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

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Figure 5.6. Graph showing solution using the two-level scheme on the verge of going
unstable when using only one iteration............................................................................103
Figure 5.7. Graphs showing error measures using the two-level semi-implicit scheme on the
hydrograph test problem.....................................................................................................105
Figure 5.8. Graph showing solution using two iterations with the two-level scheme
that is stabilized, and greatly improved, with an additional iteration ..........................107
Figure 5.9. Graphs showing error measures using the three-level semi-implicit scheme
on the hydrograph test problem for different values of the computational
space interval........................................................................................................................108
Figure 5.10. Graphs showing error measures using the three-level semi-implicit scheme
on the hydrograph test problem for different values of the computational
time interval ...........................................................................................................................110
Figure 5.11. Graph showing comparison of computed hydrographs with the leapfrog
semi-implicit scheme and the two-level semi-implicit scheme ...................................112
Figure 5.12. Graph showing comparison of solutions to the hydrograph problem with the
leapfrog semi-implicit scheme and the four-point model of Schaffranek and
others (1981) using a space step of 5,000 feet and a time step of 5 minutes
for all simulations .................................................................................................................114
Figure 5.13. Channel characteristics and initial flow conditions for the 1- dimensional
tidal test problem..................................................................................................................115
Figure 5.14. Graph showing water surface elevations at six locations along the
tidal channel..........................................................................................................................116
Figure 5.15. Graph showing solution of the tidal problem for water surface elevation and
velocity at the location x = 180,000 feet ............................................................................117
Figure 5.16. Graphs showing solutions for water surface elevation and velocity computed
with the two-level semi-implicit scheme using a space step of 6,000 feet
and a time step of (A) 60 minutes and (B) 30 minutes ....................................................118
Figure 5.17. Graph showing solution for volumetric transport using the two-level
semi-implicit scheme...........................................................................................................119
Figure 5.18. Graphs showing solutions using one iteration of the two-level scheme with
(A) advection terms turned on and (B) advection terms turned off .............................122
Figure 5.19. Graphs showing solutions using the three-level semi-implicit, leapfrog scheme
(no trapezoidal step) (A) with no smoothing and (B) with smoothing..........................124
Figure 5.20. Graphs showing solutions using the three-level semi-implicit scheme for the
tidal test problem..................................................................................................................125
Figure 5.21. Graphs showing comparison of error measures between the two-level scheme
with two iterations and the three-level leapfrog-trapezoidal scheme with one
iteration of the trapezoidal step .........................................................................................128
Figure 5.22. Diagram of (A) the test problem for an oscillating wave (seiche) in a rectangular
basin and (B) the horizontal computational grid for the numerical solution ..............130
Figure 5.23. Graphs showing computed and analytical solutions for the 3-dimensional (3-D)
seiching test problem assuming an ideal (non-viscous) fluid and using
the 3-D, explicit, leapfrog scheme for the numerical solution......................................132
Figure 5.24. Graphs showing computed and analytical solutions for the 3-dimensional (3-D)
seiching test problem assuming an ideal (non-viscous) fluid and using
the 3-D, semi-implicit, leapfrog scheme for the numerical solution............................133
Figure 5.25. Graphs showing 20-hour time history of computed and analytical solutions
for the seiching of an ideal fluid.........................................................................................134
Figure 5.26 Graphs showing mid-basin solutions for the 3-dimensional seiching problem
using an ideal fluid and three different time steps for the semi-implicit
leapfrog scheme...................................................................................................................135
Figure 5.27. Graphs showing mid-basin solutions for the 3-dimensional seiching problem
using an ideal fluid and three different time steps for the semi-implicit
leapfrog-trapezoidal scheme .............................................................................................136
Figure 5.28. Sample velocity distribution after 0.6 hour of simulation for the 3-dimensional
seiching problem using a viscous fluid.............................................................................138