Analysis of stationary iterative methods for the discrete convection-diffusion equation with a 9-point compact scheme

Samir Karaa, Jun Zhang

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We study the convergence of point and line stationary iterative methods for solving the linear system arising from a fourth-order 9-point compact finite difference discretization of the two-dimensional convection-diffusion equation with constant coefficients. We present new techniques to bound the spectral radii of iteration matrices in terms of the cell Reynolds numbers. We also derive analytic formulas for the spectral radii for special values of the cell Reynolds numbers and study asymptotic behaviors of the analytic bounds. The results provide rigorous justification for the numerical experiments conducted elsewhere, which show good stability for the fourth-order compact scheme. In addition, we compare the 9-point scheme with the traditional 5-point difference discretization schemes and conduct some numerical experiments to supplement our analyses.

Original languageEnglish
Pages (from-to)447-476
Number of pages30
JournalJournal of Computational and Applied Mathematics
Volume154
Issue number2
DOIs
Publication statusPublished - May 15 2003

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Compact Scheme
Convection-diffusion Equation
Discrete Equations
Iterative methods
Reynolds number
Iteration
Spectral Radius
Linear systems
Fourth Order
Experiments
Numerical Experiment
Discretization Scheme
Cell
Difference Scheme
Justification
Finite Difference
Discretization
Asymptotic Behavior
Linear Systems
Convection

Keywords

  • Convection-diffusion equation
  • Fourth-order compact scheme
  • Iterative methods
  • Linear systems
  • Spectral radius

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

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AB - We study the convergence of point and line stationary iterative methods for solving the linear system arising from a fourth-order 9-point compact finite difference discretization of the two-dimensional convection-diffusion equation with constant coefficients. We present new techniques to bound the spectral radii of iteration matrices in terms of the cell Reynolds numbers. We also derive analytic formulas for the spectral radii for special values of the cell Reynolds numbers and study asymptotic behaviors of the analytic bounds. The results provide rigorous justification for the numerical experiments conducted elsewhere, which show good stability for the fourth-order compact scheme. In addition, we compare the 9-point scheme with the traditional 5-point difference discretization schemes and conduct some numerical experiments to supplement our analyses.

KW - Convection-diffusion equation

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KW - Linear systems

KW - Spectral radius

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