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A-stable Runge--Kutta methods for semilinear evolution equations - Stability under Galerkin truncation

Wulff, C (2010) A-stable Runge--Kutta methods for semilinear evolution equations - Stability under Galerkin truncation UNSPECIFIED. arxiv.

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Abstract

We consider semilinear evolution equations for which the linear part is normal up to a bounded perturbation and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We apply a class of implicit, A-stable Runge-Kutta methods, which includes Gauss-Legendre collocation methods, to such equations. In a previous paper, we have shown regularity of the semiflow and its time-semidiscretization, and proved their convergence as the temporal stepsize tends to zero. In this paper, we show that these results persist under spatial spectral Galerkin truncation and provide explicit stability estimates for the Galerkin truncation of the semiflow, the Runge-Kutta method, and their derivatives on a scale of Hilbert spaces. In particular, we analyze the dependency of the order of convergence on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation on the circle.

Item Type: Monograph (UNSPECIFIED)
Authors :
AuthorsEmailORCID
Wulff, CUNSPECIFIEDUNSPECIFIED
Date : 27 July 2010
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 28 Mar 2017 13:10
Last Modified : 28 Mar 2017 13:10
URI: http://epubs.surrey.ac.uk/id/eprint/805289

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