Advertisement

Computing

, Volume 44, Issue 4, pp 331–356 | Cite as

On error structures and extrapolation for stiff systems, with application in the method of lines

  • W. Auzinger
Article

Abstract

In this paper, which carries on the considerations in [1], the structure of the global error is studied for some time discretization schemes, applied to a class of stiff initial value problems as they typically arise from the semi-discretization of parabolic initial/boundary value problems (method of lines). The implicit Euler and trapezoidal schemes and a locally one-dimensional splitting method are considered, and ‘perturbed’ asymptotic error expansions are derived which are valid independent of the stiffness (independent of the meshwidth in space). The key point within the analysis is a careful, quantitative description of the remainder term in such an expansion. The results are applicable in the method of lines setting and enable the prediction of the behavior of extrapolation algorithms for the class of problems under consideration. These theoretical considerations are illustrated by numerical examples.

AMS Subject Classifications

65L05 65M20 

Key words

Stiff systems method of lines error structures extrapolation 

Über Fehlerstrukturen und Extrapolation bei steifen Systemen, mit Anwendung bei der Linienmethode

Zusammenfassung

In dieser Arbeit wird, als Fortführung der Betrachtungen im [1] die Struktur des globalen Fehlers einiger Zeit-Diskretisierungsschemata bei Anwendung auf eine Klasse steifer Anfangswertprobleme studiert, wie sie typischerweise bei der Semi-Diskretisierung parabolischer Anfangs/Randwertprobleme auftreten (Linienmethode). Im Mittelpunkt der Betrachtungen stehen das implizite Eulerverfahren, die implizite Trapezregel und ein lokal eindimensionales Zwischenschritt-Schema, und es werden ‘gestörte’ asymptotische Entwicklungen hergeleitet, deren Gültigkeit unabhängig von der Steifheit (unabhängig von der Gitterfeinheit der Raum-Diskretisierung) besteht. Der entscheidende, Punkt in der Analyse besteht in einer sorgfältigen, quantitativen Beschreibung des Restgliedes einer solchen Entwicklung. Die Resultate sind im Kontext der Linienmethode anwendbar und erlauben eine Vorhersage über das Verhalten von Extrapolationsverfahren bei der betrachteten Problemklasse. Die theoretischen Betrachtungen werden durch numerische Beispiele illustriert.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. Auzinger, On the error structure of the implicit Euler scheme applied to stiff system of differential equations, Computing43, 115–131 (1989).Google Scholar
  2. [2]
    W. Auzinger, R. Frank, F. Macsek, Asymptotic error expansions for stiff equations: The implicit Euler scheme, SIAM J. Numer. Anal.27, 1990.Google Scholar
  3. [3]
    W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: An analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math.56, 469–499 (1989).Google Scholar
  4. [4]
    P. Brenner, M. Crouzeix, V. Thomée, Single step methods for inhomogeneous linear differential equations in Banach space RAIRO Numer. Anal.16, 5–26 (1982).Google Scholar
  5. [5]
    M. Crouzeix, P. A. Raviart, Approximation d'équations d'évolution linéaires par des méthodes multi-pas, Etude numérique, des grands systèmes, Rencontre INRIA, Novosibirsk, Dunod, Paris, 1978.Google Scholar
  6. [6]
    A. R. Gourlay, J. Ll. Morris, The extrapolation of first order methods for parabolic partial differential equations II, SIAM J. Numer. Anal.17, 641–655 (1980).Google Scholar
  7. [7]
    W. B. Gragg, Repeated extrapolation to the limit in the numerical, solution of ordinary differential equations, Ph.D. Thesis. UCLA 1963.Google Scholar
  8. [8]
    E. Hairer, Ch. Lubich, Extrapolation at stiff differential equations, Numer. Math.52, 377–400 (1988).Google Scholar
  9. [9]
    W. H. Hundsdorfer, Local and global order reduction for some LOD schemes, Report NM-R8914, Dept. of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1989.Google Scholar
  10. [10]
    W. H. Hundsdorfer, J. G. Verwer, Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems, Math. Comp.53, 81–101 (1989).Google Scholar
  11. [11]
    T. Kato, Perturbation theory for linear operators, Springer, Berlin-Heidelberg-New York, 1966.Google Scholar
  12. [12]
    J. D. Lawson, J. Ll. Morris, The extrapolation of first order methods for parabolic partial differential equations I, SIAM J. Numer. Anal.15, 1212–1224 (1978).Google Scholar
  13. [13]
    M.-N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp.33, 919–931 (1979).Google Scholar
  14. [14]
    G. I. Marchuk, V. V. Shaidurov, Difference methods and their extrapolation, Springer Applications of Mathematics 19, 1983.Google Scholar
  15. [15]
    R. Rannacher, Discretization of the heat equation with singular initial data, ZAMM62, T346-T348, 1982.Google Scholar
  16. [16]
    R. D. Richtmeyer, K. W. Morton, Difference, methods for initial value problems, Interscience, 1967.Google Scholar
  17. [17]
    J. M. Sanz-Serna, J. G. Verwer, Stability and convergence at the PDE/stiff ODE interface, Applied Numerical Mathematics5, 117–132 (1989).Google Scholar
  18. [18]
    H. J. Stetter, Asymptotic expansions for the error of discretization algorithms for non-linear functional equations, Numer. Math.7, 18–31 (1965).Google Scholar
  19. [19]
    V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics 1054, Springer, 1984.Google Scholar
  20. [20]
    J. G. Verwer, H. B. De Vries, Global extrapolation of a first order, splitting method, SIAM J. Sci. Stat. Comput.6, 771–780 (1985).Google Scholar
  21. [21]
    N. N. Yanenko, The method of fractional steps, Springer, Berlin-Heidelberg-New York, 1971.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • W. Auzinger
    • 1
  1. 1.Institut für Angewandte und Numerische MathematikTechnische Universität WienWienAustria

Personalised recommendations