Atmospheric Dispersion with a Large-Eddy Simulation
247
The spline derivatives are obtained by solving the tri-diagonal algebraic
system (Ahlberg et al., 1967; Price and Mac Pherson, 1973):
1
1
1
1
1
1
3
3
2
(
)
(
)
2
2
2
2
j
h
h
h
h
h
h
h
j
j
j
j
j
j
P
P
P
C C
C
C
x
x
−
+
−
+
+
+
=
−
+
−
Δ
Δ
(9.21)
Using Equations 9.20 through 9.21, Equations 9.18 and 9.19 become,
respectively
1
2
2
, ,
1
1, ,
, ,
, ,
3
1
1, ,
, ,
2
3 (
)
2(
)
h
h
h
h
h
h
h
j k m
j j
j
j
j j
j k m
j k m
j k m
h
h
h
h
j
j
j j
j km
jkm
C
C
P h
P h
P h
C
C
P h P h
C
C
+
−
−
−
−
⎡
⎤
=
−
α+
+
+
−
α
⎣
⎦
⎡
⎤
−
+
+
−
α
⎣
⎦
(9.22a)
1
2
2
, ,
1
1
1
1
, ,
1, ,
, ,
3
1
1
1
,,
1 ,,
2
3 (
)
2(
)
h
h
h
h
h
h
h
j k m
j j
j
j
j j
j k m
j k m
j k m
h
h
h
h
j j
j
j
j k m
j k m
C
C
P h
P h
P h
C
C
P h
P h
C
C
+
+
+
+
+
+
+
+
+
+
⎡
⎤
=
+
α−
+
+
−
α
⎣
⎦
⎡
⎤
+
+
+
−
α
⎣
⎦
(9.22b)
(b) After each advective step, a fi lter operation is applied to the intermediate fi eld to remove any negative concentration that is usually produced by
advection:
1
1
2
2
, ,
, ,
h
h
j k m
j k m
C
F C
+
+
⎛
⎞
= ⎜
⎟
⎝
⎠
(c) Finally, the diffusive step (operator D x ) is computed by means of the Crank–
Nicholson implicit scheme:
+
+
+
+
+
+
+
−
+
+
+
+
+
−
⎡
⎤
⎛
⎞
⎛
⎞
−
−
−
′
′ ′
⎢
⎥
⎜
⎟
⎜
⎟
−
⎝
⎠
⎝
⎠
⎢
⎥
=
⎢
⎥
Δ
Δ
⎢
⎥
⎢
⎥
⎣
⎦
⎡
⎤
−
−
−
′
′ ′
⎢
⎥
+
Δ
⎢
⎥
⎣
⎦
1
1
1
1
2
2
2
2
1
1
1, ,
, ,
, ,
1, ,
2
, ,
, ,
2
1
1
1
1
1, ,
, ,
, ,
1, ,
2
1
2
( )
(
)
(
)
1
2
( )
h
h
h
h
h
M
M
h
j km
jkm
jkm
j km
j k m
j k m
c
h
h
h
h
j k m
j k m
M
j k m
M
j k m
c
K C
C
K C
C
C
C
t
S
x
K C
C
K C
C
S
x
(9.23)
where
1/ 2
1/ 2
(
)
(
)
M
M
j
M
M
j
K
K x
K
K x
+
−
′ =
′′ =
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