It is probably worth mentioning that the scheme (I.6) is the same as the one derived by standard FD techhiques. This is because the mesh we are dealing with was regular and uniform. However this is not always the case. If the river changed geometry rather than being of a uniform cross-section throughout, the cells would have been non-uniform in shape. In such a case, working out the corrections to the FD terms would be very difficult. However the FV approach makes it much easier to evaluate the various terms for irregular shaped cells. For instance the above scheme could be easilly adapted by making A, the face area, variable. Of course for this case u would also change.
This can be extended to 3d as well. The 3d transport equation can be written in vector notation as
q over all the faces of the cell. This
result is called Gauss’ theorem in vector calculus.