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Geometric proofs of numerical stability for delay equations

Guglielmi, Nicola
Published in IMA Journal of Numerical Analysis. 2001, vol. 21, no. 1, p. 439-450
Abstract 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 $a,b in C$. 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 $a in R$ and $binC$. Furthermore, we prove that Radau IIA methods are stable on the subclass for equations where $a = alpha + i gamma$ with $alpha, gamma in R$, $gamma$ sufficiently small, and $bin C$.
Keywords Delay differential equationsRunge-Kutta methodsAsymptotic stabilityRoot locus technique
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GUGLIELMI, Nicola, HAIRER, Ernst. Geometric proofs of numerical stability for delay equations. In: IMA Journal of Numerical Analysis, 2001, vol. 21, n° 1, p. 439-450. https://archive-ouverte.unige.ch/unige:12328

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Deposited on : 2010-11-05

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