3.5 Numerical Method
91
a 4 =
φ i, j,k − φ i−1, j,k
x
(3.80)
a 5 =
φ i−1, j,k − φ i−2, j,k
x
(3.81)
Three variables are then defined to self-adaptively adjust the numerical dissipation
of the scheme.
S 1 =
13
12
(a 1 − 2a 2 + a 3 )
2
+
1
4
(a 1 − 4a 2 + 3a 3 )
2
(3.82)
S 2 =
13
12
(a 2 − 2a 3 + a 4 )
2
+
1
4
(a 2 − a 4 )
2
(3.83)
S 3 =
13
12
(a 3 − 2a 4 + a 5 )
2
+
1
4
(3a 4 − 4a 4 + a 5 )
2
(3.84)
Let β be a relatively small constant; in this study, its value is 10
−3
x. Finally,
three weight factors are defined respectively as
λ 1 =
1
10
1
(S 1 + β) 2
(3.85)
λ 2 =
6
10
1
(S 2 + β) 2
(3.86)
λ 3 =
3
10
1
(S 3 + β) 2
(3.87)
Let the weight variable be
w i =
λ i
λ 1 + λ 2 + λ 3
(3.88)
Then according to different upwind directions, the convective item
∂φ
∂ x
can be
calculated using the following equation
∂φ
∂ x
= w
1
1
3
a 1 −
7
6
a 2 +
11
6
a 3
+ w
1 −
1
6
a 2 +
5
6
a 3 +
1
3
a 4
+ w 3
1
3
a 3 +
5
6
a 4 −
1
6
a 5
(3.89)
Similarly,
∂φ
∂ y
and
∂φ
∂z
can also be calculated according to the above mentioned
high-precision WENO scheme.