Abstract
In the numerical solution of delay differential equations by a continuous explicit Runge-Kutta method a difficulty arises when the delay vanishes or becomes smaller than the stepsize the method would like to use. In this situation the standard explicit sequential process of computing the Runge-Kutta stages becomes an implicit process and an iteration scheme must be adopted. We will consider alternative iteration schemes and investigate their order.
Zusammenfassung
Beim numerischen Lösen von Differentialgleichungen mit nacheilendem Argument (DDEs) mit Hilfe von stetigen expliziten Runge-Kutta Methoden entstehen Schwierigkeiten, wenn die Argumentverzögerung verschwindet oder zumindest kleiner als die Verfahrensschrittweite wird. In dieser Situation wird der herkömmlich explizite und sequentielle Prozeß der Stufenberechnungen des RK-Schemas ein impliziter und muß überdies iteriert werden. In dieser Arbeit werden einige Iterationsmethoden untersucht und deren Ordnung bestimmt.
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Calvo, M., Montijano, J. I., Rández, L.: A fifth order interpolant for the Dormand and Prince Runge-Kutta method. J. Comp. Appl. Math.29, 91–100 (1990).
Enright, W. H.: A new error-control for initial value solvers. Appl. Math. Comput.31, 288–301 (1989).
Enright, W. H.: The relative efficiency of alternative defect control schemes for high order continuous Runge-Kutta formulas. SIAM J. Numer. Anal.30, 1419–1445 (1993).
Enright, W. H., Jackson, K. R., Nørsett, Thomsen, P. G.: Interpolants for Runge-Kutta formulas. ACM Trans. Math. Software12, 193–218 (1986).
Enright, W. H., Hu, M.: Interpolating Runge-Kutta methods for vanishing delay differential equations. Rep. 292, Dept. of Computer Science, Univ. of Toronto, Canada, 1994.
Hayashi, H., Enright, W. H.: A new algorithm for vanishing delay problems, CAMS annual meeting, May 30–June 2, 1993, York University (invited oral presentation).
Horn, M. K.: Fourth- and fifty-order, scaled Runge-Kutta algorithms for treating dense output. SIAM J. Numer. Anal.20, 558–568 (1983).
Hull, T. E., Enright, W. H., Jackson, K. R.: User's guide for DVERK — A subroutine for solving nonstiff ODE's. Rep. 100, Dept. of Computer Science, Univ. of Toronto, 1976.
Karoui, A., Vaillancourt, R.: A numerical method for vanishing-lag delay differential equations. Private communication, 1993.
Neves, K. W.: Automatic integration of functional differential equations: An approach, ACM Trans. Math. Software1, 357–368 (1986).
Neves, K. W., Thomson, S.: Solution of systems of functional differential equations with state dependent delays. Technical Report TR-92-003, Computer Science, Radford University, 1992.
Owren, B., Zennaro, M.: Derivation of efficient, continuous, explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput.13, 1488–1501 (1992).
Paul, C. A. H.: Developing a delay differential equation solver. Appl. Numer. Math.9, 403–414 (1992).
Sharp, P. W., Smart, E.: Private communication (1990).
Tavernini, L.: One-step methods for the numerical solution of Volterra functional differential equations. SIAM J. Numer. Anal.8, 786–795 (1971).
Verner: Differentiable interpolants for high-order Runge-Kutta methods. SIAM J. Numer. Anal.30, 1446–1466 (1993).
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This work was supported by Natural Science and Engineering Research Council of Canada and the Information Technology Research Center of Ontario.
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Enright, W.H., Hu, M. Interpolating Runge-Kutta methods for vanishing delay differential equations. Computing 55, 223–236 (1995). https://doi.org/10.1007/BF02238433
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DOI: https://doi.org/10.1007/BF02238433