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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

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


Authors: C. González and C. Palencia
Journal: Math. Comp. 68 (1999), 73-89
MSC (1991): Primary 65J10, 65M12, 65M15
DOI: https://doi.org/10.1090/S0025-5718-99-01018-2
MathSciNet review: 1609666
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider an abstract time-dependent, linear parabolic problem

\begin{displaymath}u'(t) = A(t)u(t), \qquad u(t_0) = u_0, \end{displaymath}

where $A(t) : D \subset X \to X$, $t \in J$, is a family of sectorial operators in a Banach space $X$ with time-independent domain $D$. 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 $L>0$ and $\alpha \in (0,1)$. Our results are applicable to the analysis of parabolic problems in the $L^p$, $p \ne 2$, norms.


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Additional Information

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: https://doi.org/10.1090/S0025-5718-99-01018-2
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
Article copyright: © Copyright 1999 American Mathematical Society