Abstract
Implicit Runge-Kutta methods are known as highly accurate and stable methods for solving differential equations. However, the iteration technique used to solve implicit Runge-Kutta methods requires a lot of computational efforts. To lessen the computational effort, one can iterate simultaneously at a number of points along the t-axis. In this paper, we extend the PDIRK (Parallel Diagonal Iterated Runge-Kutta) methods to delay differential equations (DDEs). We give the region of convergence and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector. It is proved that PDIRK methods to DDEs are efficient, and the diagonal matrix D of the PDIRK methods for DDES can be selected in the same way as for ordinary differential equations (ODEs).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. J. van der Houwen and B. P. Sommeijer, Iterated Runge-Kutta methods on parallel computers, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 1000-1028.
P. J. van der Houwen, B. P. Sommeijer and W. A. van der Veen, Parallel iteration across the steps of high order Runge-Kutta methods for nonstiffi nitial value problems, J. Comput. Appl. Math., 60 (1995), pp. 309-329.
P. J. van der Houwen, B. P. Sommeijer and W. A. van der Veen, Parallelism across the steps in iterated Runge-Kutta methods for nonstiffi nitial value problems, Numer. Algorithms, 8 (1994), pp. 293-312.
W. A. van der Veen, Step-parallel algorithms for stiffi nitial value problems, Comput. Math. Applic., 30:11 (1995), pp. 9-23.
J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, Wiley, New York, 1987.
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, New York, 1987.
M. Zennaro, P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49 (1986), pp. 305-318.
K. J. In 't Hout, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equation, BIT, 32 (1992), pp. 634-649.
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
P. Lancaster, Theory of Matrices, Academic Press, New York, 1984.
E. Hairer and G. Wanner, Solving ordinary diffferential equations II: Stiff and differential-algebraic problems, Springer-Verlag, Berlin, 1996.
W. A. van der Veen, Parallelism in the Numerical Solution Ordinary and Implicit Differential Equations, Centrum voor Wiskunde en Informatica (CWI), Amsterdam, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ding, X., Liu, M. Convergence Aspects of Step-Parallel Iteration of Runge-Kutta Methods for Delay Differential Equations. BIT Numerical Mathematics 42, 508–518 (2002). https://doi.org/10.1023/A:1021925109236
Issue Date:
DOI: https://doi.org/10.1023/A:1021925109236