Introduction Upwind-biased discrete approximations have a distinguished history in numerical fluid mechanics, dating at least to von Neumann and Rirhtmyer (1950). Lately, however, upwinding has come under fire in water resources engineering. Among the most effective critics of upwind techniques are Gresho and Lee (1980), who take umbrage at the smearing of steep gradients in solutions of partial differential equations. While this viewpoint has cogency, a blanket condemnation of upwinding would be injudicious. There exist fluid flows for which upstream-biased discretizations are not only valid but in fact mathematically more appropriate than central approximations having higher-order accuracy.
Figures 1 and 2 illustrate the source of the controversy. Both plots show numerical solutions to a convection-dominated species transport equation using finite-element collocation. Figure 1, the result of a centered scheme, shows a solution having unrealistic wiggles near the concentration front; Figure 2, from an upwind scheme, exhibits nonphysical smearing. The wiggles in the centered scheme disappear altogether when the spatial step Dx is small enough, whereas the smearing associated with upstream weighting decreases continuously with Dx. Gresho and Lee argue that the wiggles indicate an inappropriate spatial grid and that suppressing them via upwinding eliminates useful symptoms in favor of a less informative flaw, smearing.
Were wiggles the only difficulty with centered schemes, proscribing upwind methods might be in order. However, as we shall see, for certain types of equations centered schemes can fail to converge. This difficulty is not symptomatic of an unsuitable grid; rather, it betrays an inability of centered schemes to impose proper uniqueness criteria. For such equations, upwinding can be reasonable.
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