A critical evaluation of upstream differencing applied to problems involving fluid flow

In approximating the convection terms in conservation equations, upstream differences are sometimes used. However, objections, based partly on the appearance of a false or numerical diffusion, are often raised. This paper shows that the false diffusion normally associated with upstream differences is a poor indicator of the total error in approximating the convection terms, and a new definition for false diffusion is proposed. The limiting conditions under which the upstream approximation is valid are then determined; several numerical experiments are performed to determine how much one may depart from these ideal conditions before the upstream approximation leads to unacceptable errors. It is concluded that the upstream approximation is valid only under restricted conditions, but that these include many problems of practical interest.