Geoscience Reference
In-Depth Information
Figure 4.14
1-D finite volume grid.
Integrating Eq. (4.103) over the control volume centered at point
P
shown in
Fig. 4.14 yields
d
dx
d
dx
(ρ
u
φ)
−
(ρ
u
φ)
=
e
−
(4.104)
e
w
w
To complete the discretization, the convection flux
ρ
u
φ
and diffusion flux
d
φ/
dx
at faces
w
and
e
are determined using the schemes described below.
Central scheme
The central scheme adopts a piecewise linear profie for
φ
, as shown in Fig. 4.15. Thus,
the values of
φ
at cell faces are given as the average of two neighboring nodal values:
1
2
(φ
1
2
(φ
φ
=
+
φ
)
,
φ
=
+
φ
)
(4.105)
w
P
W
e
E
P
and the diffusion fluxes are determined by
d
dx
w
=
(φ
−
φ
)
d
dx
e
=
(φ
−
φ
)
w
P
W
e
E
P
,
,
(4.106)
x
w
x
e
where
x
w
and
x
e
are the distances from
W
to
P
and from
P
to
E
, respectively.
Figure 4.15
Piecewise linear profie in central scheme.
