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 :
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 .
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).