, Volume 31, Issue 4, pp 371–381 | Cite as

Nonlinear contractivity of a class of semi-implicit multistep methods

  • K. Strehmel
  • R. Weiner


The stability and contractivity of generalized linear multistep methods are studied for a large class of nonlinear stiff initial value problems. These methods are characterized by the fact that the coefficients of the integration formulas are matrices depending on the Jacobian or on an approximation to the Jacobian. Conditions for the parameters of such a multistep method are given which ensure that the method gives contractive numerical solutions over a large class of nonlinear dissipative systems for sufficiently small stepsizesh, where the restriction onh is not due to the stiffness of the problem. Stability and contractive properties of special methods of this class are reported.

AMS Subject Classifications

65 L 05 

Key words

Numerical analysis linear multistep methods stiff problems nonlinear stability 

Nichtlineare Kontraktivität einer Klasse von semi-impliziten Mehrschritt-Verfahren


Es wird die Stabilität und Kontraktivität von verallgemeinerten linearen Mehrschritt-Verfahren für eine große Klasse nichtlinearer steifer Anfangswertprobleme untersucht. Lie Koeffizienten dieser Verfahren sind Matrizen, die von der Jacobi-Matrix oder einer Näherung für diese abhängen. Es werden Bedingungen für die Parameter solcher Mehrschritt-Verfahren angegeben, die sicherstellen, daß das Verfahren numerische Lösungen erzeugt, die über eine große Klasse nichtlinearer dissipativer Systeme bei hinreichend kleinen Schrittweitenh kontraktiv sind, wobei die Einschränkung fürh nicht mit der Steifheit des Problems zusammenhängt. Über die Stabilität und Kontraktivität von speziellen Verfahren aus dieser Klasse wird berichtet.


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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • K. Strehmel
    • 1
  • R. Weiner
    • 1
  1. 1.Department of MathematicsMartin-Luther-University Halle-WittenbergHalleGerman Democratic Republic

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