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

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94 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation The von Neumann (or Fourier) method (O’Brien and others, 1950) is a widely used tool for studying the stability of numerical schemes that are used in solving linear differential equations. It is based on representing the solution of a difference equation as a finite Fourier series and studying the decay or amplification of each mode individually to determine stability or instability. By introducing the concept of a complex propagation factor, Leendertse (1967) extended the von Neumann method for use in analyz- ing the convergence of linear numerical schemes. Using two-time-level, finite-difference discretizations of the linear shallow water equations, Casulli and Cheng (1990) and Casulli and Cattani (1994) used the von Neumann method to study the stability of 1-D and 3-D semi-implicit methods, respectively. Cunge and others (1980) present a von Neumann convergence analysis in the form of amplitude and phase error portraits for the 1-D semi-implicit scheme of Abbott-Ionescu. Using a finite-element discretization of the 1-D linear shallow-water equations, Gray and Lynch (1977) applied the von Neumann method to analyze the convergence of both two- and three-time-level semi-implicit schemes. Although the von Neumann method is indeed useful, it is fundamentally incomplete as it does not take into account the full effects of nonlinearity, iteration schemes, and interpolations on the properties of stability and convergence. In this chapter, numerical experiments are used to investigate the stability and convergence of the full nonlinear form of the 1-D and 3-D semi-implicit schemes presented in Chapter 4. An emphasis is placed on studying the effect of iteration. 43 Instances of nonlinear instability are shown to occur in the 1-D experiments that are not revealed by the linear von Neumann analysis. Iter- ation is shown to be effective in suppressing the instabilities and obtaining a smooth solution. Because adding iterations does increase the computer run time of a model simulation, finding the minimum number of iterations that are needed to achieve a stable and accurate solution is an important goal. 5.2 One-Dimensional Experiments The two-level and three-level semi-implicit schemes for one dimension are tested below using two numerical experiments. The first experiment routes a hydrograph down a rectangular channel with an initially shallow depth of flow. The experiment rep- resents an inland river modeling problem that is strongly nonlinear and has high frictional resistance. The second experiment sim- ulates the propagation of a repeating semidiurnal tide into a closed basin of constant depth. The tidal flow experiment is influenced only a little by frictional resistance but is affected by nonlinear advection and the rapidly changing bidirectional flow that is typical in estuaries. 5.2.1 Hydrograph Routing This experiment was first designed by the U.S. Geological Survey (USGS) in the early 1980s to test and compare three dif- ferent 1-D finite-difference flow models, including the two widely used models developed by the National Weather Service (Fread, 1978) and the U.S. Geological Survey (Schaffranek and others, 1981). 44 The experiment has since been used by USGS researchers (for example, DeLong, 1993, fig. 2) to test other flow models that are based on numerical schemes of varying accuracies. The 43 In the case of the three-time-level scheme, iteration is used here to mean the addition during each time step of one or more trapezoidal-scheme steps after the leapfrog- scheme step. 44 The 1-D models by Fread (1978) and Schaffranek and others (1981) are based on the fully implicit four-point scheme by Preissmann (1961); there are a few differences between the two models in the details of how the four-point scheme is implemented.
5. Numerical Experiments 95 solution to the fully nonlinear test problem is well known from these past model applications; the convergence characteristics of these previously tested models vary widely, depending on their numerical accuracy. The channel characteristics and initial flow conditions for the hydrograph experiment are given in figure 5.1. The channel is rectangular; top width B = 100 ft (30.5 m), 45 bed slope S 0 = 0.001, and length L = 150,000 ft (45,720 m). The Manning resistance coefficient for the channel is 0.045. The initial steady flow is Q 0 = 250 ft 3 /s (7.1 m 3 /s), and the initial flow depth according to Manning’s formula is H 0 = 1.688 ft (0.51 m). The channel is considered wide enough so that the hydraulic radius in equation 4.3 can be approximated by the flow depth. and For the test problem, the discharge at the upstream boundary of the channel (x = 0) is described by 750 Qt () = 250 + -------- ⎛1– cos----- πt⎞ 0 < t < 150 π ⎝ 75⎠ Qt () = 250 t ≥ 150 , where the time t is expressed in minutes from the start of the simulation and discharge is in cubic feet per second (ft 3 /s). At the downstream boundary of the channel (x = 150,000), the depth of flow is constant at H 0. The simulation is run for 500 minutes. The results of the hydrograph routing are examined at x = 50,000 ft (15,240 m) to avoid any effect from the downstream boundary condition on the computed hydrograph. (The water level at the downstream boundary does not rise until the hydrograph passes 50,000 ft.) Q = Q (t ) Datum x 1 B H 0 Δ Q 0 1 ⁄ S0 L Channel characteristics B = 100 feet L = 150,000 feet S 0 = 0.001 Manning’s n = 0.045 Initial flow conditions Q 0 = 250 cubic feet per second H 0 = 1.688 feet Δ Q = Q 0 Figure 5.1. Channel characteristics and initial flow conditions for the 1-dimensional hydrograph test problem. 45 The 1-D numerical experiments, as originally designed, were posed in English units; these units are adopted here for consistency. The metric unit conversions are shown in parentheses following the English units.
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