Solution of the system of equations
A Newton-Raphson iteration procedure was used to solve the system of equa-
tions given above. If an equation is designated as
(
)
F = F dj , ηj , uj , υj , dj+1 , ηj+1 , uj+1 , υj+1
(130)
then for any two points `j' and `j + 1' at any time, it can be expanded as a Taylor
Series
F
F
F
F
m+1
F=mF +
∆ηj+1 +
∆ηj +
∆dj +
∆dj+1 +
ηj+1
ηj
dj
dj+1
m
m
m
m
(131)
F
F
F
F
∆υj+1
∆υj +
∆uj +
∆uj+1 +
υj+1
υj
uj
uj+1
m
m
m
m
where `m' and `m + 1' indicate the iteration level and `j' and `j + 1' indicate the x-
location. The partial derivatives are evaluated on the basis of the values of the vari-
ables after the `mth' iteration. Therefore, each equation can be transformed into a
linear equation in terms of the eight unknowns: ∆dj, ∆ηj, ∆uj, ∆υj, ∆dj+1, ∆ηj+1, ∆uj+1,
and ∆υj+1. The goal of the method is to solve for the change between `m' and `m + 1'
such that m + 1F ⇒ 0. It can also be seen that mF is a function of the final values of the
variables at time `n' and the values for the `mth' iteration at time `n + 1,' which are
all known values. The transformed equations are then put into the form
K =m+1F-mF
(132)
= a∆dj + b∆ηj + c∆uj + d∆υj + e∆dj+1 + f∆ηj+1 + g∆uj+1 + h∆υj+1
where
F
F
F
F
a=
b=
c=
d=
dj ,
ηj ,
uj ,
υj ,
F
F
F
F
e=
f=
g=
h=
,
ηj+1 ,
uj+1 ,
υj+1 .
dj+1
Since K =m+1F-mF , and the goal is to achieve
m+1
F⇒0
K =-mF .
(133)
For example, the conservation of water mass equation becomes
(d
) - (d
) + u θ(d
) + (1 - θ)(d
) +
n+1
n+1
n+1
n+1
n
n
n
n
j+1 + dj
j+1 - dj
j+1 + dj
j+1 - dj
F=
2∆t
∆x
(
)
(
) = 0
θ un+1 - un+1 + (1 - θ) un - un
(134)
j+1
j+1
d
j
j
∆x
and the derivatives are evaluated as
47
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