Abstract
This paper considers iterative solution techniques for ordinary differential equations and investigates the speed of convergence of various processes. The iterates are defined as solutions of a sequence of differential equations. The solutions are the “wave-forms.” Waveform iterations are superlinearly convergent so a measure of the speed of convergence is defined and this is used to compare the value of various waveform methods. This measure is the rate of increase of the order of accuracy. The speed of the waveform Gauss Seidel method depends on the numbering of the equations. The numbering of the equations corresponds to a numbering of the directed graph specifying equation dependencies. We show how to compute the rate of order increase from the structure of the numbered graph and hence the optimum numbering, that is, the one which maximizes the speed of convergence.
Work supported in part by US DOE under grant DOE DEFG02-87ER25026 and by NSF under grant DMS 87-03226
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Gear, C. W., “Automatic multirate methods for ordinary differential equations”, Proc. IFIP 1980, 717–722, North-Holland Publishing Company.
Gear, C. W. and Wells, D. R., “Multirate linear multistep methods”, BIT 24 (1984), 484–502.
Juang, F. L., “Waveform methods for ordinary differential equations,” PhD Thesis, Department of Computer Science, University of Illinois at Urbana- Champaign, November, 1989.
Lelarasmee, E., “The waveform relaxation methods for the time domain analysis of large scale nonlinear dynamical systems”, Ph.D. dissertation, University of California, Berkeley.
Lelarasmee, E., Ruehli, A. E., and Sangiovanni-Vincentelli, A. L., “The waveform relaxation method for time-domain analysis of large scale integrated circuits”, IEEE Trans. on CAD of IC and Sys. Vol. 1, No. 3, pp. 131–145, July 1982.
Nevanlinna, O., “Remarks on Picaxd-Lindelöf iteration”, REPORT-MAT-A254, Helsinki University of Technology, Institute of Mathematics, Finland, December 1987.
Wells, D. R., “Multirate linear multistep methods for the solution of systems of ordinary differential equations”, Report No. UIUCDCS-R-82-1093, Department of Computer Science, University of Illinois at Urbana-Champaign, 1982.
White, J., Odeh, F., Sangiovanni-Vincentelli, A.L., and Ruehli, A., “Waveform relax-ation: theory and practice”, Memorandum No. UCB/ERL M85/65, 1985, Electronics Research Laboratory, College of Engineering, University of California, Berkeley.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Kluwer Academic Publishers
About this chapter
Cite this chapter
Gear, C.W., Juang, FL. (1991). The Speed of Waveform Methods for Odes. In: Spigler, R. (eds) Applied and Industrial Mathematics. Mathematics and Its Applications, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1908-2_4
Download citation
DOI: https://doi.org/10.1007/978-94-009-1908-2_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7351-6
Online ISBN: 978-94-009-1908-2
eBook Packages: Springer Book Archive