (1984). Phase properties of high order, almost P-stable formula, BIT 24, 225-238.

    Google Scholar 

  • van der Houwen, P. J. (1977). Construction of Integration Formulas for Initial Value Problems, North-Holland, Amsterdam.

    Google Scholar 

  • van der Houwen, P. J., and Sommeijer, B. P. (1987a). Predictor-corrector methods for periodic second-order initial value problems, IMA J. Number. Anal. 7, 407-422.

    Google Scholar 

  • van der Houwen, P. J., and Sommeijer, B. P. (1987b). Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Number. Anal. 24, 595-617.

    Google Scholar 

  • van der Houwen, P. J., and Sommeijer, B. P. (1989a). Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems, SIAM J. Number. Anal. 26, 414-429.

    Google Scholar 

  • van der Houwen, P. J., and Sommeijer, B. P. (1989b). Phase-lag analysis of implicit Runge-Kutta methods, SIAM J. Number. Anal. 26, 214-229.

    Google Scholar 

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Authors and Affiliations

  1. Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR-671 00, Xanthi, Greece

    T. E. Simos

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Simos, T.E. Some Modified Runge-Kutta Methods for the Numerical Solution of Initial-Value Problems with Oscillating Solutions. Journal of Scientific Computing 13, 51–63 (1998). https://doi.org/10.1023/A:1023204727185

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  • DOI: https://doi.org/10.1023/A:1023204727185

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