5.6 Movement of Tracers
105
In all schemes, the firs term on the right-hand side of the latter equation can be
formulated as:
− Δt
∂(uB )
∂x
= C w B w − C e B e
(5.22)
where the indices “w” and “e” refer to east and west faces of the control volume
and:
C w = u
n
k−1 Δt/Δx and C e = u
n
k Δt/Δx
(5.23)
are so-called Courant numbers. In a next step, we can split the u component into
positive and negative components:
u
+
= 0.5(u + |u|) and u
−
= 0.5(u − |u|)
(5.24)
and rewrite (5.21) in the form:
− Δt
∂(uB )
∂x
= C
+
w B
+
w + C
−
w B
−
w − C
+
e B
+
e − C
−
e B
−
e
(5.25)
The objective of any finite-di ference Eulerian advection scheme is to interpolate
the volume-averaged values of B to obtain the effective face values B e and B w .
Here we use so-called Total Variation Diminishing schemes or TVD schemes that
are based on the requirement:
k
B
n+1
k+1 − B
n+1
k
≤
k
B
n
k+1 − B
n
k
(5.26)
For the TVD schemes used here, described by Fringer et al. (2005), the face
values of B are computed with the upwind values plus the addition of a higher order
term with:
B
+
e = B
n
k + 0.5Ψ
r
+
k
1 − C
+
e
B
n
k+1 − B
n
k
B
−
e = B
n
k+1 − 0.5Ψ
r
−
k
1 + C
−
e
B
n
k+1 − B
n
k
B
+
w = B
n
k−1 + 0.5Ψ
r
+
k−1
1 − C
+
w
B
n
k − B
n
k−1
B
−
w = B
n
k − 0.5Ψ
r
−
k−1
1 + C
−
w
B
n
k − B
n
k−1
where the r parameters are given by:
r
+
k =
B
n
k − B
n
k−1
B
n
k+1 − B
n
k
and r
−
k =
B
n
k+2 − B
n
k+1
B
n
k+1 − B
n
k
The limiting function Ψ define the particular scheme that is used. A few selected
options are given in the following.