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

Authors:
Samir Karaa;Jun Zhang
Affiliations:
Laboratoire MIP, CNRS, UMR 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France;Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, KY
Venue:
Journal of Computational and Applied Mathematics
Year:
2003

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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.