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Geometric proofs of numerical stability for delay equations

Guglielmi, Nicola ; Hairer, Ernst

In: IMA Journal of Numerical Analysis, 2001, vol. 21, no. 1, p. 439-450

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    Title
    • Geometric proofs of numerical stability for delay equations
    Author
    • Guglielmi, Nicola. Département di Matematica Pura e Applicata, Università dell'Aquila, via Vetoio (Coppito), I‐67010, L'Aquila, Italy, guglielm@univaq.it
    • Hairer, Ernst. Departimento de Mathématiques, Université de Genève, CH‐1211 Genève 24, Switzerland, e-mail
    Document Type
    • Postprint
    Language
    • English
    Published in
    • IMA Journal of Numerical Analysis, 2001, vol. 21, no. 1, p. 439-450. Oxford University Press
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:299997
    Summary
    In this paper, asymptotic stability properties of implicit Runge-Kutta methods for delay differential equations are considered with respect to the test equation y′ (t) = a y (t) + b y(t − 1) with a, b ∈ ∁. In particular, we prove that symmetric methods and all methods of even order cannot be unconditionally stable with respect to the considered test equation, while many of them are stable on problems where a ∈ ℜ and b ∈ ∁. Furthermore, we prove that Radau‐IIA methods are stable for the subclass of equations where a = α + iγ with α, γ ∈ ℜ, γ sufficiently small, and b ∈ ∁