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A-stable Runge-Kutta methods for semilinear evolution equations

Oliver, M and Wulff, C (2012) A-stable Runge-Kutta methods for semilinear evolution equations Journal of Functional Analysis, 263 (7). pp. 1981-2023.

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We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, A-stable Runge-Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation. © 2012 Elsevier Inc.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Date : 1 October 2012
Identification Number : 10.1016/j.jfa.2012.06.022
Additional Information : NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Functional Analysis, 263(7), October 2012, DOI 10.1016/j.jfa.2012.06.022.
Depositing User : Symplectic Elements
Date Deposited : 18 Nov 2014 11:37
Last Modified : 19 Feb 2015 02:33

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