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Stability of Runge-Kutta methods
for quasilinear parabolic problems

Authors: C. González and C. Palencia
Journal: Math. Comp. 69 (2000), 609-628
MSC (1991): Primary 65M12, 65M15, 65M20, 65L06, 65J10, 65J15
Published electronically: May 20, 1999
MathSciNet review: 1659851
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a quasilinear parabolic problem

\begin{displaymath}u'(t) = Q\big( u(t) \big) u(t), \qquad u(t_0) = u_0 \in \mathcal{D}, \end{displaymath}

where $Q(w) : \mathcal{D}\subset X \to X$, $w \in W \subset X$, is a family of sectorial operators in a Banach space $X$ with fixed domain $\mathcal{D}$. This problem is discretized in time by means of a strongly A($\theta$)-stable, $0 < \theta \le\pi/2$, Runge-Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of $Q(w)$ with respect to $w$. Our results are useful for studying in $L^p$ norms, $1 \le p \le + \infty$, many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.

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Additional Information

C. González
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain

C. Palencia
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain

Received by editor(s): March 12, 1997
Received by editor(s) in revised form: February 23, 1998, and June 9, 1998
Published electronically: May 20, 1999
Article copyright: © Copyright 2000 American Mathematical Society

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