On Convergence and Performance of Iterative Methods
for Solving Variable Coefficient Convection-Diffusion
Equation with a Fourth-Order Compact Difference 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 Computer Simulation
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA


We conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection-diffusion equation discretized by a fourth-order compact difference scheme. Several conditions are formulated under which the coefficient matrix is guaranteed to be an M-matrix. We further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods, multigrid method, and preconditioned GMRES. Three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions, the orderings of the grid points, and the magnitude of the convection coefficients.

Key words: Convection-diffusion equation, Fourth-order compact scheme, Iterative methods, grid ordering, multicoloring

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This paper has been published in Computers & Mathematics with Applications, Vol. 44, pp. 457-479 (2002).

Technical Report No. 315-01, Department of Computer Science, University of Kentucky, Lexington, KY, 2001. 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.