We conduct experimental study on issues of computing numerical solution of the two dimensional convection-diffusion equation discretized by three finite difference schemes: the traditional central difference scheme, the standard upwind scheme and the fourth-order compact scheme. We study the computed accuracy achievable by each scheme, the algebraic properties of the coefficient matrices arising from different schemes and the performance of the Gauss-Seidel iterative method and the preconditioned GMRES iterative method for solving linear systems arising from these schemes. At the beginning of this paper, we raise several questions concerning the applications of these discretization schemes. At the end of the paper, we answer these questions and give our recommendations as which scheme may be used under what conditions. Key words: Convection-diffusion equation, finite difference discretization, iterative methods, preconditioning
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