Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-28T17:25:35.687Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  16 May 2014

Marcel Oliver
Affiliation:
School of Engineering and Science, Jacobs University, 28759 Bremen, Germany, (oliver@member.ams.org)
Claudia Wulff
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK, (c.wulff@surrey.ac.uk)

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 semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow 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. We then estimate the Galerkin truncation error for the semi-flow 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 Schrodinger equation.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)