Abstract
The computational work required to implement implicit Runge-Kutta methods is often dominated by the cost of solving large sets of nonlinear equations. As an alternative to modified Newton methods, iteration schemes, which sacrifice superlinear convergence for reduced linear algebra costs, have been proposed. A new scheme of this type is considered here. This scheme avoids expensive vector transformations, is computationally more efficient, and gives improved performance.
Zusammenfassung
Der Rechenaufwand bei der Implementierung impliziter Runge-Kutta-Verfahren wird oft vom Aufwand für die Lösung großer nichtlinearer Gleichungssysteme dominiert. Als Alternative zu modifizierten Newton-Verfahren sind Iterationsverfahren vorgeschlagen worden, die auf die superlineare Konvergenz zugunsten einer Reduktion der Kosten im Bereich der linearen Algebra verzichten. Hier wird eine neue Methodik dieser Art betrachtet, die teure Vektortransformationen vermeidet, rechnerisch effizient ist und zu verbesserten Leistungsmerkmalen führt.
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References
Bickart, T. A.: An efficient solution process for implicit Runge-Kutta methods. SIAM J. Numer. Anal.14, 1022–1027 (1977).
Butcher, J. C.: On the implementation of implicit Runge-Kutta methods. BIT16, 237–240 (1976).
Butcher J. C.: Some implementation schemes for implicit Runge-Kutta methods. Proc., Dundee Conference on Numerical Analysis. Springer, Berlin Heidelberg New York (Lecture Notes in Mathematics, Vol. 773, pp. 12–24).
Cash, J. R.: On a class of implicit Runge-Kutta procedures, J. Inst. Maths. Applic.19, 455–470 (1977).
Chipman, F. H.: The implementation of Runge-Kutta implicit processess. BIT13, 391–393 (1973).
Collings, A. G., Tee, G. J., An analysis of Euler and implicit Runge-Kutta numerical integration shemes for structural dynamic problems. Proc., Sixth Australasian Conference on the Mechanics of Structures and Materials 1977,1, pp. 147–154.
Cooper, G. J., Butcher, J. C.: An iteration scheme for implicit Runge-Kutta methods. IMA J. Numer. Anal.3, 127–140 (1983).
Enright, W. H.: Improving the efficiency of matrix operations in the numerical solution of ODEs. Technical report 98 (1976), Computer Science Dept., Univ. of Toronto.
Frank, R., Ueberhuber, C. W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. BIT17, 146–159 (1977).
Gear, C. W.: The automatic integration of stiff ordinary differential equations. Proc., IFIP Congress, 1968, pp. 187–193.
Gladwell, I., Thomas, R. M.: Efficiency of methods for second order problems. Numerical Analysis Report 129 (1987), Dept. of Mathematics, Univ. of Manchester.
Varah, J. M.: On the efficient implementation of implicit Runge-Kutta methods. Maths. Comput.33, 557–561 (1979).
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Cooper, G.J., Vignesvaran, R. A scheme for the implementation of implicit Runge-Kutta methods. Computing 45, 321–332 (1990). https://doi.org/10.1007/BF02238800
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DOI: https://doi.org/10.1007/BF02238800