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On Error Growth Functions of Runge-Kutta Methods

Zennaro, Marino
Published in Applied Numerical Mathematics. 1996, vol. 22, no. 1-3, p. 205-216
Abstract This paper studies estimates of the form $| y_1 -widehat y_1 | le arphi (hu )|y_0-widehat y_0|$, where $y_1,widehat y_1$ are the numerical solutions of a Runge-Kutta method applied to a stiff differential equation satisfying a one-sided Lipschitz condition (with constant $u$). An explicit formula for the optimal function $arphi (x)$ is given, and it is shown to be superexponential, i.e., $arphi (x_1) arphi (x_2) le arphi (x_1+x_2)$ if $x_1$ and $x_2$ have the same sign. As a consequence, results on asymptotic stability are obtained. Furthermore, upper bounds for $arphi (x)$ are presented that can be easily computed from the coefficients of the method.
Keywords Runge-Kutta methodsError growth functionsStiff differential equationsB-stabilityAsymptotic stabilitySuperexponential functions
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HAIRER, Ernst, ZENNARO, Marino. On Error Growth Functions of Runge-Kutta Methods. In: Applied Numerical Mathematics, 1996, vol. 22, n° 1-3, p. 205-216. https://archive-ouverte.unige.ch/unige:12492

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Deposited on : 2010-11-16

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