“The Numerical Solution of Differential and Differential-Algebraic Systems”

Authors: Syvert P. Nørsett,
Affiliation: NTNU
Reference: 1985, Vol 6, No 3, pp. 141-152.

Keywords: Initial value systems, algebraic constraints, stiff systems, discontinuities, index

Abstract: Systems of ordinary differential equations (ODE) or ordinary differential/algebraic equations (DAE) are well-known mathematical models. The numerical solution of such systems are discussed. For (ODE) we mention some available codes and stress the need of type insensitive versions. Further the term stiffness is redefined, and ideas on handling discontinuities are presented. The paper ends with a discussion of index for DAE.

PDF PDF (1186 Kb)        DOI: 10.4173/mic.1985.3.3

DOI forward links to this article:
[1] G.L. Jones (1990), doi:10.1016/0098-1354(90)87054-S
[2] John D. Perkins (1986), doi:10.4173/mic.1986.2.2
[3] W. Marquardt, P. Holl and E.D. Gilles (1987), doi:10.1016/S1474-6670(17)55465-8
[4] Hubert B. Keller (1987), doi:10.1007/978-3-642-73000-9_26
[1] BUTCHER, J.C. (1963). Coefficients for the study of Runge-Kutta integration processes, J. Australian Math. Soc. 3, 185-201 doi:10.1017/S1446788700027932
[2] COURANT, R., FRIEDRICHS, K.O. (1948). Supersonic Flow and Shock Waves, Springer-Verlag, New-York-Heidelberg-Berlin.
[3] CURTISS, C.F., HIRSCHFELDFR, J.O. (1952). Integration of stiff equations, Proc. Nat. Acad. Science, U.S.A., 38, 235-243 doi:10.1073/pnas.38.3.235
[4] DAHLQUIST, G. (1956). Numerical integration of ordinary differential equations, Math. Scandinavica, 4, 33-50.
[5] DAHLQULST, G. (1963). A special stability problem for linear multistep methods, BIT, 3, 27-43 doi:10.1007/BF01963532
[6] ENRIGHT, W., JACKSON, K., NØRSETT, S.P., THOMSON, P.G. (1985). Continuous Runge-Kutta methods for dense output and discontinuities, Report No. 180/85, Dept. of Computer Science, University of Toronto, Canada.
[7] EULER (1913). Opera mania, series prima Vol. 11, Leipzig and Berlin.
[8] GEAR, C.W., PETZOLD, L.R. (1982). ODE Methods for the Solution of Differential/Algebraic Systems, Report No. SAND82-8051, Sandia National Laboratories, Livermore, CA, U.S.A.
[9] NØRRSET, S.P., THOMSEN, P.G. (1984). Embedded SDIRK-methods of basic order three, BIT, 24, 634-646.
[10] NØRSETT, S.P., THOMSEN, P.G. (1985). Switching between modified Newton and fixpoint iteration for implicit ODE-solvers, Submitted BIT doi:10.1007/BF01934920
[11] PETZOLD, L.R. (1982). Differential/algebraic equations are not ODEs, SIAM J. Sci. Stat. Comput., 3, 367-384 doi:10.1137/0903023
[12] SHAMPINE, L.F. (1981). Type insensitive ODE codes based on implicit A-stable formulas, Math. Comp., 36, 499-510 doi:10.2307/2007655
[13] SINCOREE, R.F., DEMBART, B., EPTON, M.A., MANKE, J.W., YIP, E.L. (1979). Solvability of Large-Scale Descriptor Systems, Report Boeing Computer Services Company, Seattle, WA, U.S.A.

  title={{The Numerical Solution of Differential and Differential-Algebraic Systems}},
  author={Nørsett, Syvert P.},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}