Mathematics of Computation

Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

Author(s): C. González; C. Palencia.
Journal: Math. Comp. 68 (1999), 73-89.
MSC (1991): Primary 65J10, 65M12, 65M15
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where $A(t) : D \subset X \to X$ , $t \in J$ , is a family of sectorial operators in a Banach space

with time-independent domain

. This problem is discretized in time by means of an A( $\theta$ ) strongly stable Runge-Kutta method, $0 < \theta <\pi/2$ . We prove that the resulting discretization is stable, under the assumption

$\begin{displaymath}\| (A(t) - A(s) )x \| \le L|t-s|^\alpha (\|x\|+ \| A(s)x\|), \qquad x\in D, \,t,\,s \in J, \end{displaymath}$

where

and $\alpha \in (0,1)$ . Our results are applicable to the analysis of parabolic problems in the

, $p \ne 2$ , norms.

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Retrieve articles in Mathematics of Computation with MSC (1991): 65J10, 65M12, 65M15

C. González
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: cesareo@mac.cie.uva.es

C. Palencia
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: palencia@cpd.uva.es

DOI: 10.1090/S0025-5718-99-01018-2
PII: S 0025-5718(99)01018-2
Keywords: Parabolic problems, time-dependent, H\"older, Banach space, resolvents, sectorial, stability, Runge--Kutta
Received by editor(s): June 5, 1996
Received by editor(s) in revised form: September 4, 1996
Copyright of article: Copyright 1999, American Mathematical Society

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