246
Air Pollution and Turbulence: Modeling and Applications
In the next paragraph, we consider a method based on cubic-spline interpolations
for the advective terms (operators A i ), which are usually the most diffi cult to implement, and a Crank–Nicholson implicit scheme for the diffusive terms (operators D i ).
9.3.3 DETAILS OF THE LOD METHOD
The fi nite difference algorithm for Equation 9.17a, or its reverse Equation 9.17b, contains three steps, one for each direction. In the following, we only show the numerical scheme for the x direction, as the same scheme is used for the other directions
with the appropriate boundary conditions.
We use the notation j ∈ [1, N x ], k ∈ [1, N y ], m ∈ [1, N z ] for increments in the (x, y, z)
Cartesian space, so we have
0
0
0
j
k
m
x
x j x
y
y k y
z
z m z
= + Δ
= + Δ
= + Δ
where Δx, Δy, and Δz are the grid sizes and (N x , N y , N z ) are the number of grid points
in the x-, y-, and z-directions, respectively. For each direction, the scheme (Equations
9.17a and 9.17b) contains three substeps: (a) the advective part, (b) the fi ltering procedure, and (c) the diffusive part.
(a) The advection is computed using a quasi-Lagrangian cubic-spline method
(Long and Pepper, 1981; Pielke, 1984), so for operator A x , we have
+
=
−αΔ
≥
1
2
, ,
, ,
(
) , i f
0
h
h
h
j
j k m
j k m
C
S x
x
u
(9.18)
+
=
+αΔ
<
1
2
, ,
, ,
(
) , i f
0
h
h
h
j
j k m
j k m
C
S x
x
u
(9.19)
with
, ,
h
j k m
t
u
x
Δ
α =
Δ
, where the superscript h denotes an intermediate fi ctitious time step between n and n+1. This is called the fractional steps (FS)
technique.
The interpolation function (cubic spline) S can be expressed in terms of
the spline derivatives
, ,
h
h
j
j k m
C
P
x
⎛
⎞
∂
= ⎜
⎟
∂
⎝
⎠
as
2
2
1
1
1
2
2
2
2
1
1
1
3
3
(
) (
)
(
) (
)
( )
(
) 2(
)
(
) 2(
)
j
j
j
j
h
h
h
j
j
j
j
j
j
j
j
j
j
h
h
j
j
j
j
x x x x
x x
x x
S x
P
P
h
h
x x
x x
h
x x
x x
h
C
C
h
h
−
−
−
−
−
−
−
−
−
−
=
−
⎡
⎤
⎡
⎤
−
−
+
−
− +
⎣
⎦
⎣
⎦
+
+
(9.20)
where h j = x j − x j−1 .
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
In the next paragraph, we consider a method based on cubic-spline interpolations
for the advective terms (operators A i ), which are usually the most diffi cult to implement, and a Crank–Nicholson implicit scheme for the diffusive terms (operators D i ).
9.3.3 DETAILS OF THE LOD METHOD
The fi nite difference algorithm for Equation 9.17a, or its reverse Equation 9.17b, contains three steps, one for each direction. In the following, we only show the numerical scheme for the x direction, as the same scheme is used for the other directions
with the appropriate boundary conditions.
We use the notation j ∈ [1, N x ], k ∈ [1, N y ], m ∈ [1, N z ] for increments in the (x, y, z)
Cartesian space, so we have
0
0
0
j
k
m
x
x j x
y
y k y
z
z m z
= + Δ
= + Δ
= + Δ
where Δx, Δy, and Δz are the grid sizes and (N x , N y , N z ) are the number of grid points
in the x-, y-, and z-directions, respectively. For each direction, the scheme (Equations
9.17a and 9.17b) contains three substeps: (a) the advective part, (b) the fi ltering procedure, and (c) the diffusive part.
(a) The advection is computed using a quasi-Lagrangian cubic-spline method
(Long and Pepper, 1981; Pielke, 1984), so for operator A x , we have
+
=
−αΔ
≥
1
2
, ,
, ,
(
) , i f
0
h
h
h
j
j k m
j k m
C
S x
x
u
(9.18)
+
=
+αΔ
<
1
2
, ,
, ,
(
) , i f
0
h
h
h
j
j k m
j k m
C
S x
x
u
(9.19)
with
, ,
h
j k m
t
u
x
Δ
α =
Δ
, where the superscript h denotes an intermediate fi ctitious time step between n and n+1. This is called the fractional steps (FS)
technique.
The interpolation function (cubic spline) S can be expressed in terms of
the spline derivatives
, ,
h
h
j
j k m
C
P
x
⎛
⎞
∂
= ⎜
⎟
∂
⎝
⎠
as
2
2
1
1
1
2
2
2
2
1
1
1
3
3
(
) (
)
(
) (
)
( )
(
) 2(
)
(
) 2(
)
j
j
j
j
h
h
h
j
j
j
j
j
j
j
j
j
j
h
h
j
j
j
j
x x x x
x x
x x
S x
P
P
h
h
x x
x x
h
x x
x x
h
C
C
h
h
−
−
−
−
−
−
−
−
−
−
=
−
⎡
⎤
⎡
⎤
−
−
+
−
− +
⎣
⎦
⎣
⎦
+
+
(9.20)
where h j = x j − x j−1 .
© 2010 by Taylor and Francis Group, LLC