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An Analysis of Rosenbrock Methods for Nonlinear Stiff Initial Value Problems

J. G. Verwer
SIAM Journal on Numerical Analysis
Vol. 19, No. 1 (Feb., 1982), pp. 155-170
Stable URL: http://www.jstor.org/stable/2157190
Page Count: 16
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An Analysis of Rosenbrock Methods for Nonlinear Stiff Initial Value Problems
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Abstract

The paper presents an analysis of the Rosenbrock integration method applied to a stiff system of the form \begin{equation*}\tag{(1)} \dot{x} = f(t, x, y, \varepsilon) + \varepsilon^{-1} A(t)y, \quad \dot{y} = g(t, x, y, \varepsilon) + \varepsilon^{-1} \mu(t)By.\end {equation*} This equation possesses the following desirable model properties. (a) It permits the simultaneous occurrence of smooth and transient solution components. (b) It contains a small parameter admitting a transition to arbitrarily high stiffness. (c) The Jacobian matrix has a time-dependent eigensystem. (d) It contains nonlinear terms. Provided certain assumptions have been satisfied, a characteristic of (1) is that for given initial vectors x(0) = x0, y(0) = y0 \begin{equation*}\tag{(2)}\|x(t, \varepsilon)\| = O(1), \quad \|y(t, \varepsilon)\| = O(\varepsilon), \quad \varepsilon \rightarrow 0, \quad t \in(0, T], \quad T \text{finite}.\end {equation*} Our analysis will be directed towards obtaining criteria which guarantee a similar behavior for finite sequences of Rosenbrock approximations. By way of comparison, we also pay attention to D-stability properties of the Rosenbrock method. The property of D-stability, as introduced by van Veldhuizen, applies to the first variational equation of (1).

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