Embedded diagonally implicit runge-kutta algorithmson parallel computers

Mathematics of Computation

Volumn 58, Issue 197, 1992, Pages 135-159

  Van Der Houwen, P J ^a   Sommelier, B P ^a   Couzy, W ^a  

^a CWI   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84968504322     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-1992-1106986-8     Document Type: Article

References (24)

1. R. Alexander, Diagonally implicit Runge-Kutta methods for stiff ODE's, SIAM J. Numer. Anal. 14(1977), 1006-1021.
2. K. Burrage, The error behaviour of a general class of predictor-corrector methods, CMSR Report, University of Liverpool, 1989.
3. J. C. Butcher, The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, Wiley, New York, 1987.
4. J. R. Cash, Diagonally implicit Runge-Kutta formulae with error estimates, J. Inst. Math. Appl. 24 (1979), 293-301.
5. 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.
6. M. Crouzeix, Sur l'approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, Ph. D. Thesis, Université de Paris, 1975.
7. K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monograph, no. 2, North-Holland, Amsterdam-New York-Oxford, 1984.
8. W. H. Enright, T. E. Hull, and B. Lindberg, Comparing numerical methods for stiff systems of ODEs, BIT 15 (1975), 10-48.
9. B. A. Gottwald and G. Wanner, A reliable Rosenbrock integrator for stiff differential equations, Computing 26 (1981), 355-360.
10. E. Hairer, Ch. Lubich, and M. Roche, Error of Runge-Kutta methods for stiff problems studied via differentiala lgebraice quations, BIT 28 (1988), 678-700.
11. A. C. Hindmarsh, LSODE and LSODI, two new initial value ordinary differential equation solvers, ACM/SIGNUM Newsletter (4) 15 (1980), 10-11.
12. P. J. van der Houwen and B. P. Sommeijer, Variable step iteration of high-order Runge-Kuttamethods on parallel computers, J. Comp. Appl. Math. 29 (1990), 111-127.
13. P. J. van der Houwen and B. P. Sommeijer, Iterated Runge-Kutta methods on parallel computers, SIAM J. Sci. Statist. Comput. 12(1991), 1000-1028.
14. P. J. van der Houwen, B. P. Sommeijer, and W. Couzy, Embedded diagonally implicit Runge-Kutta algorithms on parallel computers, Report NM-R8912, Centre for Mathematics and Computer Science, Amsterdam, 1989.
15. A. Iserles and S. P. Narsett, On the theory of parallel Runge-Kutta methods, IMA J. Numer. Anal. 10 (1990), 463-488.
16. K. Jackson and S. P. Narsett, Parallel Runge-Kutta methods, manuscript, 1988.
17. P. Kaps, Rosenbrock-type methods, Numerical Methods for Stiff Initial Value Problems (G. Dahlquist and R. Jeltsch, eds.), Bericht Nr. 9, Inst, für Geometrie und Praktische Mathematik der RWTH Aachen, 1981.
18. I. Lie, Some aspects of parallel Runge-Kutta methods, Report No. 3/87, Division of Numerical Mathematics, University of Trondheim, 1987.
19. S. P. Norsett, Semi-explicit Runge-Kutta methods, Report Mathematics and Computation No. 6/74, Dept. of Mathematics, University of Trondheim, 1974.
20. S. P. Norsett, C-polynomials for rational approximation to the exponential function, Numer. Math. 25 (1975), 39-56.
21. S. P. Nersett and P. G. Thomsen, Embedded SDIRK-methodso f basic order three, BIT 24 (1984), 634-646.
22. K. Stewart, Avoiding stability-induced inefficiencies in BDF methods, J. Comput. Appl. Math. 29(1990), 357-367.
23. H. W. Tarn, Parallel methods for the numerical solution of ordinary differential equations, Report NO. UIUCDCS-R-89-1516, Computer Science Department, University of Illinois, 1989.
24. A. Wolfbrandt, A study of Rosenbrock processes with respect to order conditions and stiff stability, Ph. D. Thesis, Chalmers University of Technology, Göteborg, 1977.