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Numerische Mathematik

, Volume 42, Issue 3, pp 271–290 | Cite as

Contractivity in the numerical solution of initial value problems

  • M. N. Spijker
Article

Summary

Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|≦|U(0)| (t≧0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh>0 the numerical process is contractive.

In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp≦1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp>1 as well.

Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.

Subject Classifications

AMS(MOS): 65J10, 65L20, 65M10 CR: 5.17 

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • M. N. Spijker
    • 1
  1. 1.Universiteit van LeidenLeidenThe Netherlands

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