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

5. Numerical Experiments 113
two-level scheme. In the simulations for figure 5.9, the basic leapfrog scheme yielded stable solutions using a Δt of 5 minutes only
for spatial step sizes that are greater than or equal to 1,000 ft (305 m). In figure 5.10, the leapfrog solutions were stable only up to
a time step of 5 minutes for the space step of 2, 000 ft (610 m). The leapfrog-trapezoidal solutions in figure 5.10 with one iteration
of the trapezoidal step were stable up to a time step of 15 minutes; with four iterations, the solutions were stable up to a time step
of 20 minutes.
In comparing the error measures in figure 5.10 with those in figure 5.7, it is significant that the addition of the trapezoidal step
to the leapfrog scheme gave solutions that generally have equal or smaller errors than solutions using two iterations of the two-
level scheme. In particular, for the large time steps of 10 and 15 minutes, the error measures are noticeably smaller in figure 5.10
for the solutions using the leapfrog-trapezoidal scheme (1 iteration) than in figure 5.7 for the solutions using the two-level scheme
(niter = 2). The leapfrog-trapezoidal scheme solution for Δt = 15 minutes and Δx = 2,000 ft (610 m) also did not exhibit the oscil-
lations shown in figure 5.8 for the solution using the two-level scheme (niter = 2).
5.2.1.3 Remarks
Based on the results of the hydrograph numerical experiment, the three-level scheme is more accurate than the two-level
scheme. The two-level scheme without using iterations is only first-order accurate and requires an extremely small numerical grid
size to reduce the error in the hydrograph solutions to an acceptable level. Using two iterations, it is possible to raise the accuracy
of the two-level scheme to second order, but the same order of accuracy can be achieved at approximately half the computational
cost by using the three-level, leapfrog scheme. The use of iteration in the solutions for either scheme extends the range of compu-
tational intervals and Courant numbers for which the solutions are stable. If a single iteration of the trapezoidal step is used in the
three-level scheme (so that, for example, a stable solution is obtained for a large time step), generally the scheme still is more accu-
rate than the comparable two-level scheme using two iterations. The use of 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 hydrograph problem, consid-
ering the extra computational cost incurred for the minimal improvement in accuracy that is gained; if the extra iterations are
needed to stabilize a particular solution, then the time step and Courant numbers may be too large on the basis of accuracy
considerations anyway.
As noted previously, the hydrograph routing problem has been used by the USGS to test three 1-D models (Fread, 1978;
Schaffranek and others, 1981; and Delong, 1993) that are based on the popular four-point finite-difference scheme. A few com-
ments can be made regarding how the semi-implicit, three-level scheme presented here compares with the four-point scheme. The
four-point scheme, first introduced by Preissmann (1961), is a fully implicit scheme in which both of the independent variables are
computed at the same grid points rather than at different ones as in the case of a staggered grid. The scheme is nearly second-order
accurate in the space and time integration methods so long as a finite-difference weighting factor θ is chosen close to 0.5 (for
unconditional linear stability, θ must be greater than or equal to 0.5). The overall accuracy (convergence characteristics) of the
four-point and semi-implicit schemes are similar for solutions with gravity-wave Courant numbers greater than one (see Fread,
1974, for a detailed report on the convergence characteristics of the four-point scheme). For gravity-wave Courant numbers less
than one, the high frequency solution components of the four-point scheme have leading phase errors; these errors are the cause