(
)
(
)
+ (1  θ) ujn 1  ujn
n+1
n+1
u θ θ θ uj+1  uj
+
F
1
a=
=

+
dj 2∆t ∆x 2
(135)
∆x
F
b=
=0
(136)
ηj
(
)
(
)  d θ
+ (1  θ) djn 1  djn
n+1
n+1
F θ θ dj+1  dj
+
c=
=
uj 2
(137)
∆x
∆x
F
d=
=0
(138)
υj
(
)
(
)
+ (1  θ) ujn 1  ujn
n+1
n+1
u θ θ θ uj+1  uj
+
F
1
e=
=
+
+
2∆t ∆x 2
(139)
∆x
dj+1
F
f=
=0
(140)
ηj+1
(
)
(
) + d θ
+ (1  θ) djn 1  djn
n+1
n+1
θ θ dj+1  dj
+
F
g=
=
uj+1 2
(141)
∆x
∆x
F
h=
= 0.
(142)
υj
The same procedure was followed for the other three equations, using the nota
tions F′, F′′, F′′′, a′, a′′, a′′′, ..., K ′, K ′′, K ′′′ to distinguish among equations. The
discretized equations and their derivatives are listed in Appendix A.
For each point, there are now four unknowns (∆d, ∆η, ∆u, and ∆υ) or a total of
4N unknowns, where N is the number of cross sections. For each reach there are
four linear equations or 4(N 1) equations. Four boundary condition equations are
needed to close the system. The linear algebraic system can be represented as
[A]{∆Z} = {K} or {∆Z} = [A]1{K}
(143)
where
{∆Z} = {∆d1 , ∆η1 , ∆u1 , ∆υ1 , ∆d2 , ∆η2 , ..., ∆dN , ∆ηN , ∆uN , ∆υN }T
(144)
and
{K} = {K1 , K1 , K1′, K1′,′ K2 , K2 , K2′, K2′′, ..., KN , KN , KN , KN′}T .
′′ ′
′′ ′
′
′′ ′′
(145)
The coefficient matrix [A] is depicted in Figure 31. [A] is a banded coefficient
matrix of 11 diagonals with the x's signifying boundary conditions and the empty
spaces filled with zeros. The system is solved using a decompositionback
substitution scheme to obtain the ∆Z terms. Subsequently, for the next iteration
48