Abstract
In this paper, we study diagonally implicit Runge-Kutta-Nyström methods (DIRKN methods) for use on parallel computers. These methods are obtained by diagonally implicit iteration of fully implicit Runge-Kutta-Nyström methods (corrector methods). The number of iterations is chosen such that the method has the same order of accuracy as the corrector, and the iteration parameters serve to make the method at least A-stable. Since a large number of the stages can be computed in parallel, the methods are very efficient on parallel computers. We derive a number of A-stable, strongly A-stable and L-stable DIRKN methods of orderp withs * (p) sequential, singly diagonal-implicit stages wheres *(p)=[(p+1)/2] ors * (p)=[(p+1)/2]+1,[°] denoting the integer part function.
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References
K. Burrage, A study of order reduction for semi-linear problems, Report, University of Auckland (1990).
J.C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods (Wiley, New York, 1987).
J.R. Cash, Diagonally implicit Runge-Kutta formulae with error estimates, J. Inst. Math. Appl. 24 (1979) 293–301.
J.R. Cash and C.B. Liem, On the design of a variable order, variable step diagonally implicit Runge-Kutta algorithm, J. Inst. Math. Appl. 26 (1980) 87–91.
G.J. Cooper and A. Sayfy, Semiexplicit A-stable Runge-Kutta methods, Math. Comp. 33 (1979) 146, 541–556.
M. Crouzeix, Zur l'approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, Ph.D. Thesis, Université de Paris, France (1975).
E. Fehlberg, Klassische Runge-Kutta-Nyström Formeln mit Schrittweiten-Kontrolle für Differentialgleichungenx″=f(t,x), Computing 10 (1972) 305–315.
E. Hairer and G. Wanner,Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Series in Comp. Math., vol. 14 (Springer, Berlin, 1991).
P.J. van der Houwen, B.P. Sommeijer and W. Couzy, Embedded diagonally implicit Runge-Kutta algorithms on parallel computers, Math. Comp. 58 (1992) 197, 135–159.
P.J. van der Houwen, B.P. Sommeijer and Nguyen huu Cong, Stability of collocation-based Runge-Kutta-Nyström methods, BIT 31 (1991) 469–481.
P.J. van der Houwen, B.P. Sommeijer and Nguyen huu Cong, Parallel diagonally implicit Runge-Kutta-Nyström methods, J. Appl. Numer. Math. 9 (1992) 111–131.
A. Iserles and S.P. Nørsett, On the theory of parallel Runge-Kutta methods, IMA J. Numer. Anal. 10 (1990) 463–488.
L. Kramarz, Stability of collocation methods for the numerical solution ofy″=f(x,y), BIT 20 (1980) 215–222.
Nguyen huu Cong, A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers, Report NM-R9208, Centre for Mathematics and Computer Science, Amsterdam (1992).
S.P. Nørsett, Semi-explicit Runge-Kutta methods, Report Mathematics and Computation No. 6/74, Dept. of Mathematics, University of Trondheim, Norway (1974).
S.P. Nørsett and P.G. Thomsen, Embedded SDIRK-methods of basic order three, BIT 24 (1984) 634–646.
P.W. Sharp, J.H. Fine and K. Burrage, Two-stage and three-stage diagonally implicit Runge-Kutta-Nyström methods of orders three and four, IMA J. Numer. Anal. 10 (1990) 489–504.
B.P. Sommeijer, Parallelism in the numerical integration of initial value problems, Thesis, University of Amsterdam (1992).
K. Strehmel and R. Weiner, Nichtlineare Stabilität und Phasenuntersuchung adaptiver Nyström-Runge-Kutta Methoden, Computing 35 (1985) 325–344.
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Communicated by C. Brezinski
These investigations were supported by the University of Amsterdam with a research grant to enable the author to spend a total of two years in Amsterdam.
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Cong, N.h. A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers. Numer Algor 4, 263–281 (1993). https://doi.org/10.1007/BF02144107
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DOI: https://doi.org/10.1007/BF02144107