Abstract
An explicit fourth order Runge-Kutta Fehlberg method for the numerical solution of first order differential equations having oscillating solutions is developed in this paper. This method is constructed using a linear homogeneous test equation with phase-lag of order either six or eight and with dissipative order six. Both the schemes are used for the numerical solution of equations describing free and weakly forced oscillations and semidiscretized hyperbolic equations. The numerical results obtained show that the new method is much more accurate than other methods proposed recently.
Zusammenfassung
In dieser Arbeit wird ein explizites RKF-Verfahren zur numerischen Lösung von Differentialgleichungen 1. Ordnung mit periodischen Lösungen entwickelt. Für eine lineare homogene Testaufgabe ergeben sich dabei eine dissipative Ordnung 6 und Phasenverschiebungen der Ordnung 6 bzw. 8. Beide Varianten werden auf Gleichungen angewandt, die freie oder schwach-erzwungene Schwingungen beschreiben, sowie auf teildiskretisierte hyperbolische Gleichungen. Die numerischen Ergebnisse erweisen das neue Verfahren als wesentlich genauer als andere kürzlich vorgeschlagene Verfahren.
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Brusa, L., Nigro, L.: A one-step method for direct integration of structural dynamic equations. Internat. J. Numer. Methods Engrg.15, 685–699 (1980).
Chawla, M. M., Rao, P. S.: 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 (1984).
Chawla, M. M., Rao, P. S.: 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 (1986).
Chawla, M. M., Rao, P. S., Neta, B.: Two-step fourth orderP-stable methods with phase-lag of order six fory″=f(t, y). J. Comput. Appl. Math.16, 233–236 (1986).
Chawla, M. M., Rao, P. S.: An explicit sixth-order method with phase-lag of order eight fory″=f(t,y). J. Comput. Appl. Math.17, 365–368 (1987).
Coleman, J. P.: Numerical methods fory″=f(x,y) via rational approximation for the cosine. IMA J. Numer. Anal.9, 145–165 (1989).
Fehlberg, E.: Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. NASA Technical Report315, USA, 1969.
Houwen, P. J. van der, Sommeijer, B. P.: Predictor-corrector methods for periodic second-order intial value problems. IMA J. Numer. Anal.7, 407–422 (1987).
Houwen, P. J. van der, Sommeijer, B. P.: Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal.24, 595–617 (1987).
Houwen, P. J. van der Sommeijer, B. P.: Diagonally implicit Runge-Kutta-Nystrom methods for oscillatory problems. SIAM J. Numer. Anal.26, 414–429 (1989).
Houwen, P. J. van der, Sommeijer, B. P.: Phase-lag analysis of implicit Runge-Kutta methods. SIAM J. Numer. Anal.26, 214–229 (1989).
Houwen, P. J. van der: Construction of integration formulas for initial value problems. Amsterdam: North-Holland 1977.
Raptis, A. D., Simos, T. E.: A four-step phase fitted method for the numerical integration of second order initial-value problems. BIT31, 89–121 (1990).
Sideridis, A. B., Simos, T. E.: Accurate numerical approximations to initial value problems with oscillating solutions in biology, Report TR/48, Informatics Laboratory, Agricultural University of Athens, Greece, 1991.
Simos, T. E., Raptis, A. D.: Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrodinger equation. Computing45, 175–181 (1990).
Simos, T. E.: A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem, Internat. J. Comput. Math.39, 135–140 (1991).
Thomas, R. M.: Phase properties of high order, almostP-stable formulae. BIT.24, 225–238 (1984).
Simos, T. E.: On the phase-lag analysis (in preparation).
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Simos, T.E., Sideridis, A.B. Accurate numerical approximations to initial value problems with periodical solutions. Computing 50, 87–92 (1993). https://doi.org/10.1007/BF02280042
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DOI: https://doi.org/10.1007/BF02280042