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IMA Journal of Numerical Analysis Advance Access originally published online on November 20, 2006
IMA Journal of Numerical Analysis 2007 27(3):507-528; doi:10.1093/imanum/drl034
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh

J. A. Mackenzie{dagger} and W. R. Mekwi{ddagger}

Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK

{dagger} Corresponding author. Email: jam{at}maths.strath.ac.uk

{ddagger} Email: ta.wmek{at}maths.strath.ac.uk

Received on 3 March 2006. Revised on 11 October 2006.


  Abstract

The aim of this paper is to investigate the stability and convergence of time integration schemes for the solution of a semi-discretization of a model parabolic problem in 1D using a moving mesh. The spatial discretization is achieved using a second-order central finite-difference scheme. Using energy techniques we show that the backward Euler scheme is unconditionally stable in a mesh-dependent L2-norm, independently of the mesh movement, but the Crank–Nicolson (CN) scheme is only conditionally stable. By identifying the diffusive and anti-diffusive effects caused by the mesh movement, we devise an adaptive {theta}-method that is shown to be unconditionally stable and asymptotically second-order accurate. Numerical experiments are presented to back up the findings of the analysis.

Key Words: adaptivity; moving meshes; ALE schemes; stability


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