zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers. (English) Zbl 0768.65041
For the numerical solution of second order differential equations y '' =f(y) the author considers fully implicit Runge-Kutta Nyström methods and applies diagonally implicit iteration for the solution of the nonlinear system. This makes the methods suitable for parallel computation. The order of convergence and the stability of the methods (for the scalar test equation y '' =λy, λ<0) are investigated, and many numerical experiments are presented.
65L06Multistep, Runge-Kutta, and extrapolation methods
65Y05Parallel computation (numerical methods)
65L20Stability and convergence of numerical methods for ODE
34A34Nonlinear ODE and systems, general
[1]K. Burrage, A study of order reduction for semi-linear problems, Report, University of Auckland (1990).
[2]J.C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods (Wiley, New York, 1987).
[3]J.R. Cash, Diagonally implicit Runge-Kutta formulae with error estimates, J. Inst. Math. Appl. 24 (1979) 293–301. · Zbl 0419.65044 · doi:10.1093/imamat/24.3.293
[4]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. · Zbl 0441.65056 · doi:10.1093/imamat/26.1.87
[5]G.J. Cooper and A. Sayfy, Semiexplicit A-stable Runge-Kutta methods, Math. Comp. 33 (1979) 146, 541–556.
[6]M. Crouzeix, Zur l’approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, Ph.D. Thesis, Université de Paris, France (1975).
[7]E. Fehlberg, Klassische Runge-Kutta-Nyström Formeln mit Schrittweiten-Kontrolle für Differentialgleichungenx”=f(t,x), Computing 10 (1972) 305–315. · Zbl 0261.65046 · doi:10.1007/BF02242243
[8]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).
[9]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.
[10]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. · Zbl 0731.65071 · doi:10.1007/BF01933263
[11]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. · Zbl 0747.65059 · doi:10.1016/0168-9274(92)90009-3
[12]A. Iserles and S.P. Nørsett, On the theory of parallel Runge-Kutta methods, IMA J. Numer. Anal. 10 (1990) 463–488. · Zbl 0712.65071 · doi:10.1093/imanum/10.4.463
[13]L. Kramarz, Stability of collocation methods for the numerical solution ofy”=f(x,y), BIT 20 (1980) 215–222. · Zbl 0425.65043 · doi:10.1007/BF01933194
[14]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).
[15]S.P. Nørsett, Semi-explicit Runge-Kutta methods, Report Mathematics and Computation No. 6/74, Dept. of Mathematics, University of Trondheim, Norway (1974).
[16]S.P. Nørsett and P.G. Thomsen, Embedded SDIRK-methods of basic order three, BIT 24 (1984) 634–646. · Zbl 0554.65053 · doi:10.1007/BF01934920
[17]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. · Zbl 0711.65057 · doi:10.1093/imanum/10.4.489
[18]B.P. Sommeijer, Parallelism in the numerical integration of initial value problems, Thesis, University of Amsterdam (1992).
[19]K. Strehmel and R. Weiner, Nichtlineare Stabilität und Phasenuntersuchung adaptiver Nyström-Runge-Kutta Methoden, Computing 35 (1985) 325–344. · Zbl 0569.65054 · doi:10.1007/BF02240198