Numerische Mathematik

, Volume 51, Issue 5, pp 501–516

One-step and extrapolation methods for differential-algebraic systems

  • P. Deuflhard
  • E. Hairer
  • J. Zugck
Article

Summary

The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.

Subject Classifications

AMS(MOS): 65L05 CR: G1.7 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bader, G., Deuflhard, P.: A Semi-Implicit Midpoint Rule for Stiff Systems of Ordinary Differential Equations. Numer. Math.41, 373–398 (1983)Google Scholar
  2. 2.
    Deuflhard, P.: Recent Progress in Extrapolation Methods for Ordinary Differential Equations. SIAM Rev.27, 505–535 (1985)Google Scholar
  3. 3.
    Deuflhard, P.: Order and Stepsize Control in Extrapolation Methods. Numer. Math.41, 399–422 (1983)Google Scholar
  4. 4.
    Ebert, K.H., Ederer, H.J., Isbarn, G., Nowak, U.: The thermal decomposition ofn-Hexane. Kinetics, mechanism, and simulation. II. Univ. Heidelberg, SFB 123: Tech. Rep. 109 (1981)Google Scholar
  5. 5.
    Esser, C., Maas, U., Warnatz, J.; Chemistry of the Auto-Ignition in Hydrocarbon-Air Mixtures up to Octane and its Relation to Engine Knock (Preliminary Abbreviated Version), submitted to 10th International Colloquium on Gas Dynamics of Explosions and Reactive Systems. Berkeley, CA (1985)Google Scholar
  6. 6.
    Gantmacher, F.: The Theory of Matrices, vol. 2, New York, Chelsea (1959)Google Scholar
  7. 7.
    Gear, C.W., Petzold, L.: ODE Methods for the Solution of Differential Algebraic Systems. SIAM J. Numer. Anal.21, 716–728 (1984)Google Scholar
  8. 8.
    Hindmarsh, A.C.: LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM-SIGNUM Newsletter15, 10–11 (1980)Google Scholar
  9. 9.
    Hairer, E., Lubich, C.: Asymptotic Expansions of the Global Error of Fixed-Stepsize Methods. Numeer. Math.45, 345–360 (1984)Google Scholar
  10. 10.
    Lötstedt, P., Petzold, L.: Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints. SANDIA Livermore: Tech. Rep. SAND 83-8877 (1983)Google Scholar
  11. 11.
    März, R.: Multistep methods for initial value problems in implicit differential-algebraic equations. Berlin, DDR, Humboldt-Universität, Sektion Mathematik: Preprint Nr. 22 (1981). Beiträge zur Numer. Math.12, 107–123 (1984)Google Scholar
  12. 12.
    März, R.: On numerical integration methods for implicit ordinary differential equations and differential-algebraic equations. Wiss. Beiträge Univ. Jena, Proc. Kolloquium “Numerische Behandlung von Differentialgleichungen”. June 1982, pp. 1–15 (1983)Google Scholar
  13. 13.
    März, R.: On correctness and numerical treatment of boundary value problems in DAE's. Humboldt-Univ. zu Berlin, Sektion Mathematik, Preprint 73 (1984)Google Scholar
  14. 14.
    Petzold, L.: Differential/Algebraic Equations are not ODE's. SANDIA Livermore: Tech. Rep. SAND 81-8668 (1981). SIAM J. Stat. Sci. Comput.3, 367–384 (1984)Google Scholar
  15. 15.
    Petzold, L.R.: Order Results for Implicit Runge-Kutta Methods Applied to Differential/Algebraic Systems. SANDIA Livermore: Tech. Rep. SAND 84-8838 (1984)Google Scholar
  16. 16.
    Petzold, L.: A Description of DASSL: a Differential/Algebraic System Solver. Proc. IMACS World Congress 1982 (to appear)Google Scholar
  17. 17.
    Rheinboldt, W.C.: Differential-algebraic Systems as Differential Equations on Manifolds. Univ. Pittsburgh, Dept. Math. Stat., Tech. Rep. ICMA-83-55 (1983). Math. Comput.43, 473–482 (1985)Google Scholar
  18. 18.
    Sincovec, R.F., Erisman, A.M., Yip, E.L., Epton, M.A.: Analysis of Descriptor Systems Using Numerical Algorithms. IEEE Trans. Autom. Control AC26, 139–147 (1981)Google Scholar
  19. 19.
    Warnatz, J.: Homogene Elementarreaktionen und deren Zusammenwirken in Verbrennungsprozessen. BWK,37 (1985) Nr. 1–2-Jan.Feb.Google Scholar
  20. 20.
    Zugck, J.: Numerische Behandlung linear-impliziter Differentialgleichungen mit Hilfe von Extrapolationsmethoden. Univ. Heidelberg, Inst. Angew. Math.: Diplomarbeit (1984)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • P. Deuflhard
    • 1
  • E. Hairer
    • 2
  • J. Zugck
    • 3
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlin 31
  2. 2.Section de MathématiquesUniversité de GenèveGenève 24Switzerland
  3. 3.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergFederal Republic of Germany

Personalised recommendations