where xj ≤ x ≤ xj+1. Similarly, for the time interval tn ≤ t ≤ tn +1
()
f xj , t = θfjn+1 + (1 - θ) fjn
(117a)
and
(
)
f xj+1 , t = θfjn+1 + (1 - θ) fjn1
(117b)
+1
+
where 0 ≤ θ ≤ 1 and 0 ≤ Ψ ≤ 1 are weighting factors. Subscripts denote the x-
location while superscripts denote time level. By use of these relations, the finite
difference equations that approximate the conservation laws expressed by the dif-
ferential eq 112, 113, 114, and 115 can be developed. When Ψ = 0.5, the resulting
equations represent the Preissmann four-point scheme of finite differences. In the
discretization, the derivatives of a function f(x,t) are estimated as
(
)(
)
fjn+1 + fjn+1 - fjn1 + fjn
+1
+
f
=
(118a)
2∆t
t
(
)
(
)
and
+ (1 - θ) fjn1 - fjn
n+1
n+1
f θ fj+1 - fj
+
=
(118b)
∆x
x
and the values of the variables, or the functions themselves, are estimated as
(
)
(
).
θ fjn+1 + fjn+1 + (1 - θ) fjn + fjn1
+1
+
f (x, t) =
(119)
2
The above rules of discretization can be used to recast the conservation of water
mass equation as
(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
2∆t
∆x
(
)
(
) = 0
θ un+1 - un+1 + (1 - θ) un - un
(120)
j+1
j+1
+d
j
j
∆x
where
(
)
(
)
θ ujn+1 + ujn+1 + (1 - θ) ujn 1 + ujn
+1
+
u=
(121)
2
and
(
)
(
).
θ djn+1 + djn+1 + (1 - θ) djn 1 + djn
+1
+
d=
(122)
2
Similarly, the conservation of ice mass equation becomes
(η
) - (η
) + υ θ (η
) + (1 - θ )(η
)
n+1
n+1
n+1
n+1
n
n
n
n
j+1 - ηj
j+1 + ηj
j+1 + ηj
j+1 - ηj
i
i
2∆t
∆x
(
)
(
) = 0
θ υn+1 - υn+1 + (1 - θ ) υn - υn
(123)
i j+1
j+1
+ η
j
i
j
∆x
45
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