## 3.10Upwind scheme

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

In the graphic above, we track the translation of a proﬁle of by equating changes in in time to the local gradient . If we apply upwind to calculate the change at point P, the gradient and no change in is correctly calculated (left).

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

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

### 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:

 (3.9)
In our 1D example, the upwind scheme calculates using and at locations U and P, separated by distance . Relating the upwind calculation to Eq. (3.9 ) gives
 (3.10)
In other words, the upwind discretisation represents but also the second derivative (and higher derivatives). is equivalent to a Laplacian, described in Sec. 2.14 , which diﬀuses with a diﬀusivity proportional to .

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, is advected at a angle, beginning with an abrupt step change from  = 1 and  = 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