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

VOLUMETRIC TRANSPORT, IN SQUARE FEET PER SECOND
100
50
-50
5. Numerical Experiments 119
Considering that time steps of 60 minutes and 30 minutes are quite large relative to the 12.4 hour period of the tidal forcing,
the solutions in figure 5.16 are reasonably good. Simulations using these two time steps were made using the two-level scheme,
trying both larger and smaller values of Δx, and the results generally were only slightly changed. Because of the long tidal wave-
length, Δx, up to the largest value of Δx = 30,000 ft (9,144 m), had relatively little effect on the solutions.
The two-level scheme was tested further by keeping Δx constant at 6,000 ft (1,828 m) and varying Δt over a wide range. The
values of Δt and the corresponding Courant numbers are given in table 5.6. The test simulations revealed that only for the smallest
choice, Δt = 1 minute, is a stable solution obtained when the two-level scheme is used with one iteration (niter = 1); the 1 minute
solution had very little error (RMS e = 0.05 percent). Each simulation using niter = 1 and Δt ≥ 5 minutes ran for only several tidal
cycles before oscillations developed that eventually grew to destroy the solutions entirely. The oscillations could be traced back to
the nonlinear advection term in the momentum equation and appeared first in the solutions at the times just prior to the high and
low water points in the tidal cycle. As a consequence of the standing wave character of the solution, at the times just before high
and low water, the volumetric transport and water surface elevation are changing in opposite directions and the magnitude of the
advection term is oscillating near zero.
0
-100
base solution
Δt = 60 minutes, Δx = 6,000 feet, niter = 2
11th M cycle 2
x = 180,000 feet
RMS error = 8.7 percent
124 126 128 130 132 134 136
TIME, IN HOURS
Figure 5.17. Solution for volumetric transport (U = uh) using the two-level semi-implicit scheme.
The solution for time step Δt = 60 minutes and space step Δx = 6,000 feet is compared with the base solution.
RMS, root mean square.