3.3 Finite volume mesh
The finite volume numerical method is closely associated with the concept of surface and volume integrals used in Chapter 2 . Below, we extract the main geometric elements from the figures used for the conservation laws.
The integrals use volumes 
and area vectors
.
The numerical method uses equivalent discrete quantities for cells
and faces:
;
,
;
.
The finite volume method relates discrete values
of fields, e.g. pressure, to
cells and faces within the mesh. For many calculations data must to
be assigned to point locations, in particular the cell centre (more
specifically centroid)
and face centre 
.
Calculating mesh data
To calculate 
, each polygonal face is
decomposed into triangles using an apex point 
. The area vector
and
centre 
are then calculated for each triangle according to
” is the cross product, Eq. (2.70
). The sum of area vectors
gives 
and 
is the area weighted sum of triangle centres over
triangles, calculated by
 |
can be
calculated from 
and 
by Gauss’s theorem, using 
to describe position
and noting 
. The surface integral (
) becomes a sum over
cell faces 
, replacing discrete values 
and 
for 
and 
respectively as
follows:
 |
, by
