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Stability under Galerkin truncation of A-stable Runge--Kutta semidiscretizations in time

Wulff, C (2014) Stability under Galerkin truncation of A-stable Runge--Kutta semidiscretizations in time Proc. Royal Soc. Edinb. A, 144 (3). pp. 603-636.

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Abstract

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow by an implicit, A-stable Runge--Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semiflow and its time-discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. Then we estimate the Galerkin truncation error for the semiflow of the evolution equation, its Runge--Kutta discretization, and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Wulff, CUNSPECIFIEDUNSPECIFIED
Date : 2014
Identification Number : 10.1017/S0308210512002028
Related URLs :
Additional Information : Copyright 2014 Cambridge University Press. Reprinted with permission.
Depositing User : Symplectic Elements
Date Deposited : 13 Nov 2014 09:38
Last Modified : 19 Feb 2015 02:33
URI: http://epubs.surrey.ac.uk/id/eprint/806396

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