Stability of Runge-Kutta methods for quasilinear parabolic problems

González, C.; Palencia, C.

doi:10.1090/S0025-5718-99-01156-4

Stability of Runge-Kutta methods for quasilinear parabolic problems
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by C. González and C. Palencia;

Math. Comp. 69 (2000), 609-628

DOI: https://doi.org/10.1090/S0025-5718-99-01156-4

Published electronically: May 20, 1999

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Abstract:

We consider a quasilinear parabolic problem \[ u’(t) = Q\big ( u(t) \big ) u(t), \qquad u(t_0) = u_0 \in \mathcal {D}, \] 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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