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

5. Numerical Experiments 103
The iteration process has a definite effect on the stability of the two-level scheme. The solutions attempted using a single iter-
ation (with Δx = 5 minutes) remained stable and free of significant oscillations only for values of Δx greater than or equal to
1,000 ft (305 m). The solution using Δx = 1,000 ft (305 m) corresponds to a maximum gravity-wave Courant number of 3.8 and
an advection Courant number of 0.68 (table 5.2). For smaller values of Δx and increasing Courant numbers, the solutions using
one iteration exhibited either severe oscillations or complete instability. Figure 5.2 shows the solution using a single iteration with
Δx = 500 ft (152 m) and Δt = 5 minutes. The solution has severe oscillations and is on the verge of going completely unstable
at the termination of the 500 minute simulation. The addition of a second iteration stabilizes the solution and removes most of
the error. For the space-centered differencing of the advection term used here, Casulli and Cheng (1990) found, using the
von Neumann analysis, the requirement that the flow be subcritical to ensure linear stability of the two-level scheme without
implementing iterations. Because the flow in the test problem remains subcritical throughout the simulation, the instabilities that
occur for high Courant numbers are a form of nonlinear instability. No instabilities were encountered in any of the solutions
attempted using two or five iterations for Courant numbers as high as Cr g = 37.5 and Cr a = 6.8.
DISCHARGE, IN CUBIC FEET PER SECOND
800
700
600
500
400
300
200
100
Upstream
boundary
50,000 feet
Base solution
Δ t = 5 minutes, Δ x = 500 feet, niter = 1
Δ t = 5 minutes, Δ x = 500 feet, niter = 2
RMS error
niter = 1 7.4 percent
niter = 2 0.2 percent
0 100 200 300 400 500
TIME, IN MINUTES
Figure 5.6. Solution using the two-level scheme on the verge of going unstable when using only one iteration (niter = 1).
A second iteration (niter = 2) stabilizes the solution and removes most of the error. The solution using two iterations is almost
indistinguishable from the base solution. Δx is the size of the computational space interval. Δt is the size of the
computational time interval. RMS, root mean square