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

MacKenzie, J.A. and Mekwi, W. (2007) An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh. IMA Journal of Numerical Analysis, 27 (3). pp. 507-528. ISSN 0272-4979

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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.