3.10 Upwind scheme

PICT\relax \special {t4ht=

The upwind scheme represents the face value eqn by the value eqn in the cell upwind of the face. The advantage of upwind is that it can guarantee boundedness of a field eqn. We can demonstrate this point by revisiting the 1D Eq. (2.32 ) in Sec. 2.9 :

@ @ -@t-+ ux @x-= 0: \relax \special {t4ht=
In the graphic above, we track the translation of a profile of eqn by equating changes in eqn in time to the local gradient eqn. If we apply upwind to calculate the change at point P, the gradient eqn and no change in eqn is correctly calculated (left).

However, linear differencing between upwind and downwind values results in eqn, so predicts a decrease in the value at P (right). The solution produces a solution with eqn, so is unbounded.

Boundedness of the conservative form of advection eqn is only guaranteed when eqn, as discussed in Sec. 2.9 . In 1D, the conservative form moves eqn inside the derivative eqn. That gradient is only zero with upwind when eqn is uniform, i.e. the 1D equivalent to eqn.

Diffusion of upwind

The upwind scheme is highly diffusive which can result in poor accuracy. Its diffusive nature can be explained by considering the following Taylor’s series expansion:

 @---- @2--- --x2 (x + x) = (x)+ @x x+ @x2 2 + ::: \relax \special {t4ht=
In our 1D example, the upwind scheme calculates eqn using eqn and eqn at locations U and P, separated by distance eqn. Relating the upwind calculation to Eq. (3.9 ) gives
 P U @ @2 x --- --x--- = -@x-+ @x2--2- + ::: \relax \special {t4ht=
In other words, the upwind discretisation represents eqn but also the second derivative eqn (and higher derivatives). eqn is equivalent to a Laplacian, described in Sec. 2.14 , which diffuses eqn with a diffusivity proportional to eqn.

PICT\relax \special {t4ht=

The upwind scheme is particularly diffusive when the flow direction is not aligned with the cells of a mesh. In the 2D box of cells above, eqn is advected at a eqn angle, beginning with an abrupt step change from eqn = 1 and eqn = 0 between the left and lower boundaries. The step rapidly diffuses along the direction of travel as shown in graph (right) and shaded area (left).

Notes on CFD: General Principles - 3.10 Upwind scheme