Abstract
In this paper we study the numerical solution of the differential/algebraic systems F(t, y, y′) = 0. Many of these systems can be solved conveniently and economically using a range of ODE methods. Others can be solved only by a small subset of ODE methods, and still others present insurmountable difficulty for all current ODE methods. We examine the first two groups of problems and indicate which methods we believe to be best for them. Then we explore the properties of the third group which cause the methods to fail.
The important factor which determines the solvability of systems of linear problems is a quantity called the global nilpotency. This differs from the usual nilpotency for matrix pencils when the problem is time dependent, so that techniques based on matrix transformations are unlikely to be successful.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Supported in part by the U.S. Department of Energy, Grant DEAC0276ER02383 and by the U.S. Department of Energy Office of Basic Energy Sciences.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
CAMPBELL, S. L., Linear time varying singular systems of differential equations, Dept. Mathematics, North Carolina State Univ., Raleigh, 1981.
DEUFLHARD, P., Order and stepsize control in extrapolation methods, Preprint No. 93, Univ. Heidelberg, 1980.
GEAR, C. W., The simultaneous numerical solution of differential-algebraic equations, IEEE Trans. Circuit Theory TC-18, (1), 1971, 89–95.
GEAR, C. W. and L. R. PETZOLD, ODE methods for differential/algebraic systems. In preparation.
GEAR, C. W., HSU, H. H. and L. PETZOLD, Differential-algebraic equations revisited, Proc. Numerical Methods for Solving Stiff Initial Value Problems, Oberwolfach, W. Germany, June 28–July 4, 1981.
Painter, J. F., Solving the Navier-Stokes equations with LSODI and the method of lines, Lawrence Livermore Laboratory Rpt. UCID-19262, 1981.
PETZOLD, L. R., Differential/algebraic equations are not ODEs, Rpt. SAND81-8668, Sandia National Laboratories, Livermore, CA, April 1981.
PETZOLD, L. R., A description of DASSL: A differential/algebraic system solver, to appear, Proceedings of IMACS World Congress, Montreal, Canada, August 1982.
STARNER, J. W., A numerical algorithm for the solution of implicit algebraic-differential systems of equations, Tech. Rpt. 318, Dept. Mathematics and Statistics, Univ. New Mexico, May 1976.
SASTRY, S. S., DESOER, C. A. and P. P. VARAIYA, Jump behavior of circuits and systems, Memorandum No. UCB/ERL M80/44, Electronics Research Laboratory, University of California-Berkeley, CA, October 1980.
SINCOVEC, R. F., DEMBART, B., EPTON, M. A., ERISMAN, A. M., MANKE, J. W. and E. L. YIP, Solvability of large-scale descriptor systems, Final Report DOE Contract ET-78-C-01-2876, Boeing Computer Services Co., Seattle, WA.
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Gear, C.W., Petzold, L.R. (1983). Differential/algebraic systems and matrix pencils. In: Kågström, B., Ruhe, A. (eds) Matrix Pencils. Lecture Notes in Mathematics, vol 973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062095
Download citation
DOI: https://doi.org/10.1007/BFb0062095
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11983-8
Online ISBN: 978-3-540-39447-1
eBook Packages: Springer Book Archive