Abstract
Five-point finite-difference approximations are considered for a model (linear, constant-coefficient) convection-diffusion equation in two dimensions. Standard difference schemes for such problems behave badly when the convective terms are dominant. A new discretization is derived from a local integral representation of the true solution. This derivation is analogous to the way that the discrete Laplacian can be derived from the mean-value property of harmonic functions, and it generalizes an approach due to Allen and Southwell [Quart. J. Mech. Appl. Math., 8 (1955), pp. 129–45]. Also discussed is how the strong upwind bias of this and other discretizations serves to make more stable some methods of the two-sweep or marching type for the direct solution of the resulting linear algebraic equations.
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