Analysis of Stationary Iterative Methods for Discrete
Convection-Diffusion Equation with a Nine-Point Compact Scheme

Samir Karaa
Laboratiore MIP, UMR 5640, Universit\'e
Paul Sabatier, 118 route de Narbonne,
31062 Toulouse Cedex 4, France


Jun Zhang
Laboratory for High Performance Scientific Computing and Numerical Simulation
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA


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.

Key words: Convection-diffusion equation, Fourth-order compact scheme, Linear systems, Iterative methods, Spectral radius

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This paper has been published in Journal of Computational and Applied Mathematics, Vol. 154, No. 2, pp. 447-476 (2003).

Technical Report No. 308-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. The second author's research work was supported in part by the U.S. National Science Foundation under grants CCR-9902022, CCR-9988165 and CCR-0043861.