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, W. Mekwi

Research output: Contribution to journalArticle

18 Citations (Scopus)

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.
LanguageEnglish
Pages507-528
Number of pages22
JournalIMA Journal of Numerical Analysis
Volume27
Issue number3
DOIs
Publication statusPublished - 2007

Fingerprint

Moving Mesh
Parabolic PDEs
Stability and Convergence
Finite Difference
Unconditionally Stable
Discretization
Mesh
Central Difference Schemes
Crank-Nicolson Scheme
θ-method
Semidiscretization
Euler Scheme
Experiments
Parabolic Problems
Time Integration
Finite Difference Scheme
Numerical Experiment
Model
Norm
Dependent

Keywords

  • adaptivity
  • moving meshes
  • ALE schemes
  • stability
  • numerical mathematics

Cite this

@article{ef00533f98894e7aadd017a3153f3c88,
title = "An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh",
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.",
keywords = "adaptivity, moving meshes, ALE schemes, stability, numerical mathematics",
author = "J.A. MacKenzie and W. Mekwi",
year = "2007",
doi = "10.1093/imanum/drl034",
language = "English",
volume = "27",
pages = "507--528",
journal = "IMA Journal of Numerical Analysis",
issn = "0272-4979",
number = "3",

}

TY - JOUR

T1 - An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh

AU - MacKenzie, J.A.

AU - Mekwi, W.

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

KW - adaptivity

KW - moving meshes

KW - ALE schemes

KW - stability

KW - numerical mathematics

U2 - 10.1093/imanum/drl034

DO - 10.1093/imanum/drl034

M3 - Article

VL - 27

SP - 507

EP - 528

JO - IMA Journal of Numerical Analysis

T2 - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 3

ER -