### WWRC 84-17

Why Upwinding is Reasonable

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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