Numerische Mathematik

, Volume 48, Issue 1, pp 85–90 | Cite as

Multi-step methods are essentially one-step methods

  • Urs Kirchgraber
Numerical Approximation of Transverse Shearing Stress in Bent Plates


In this note the geometry of multi-step methods is studied using invariant manifold theory for maps, as familiar from dynamical systems theory. This permits to associate a one-step method to each multi-step method to which the former is not only equivalent asymptotically, but equal in each step if the one-step method is used to produce the initial data of the multi-step method.

Subject Classifications

AMS(MOS): 65LO5 CR: G1.7 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Fenichel, N.: Persistence and Smoothness of Invariant Manifolds of Flows. Indiana Univ. Math. J.21, 193 (1971/72)Google Scholar
  2. Hartman, P.: Ordinary Differential Equations. New York: Wiley 1964Google Scholar
  3. Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lect. Notes in Math., 583. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  4. Iooss, G.: Bifurcations of Maps. Math. Studies, 36. Amsterdam-New York-Oxford: North Holland 1979Google Scholar
  5. Kirchgraber, U., Stiefel, E.: Methoden der analytischen Störungsrechnung und ihre Anwendungen. Leitfäden der Angew. Math. u. Mechanik, 44. Stuttgart: Teubner 1978Google Scholar
  6. Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Applications. Appl. Math. Sci., 19. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  7. Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer Tracts in Nat. Phil., 23. Berlin-Heidelberg-New York: Springer 1973Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Urs Kirchgraber
    • 1
    • 2
  1. 1.Department of MathematicsSwiss Federal Institute of Technology (ETH)ZürichSwitzerland
  2. 2.Wiskundig SeminariumVirije UniversiteitAmsterdamThe Netherlands

Personalised recommendations