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

144 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
for both the two-level and three-level schemes can be extended by using iteration, although the number of iterations should be kept
to a minimum because iteration in three dimensions is expensive. Using more than one iteration of the trapezoidal step in the three-
level scheme or more than two iterations in the two-level scheme was not worthwhile for the 1-D test experiments, considering the
extra computational expense incurred for minimal improvement in accuracy; in the few instances when extra iterations were
needed to stabilize a particular solution, the values for the time step and Courant numbers were too large anyway to retain needed
accuracy.
Although more testing of the 3-D, semi-implicit scheme is necessary, the results from the seiching experiment look promising.
Accurate results are obtained for time steps that exceed the CFL condition for the gravity waves. Mass preservation is excellent.
Because of the implicit nature of the numerical scheme, some phase errors do appear in solutions having large time steps. However,
it is unlikely that this source of error will be prevalent in real tidal simulations of estuaries. If it is, it can at least be minimized by
the use of the trapezoidal step in the scheme.
7. References
Abbott, M.B., and Ionescu, F., 1967, On the numerical computation of nearly-horizontal flows: Journal of Hydraulic Research,
v. 5, no. 2, p. 97−117.
Abraham, Gerrit, 1988, Turbulence and mixing in stratified tidal flows, in Dronkers, J.J., and van Leussen, Wim, eds., Physical
processes in estuaries: Berlin, Germany, Springer, p. 149−180.
Adams, C.E., and Weatherly, G.L., 1981, Suspended-sediment transport and benthic boundary-layer dynamics: Marine Geology,
v. 42, p. 1−18.
Amein, M.M., and Fang, C.S., 1970, Implicit flood routing in natural channels: American Society of Civil Engineers, Journal of
the Hydraulics Division, v. 96, no. 12, p. 2481−2500.
American Society of Civil Engineers, 1988, Turbulence modeling of surface water flow and transport—Part I, Prepared by the Task
Committee on Turbulence Models in Hydraulic Computations: American Society of Civil Engineers, Journal of Hydraulic
Engineering, v. 114, no. 9, p. 970−991.
Ames, W.F., 1977, Numerical methods for partial differential equations (2nd ed.): New York, Academic Press, 365 p.
Asselin, R., 1972, Frequency filters for time integrations: American Meteorological Society, Monthly Weather Review, v. 100, p.
487−490.
Axelsson, O., and Lindskog, G., 1986, On the eigenvalue distribution of a class of preconditioning methods: Numerical
Mathematics, v. 48, p. 479−498.
Backhaus, J.O., 1983, A semi-implicit scheme for the shallow water equations for application to shelf sea modeling: Continental
Shelf Research, v. 2, no. 4, p. 243−254.
Backhaus, J.O., 1985, A three-dimensional model for the simulation of shelf sea dynamics: Deutsche Hydrographische Zeitschrift
[German Hydrographic Newspaper], v. 38, p. 165−187.
Bedford, K.W., Dingman, J.S., and Yeo, W.K., 1987, Preparation of estuary and marine model equations by generalized filtering
methods, in Nihoul, J.C.J., and Jamart, B.M., eds., Three-dimensional models of marine and estuarine dynamics: Amsterdam,
Netherlands, Elsevier, p. 113−125.
Beer, F.P., and Johnston, E.R., 1962, Vector mechanics for engineers—statics and dynamics: New York, McGraw-Hill, 781 p.