In Sec. 2.8 , we described advection terms as those of the form or . The inclusion of velocity within the divergence gives the term particular characteristics that require special treatment in discretisation.
Following the ﬁnite volume principles outlined in Sec. 3.1, the discretisation approximates the surface integral by a summation over faces by
The discretisation requires calculation of the volumetric ﬂux (see Sec. 2.3 ). In the case of included in the advection term , is the mass ﬂux .
In the ﬂux calculation, the interpolation of at cell centres to at faces uses the linear scheme of Sec. 3.7 . Similarly, the linear scheme is used for the interpolation .
The advected property is transported in the direction of ﬂow velocity . Interpolation to the face of usually involves the ﬂow direction. In the graphic below a face f is positioned between two cells. Based on the ﬂow direction, the cells are labelled upwind ‘U’ and downwind ‘D’.
The linear interpolation scheme, 5 described in Sec. 3.7 , does not use the ﬂow direction but expresses in terms of adjacent cells. At ﬁrst sight, this choice of scheme is logical when considering accuracy. However, for advection, the linear scheme tends to generate unbounded solutions which are unstable.
The upwind scheme simply represents by the value in the upwind cell . It makes sense from a physical perspective since particles of ﬂuid in the upwind cell are destined to travel to the face, transporting property with them.6
While the linear scheme is generally unbounded, the upwind scheme exhibits poor accuracy. In following section we explore the behaviour of upwind before looking at schemes that oﬀer greater accuracy while attempting to maintain boundedness.