Abstract
This paper investigates iterated Multistep Runge-Kutta methods of Radau type as a class of explicit methods suitable for parallel implementation. Using the idea of van der Houwen and Sommeijer [18], the method is designed in such a way that the right-hand side evaluations can be computed in parallel. We use stepsize control and variable order based on iterated approximation of the solution. A code is developed and its performance is compared with codes based on iterated Runge-Kutta methods of Gauss type and various Dormand and Prince pairs [15]. The accuracy of some of our methods are comparable with the PIRK10 methods of van der Houwen and Sommeijer [18], but require fewer processors. In addition at very stringent tolerances these new methods are competitive with RK78 pairs in a sequential implementation.
Similar content being viewed by others
References
K. Burrage and P. M. Moss, Simplifying assumptions for the order of partitioned multivalue methods, BIT 20 (1980) 452–465.
K. Burrage, Order and stability of explicit multivalue methods, Appl. Numer. Math. 1 (1985) 363–379.
K. Burrage, Order properties of multivalue methods, IMA J. Numer. Anal. 8 (1988) 43–69.
K. Burrage, The search for the holy grail, or: predictor-corrector methods for solving ODEIVPs, Appl. Numer. Math. 11 (1993) 125–141.
K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations (Oxford University Press, New York, 1995).
J. C. Butcher, On the convergence of numerical solutions to ordinary differential equations, Math. Comp. 20 (1966) 1–10.
J. C. Butcher, The order of numerical methods for ordinary differential equations, Math. Comp. 27 (1973) 793–806.
Nguyen huu Cong and T. Mitsui, A class of explicit parallel two-step Runge-Kutta methods, in preparation.
Nguyen huu Cong, Parallel iteration of symmetric Runge-Kutta methods for non stiff initial value problems, J. Comput. Appl. Math. 51 (1994) 117–125.
Nguyen huu Cong, Explicit parallel two-step Runge-Kutta-Nyström methods, Comput. Math. Appl., to appear.
G. J. Cooper, The order of convergence of general linear methods for ordinary differential equations, SIAM J. Numer. Anal. 15 (1978) 643–661.
W. H. Enright and J. D. Pryce, Two Fortran packages for assessing initial value methods, ACM Trans. Math. Software 13 (1987) 1–27.
A. Guillou and J. L. Soulé, La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation, R.I.R.O. R-3 (1969) 17–44.
E. Hairer and G. Wanner, On the Butcher group and general multivalue methods, Computing 13 (1974) 1–15.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics (Springer, Berlin, 1987).
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics (Springer, Berlin, 1991).
P. J. van der Houwen and B. P. Sommeijer, Variable step integration of high order Runge-Kutta methods on parallel computers, Report NM-R8817, CWI, Amsterdam, The Netherlands (1988).
P. J. van der Houwen and B. P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math. 29 (1990) 111–127.
P. J. van der Houwen and Nguyen huu Cong, Parallel block predictor-corrector methods of Runge-Kutta type, Appl. Numer. Math. 13 (1993) 109–123.
T. E. Hull, W. E. Enright, B. M. Fellen and A. E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9 (1972) 603–637.
A. Iserles and S. P. Nørsett, On the theory of parallel Runge-Kutta methods, IMA J. Numer. Anal. 10 (1990) 463–488.
Z. Jackiewicz and S. Tracogna, A general class of two-step Runge-Kutta methods for ordinary differential equations, SIAM J. Numer. Anal. 1 (1988) 1–38.
K. R. Jackson and S. P. Nørsett, The potential for parallelism in Runge-Kutta methods. Part I: RK formula in standard forms, SIAM J. Numer. Anal. 32 (1995) 49–82.
V. I. Krylov, Priblizhennoe Vyschisslenie Integralov (Gos. Izd. Fiz.-Mat. Lit., Moscow, 1959). English translation: Approximate Calculation of Integrals (Macmillan, New York, 1962).
I. Lie, Some aspects of parallel Runge-Kutta methods, Report No. 3/87, University of Trondheim, Division Numerical Mathematics, Norway (1987).
I. Lie and S. P. Nørsett, Superconvergence for multistep collocation, Math. Comp. 52 (1989) 65–79.
S. P. Nørsett and H. H. Simonsen, Aspects of parallel Runge-Kutta methods, in: Workshop on Numerical Methods for Ordinary Differential Equations, L'Aquila, ed. A. Bellen, Lecture Notes in Mathematics 1386 (Springer, Berlin, 1989) 103–117.
S. Schneider, Numerical experiments with a multistep Radau method, BIT 33 (1993) 332–350.
Rights and permissions
About this article
Cite this article
Burrage, K., Suhartanto, H. Parallel iterated methods based on multistep Runge-Kutta methods of Radau type. Advances in Computational Mathematics 7, 37–57 (1997). https://doi.org/10.1023/A:1018930415863
Issue Date:
DOI: https://doi.org/10.1023/A:1018930415863