A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

4. Finite-difference Formulation 73
The solution scheme involves iteration; the symbol (~) in equation 4.8 denotes an approximate solution at time t = (n + 1)Δt taken
n + 1 n n + 1 n
from the preceding iteration. For the first (starting) iteration, it is assumed that Ũ i + 1⁄ = U
2 i + 1⁄ , ũ
2 i + 1⁄ = u
2 i + 1⁄ , and
2
H˜ n + 1 n
1
i + 1⁄ = H
2 i + 1⁄ ; in this way, the variables superscripted with n + ⁄ 2 are always known. One should use caution if this scheme
2
n + 1⁄ 2
is used with only one iteration each time step; then the scheme is only first-order accurate because the coefficients H i – 1⁄ and
2
n + 1
γ ⁄ 2
n + 1⁄ 2 n + 1⁄ 2
i – 1⁄ in equation 4.7 and the advection term ( uU )i – ( uU ) 2
i – 1 ) ⁄ Δxin
equation 4.4 are evaluated backward-in-time at the nΔt
1
time level instead of being centered at the ( n + ⁄ 2)Δt
time level. In Chapter 5 it will be demonstrated that a significant loss of
accuracy can result if the scheme is used with only one iteration.
The weighting coefficient χ was suggested by Cunge and others (1980) to improve accuracy in evaluating the friction term
when a rapid variation and reversal of the discharge can occur; in the first iteration, χ i – 1 is equal to 1.0, and in each subsequent
⁄ 2
iteration, it is defined by
χ i – 1⁄ 2
n + 1⁄ 2 n + 1⁄ 2 n n
Ui – 1 Ui ⁄ 1
2 – ⁄ – U 2 i – 1⁄ U 2 i – 1⁄ 2
----------------------------------------------------------------n
n + 1 n
U i – 1⁄ ( U 2 i – 1⁄ – U 2 i – 1⁄ )
2
= . (4.9)
Because the water surface elevation ζi and the volumetric transport U i – are computed at different nodal points, the values
1
⁄ 2
for (uU) i, (uU) i − 1 and Hi – 1⁄ in equations 4.5 and 4.7 must be obtained by some form of interpolation. The usual procedure is to
2
calculate the average of the two adjacent values on the grid. For improved accuracy here, a center-weighted average of the four
adjacent values is used. Thus, the total depth H at the point i
Hi + 1⁄ 2
+
1
⁄ 2 is
9
1
= ----- ( H
16 i + 1 + H ) – ----- ( i H
16 i + + H ) , (4.10)
2 i – 1
which applies at all points except those immediately adjacent to the boundary, where a two-point average is used.
form
Before actually solving the implicit stage of the two-level scheme, the momentum equation (eq. 4.7) is first rewritten in the
+
n 1
U i – 1⁄ 2
n + 1⁄ 2 g
n + 1⁄ 2 n + 1 n + 1 n n
n
Ri – 1 Ûi 1
⁄ 2 – ⁄ – -- ( Δt ⁄ Δx)
H 2
i 1
2
– ⁄ ( ζi – ζ
2
i – 1 + ζi – ζi
– 1)
gΔx 1 χ + 1⁄ n + 1⁄ 2 2 n n
= ⎛
– ( – i – 1⁄ )Hγi Ui
2 i – 1⁄ 1 1 Ui 1
2 – ⁄ – 2 ⁄ 2 – ⁄ )
⎝
2 (4.11)
–
n + 1⁄ 2
n
where R i – 1⁄ 1 gΔt χ + 1⁄ 2 n
+ i – 1⁄ H γ + 1⁄ 2
n
(
2
2 i – 1⁄ 2 i – 1⁄ Ui 1
2 – ⁄ ) 2
1
n 1
= . After substituting equation and a similar expression for U i + 1⁄ 2
n + 1
n + 1
continuity equation (eq. 4.6), U i + 1⁄ and Ui 1
2 – ⁄ are eliminated, and an equation of the following form is obtained:
2
+ into the
n + 1 n + 1 n + 1
Ai ζ i – 1 + Bi ζ i + Ci ζ i + 1 = Di , (4.12)