This Draft Report contains Panelist Victor M. Ponce's
contributions and recommendations
after attending the
South Florida Water Management District (SFWMD)
Peer Review of the Regional Simulation Model's (RSM),
held in West Palm Beach, Florida, on July 24-25,
2019. This review has concluded that the methodologies included in RSM are adequate for its use in south Florida. To improve and complement current efforts, the author recommends that District scientists spend additional time on the issues of numerical accuracy, particularly on the determination of the applicable Courant and cell Reynolds numbers for specific model runs. The author's experience in this area is offered to serve as a suitable framework for the analysis.
All numerical models, and RSM is no exception, have a way of becoming unstable under a certain set of circumstances. Thus, it seems appropriate, at the start, to provide a general discussion on strategies for model control to manage instabilities. A good physically based mathematical model is based on generally accepted partial differential equations describing the relevant physical processes. RSM uses 1-D and 2-D formulations of watershed, channel, reservoir, and groundwater flow, coupling them as appropriate to better represent the physical reality at the chosen level of abstraction.
All numerical models suffer from problems of stability and convergence.
Stability is related to roundoff errors; convergence to discretization
errors (O'Brien
Convergence, which is akin to accuracy (in the sense of In practice, the control of numerical instability is seen to be a careful balancing act: How to build a scheme that has enough numerical diffusion to handle the high-frequency perturbations that are responsible for the instability, while at the same time making sure that the solution itself is not being substantially affected by the artificially introduced numerical diffusion. This dilemma is at the crux of all numerical modeling. The laws of mass and momentum conservation, which underpin all physical-process modeling of unsteady flows, may be combined, through appropriate linearization, into a single second-order, convection-diffusion equation (Hayami, 1951). In one extreme, when the diffusion term vanishes, the equation becomes hyperbolic; in the other extreme, when the convection term vanishes, the equation becomes parabolic.
Numerical models of hyperbolic systems are subject to the
Courant law, which expresses the ratio of physical celerity (
The properties of numerical schemes may be analyzed using various tools
of advanced mathematics.
In hyperbolic systems, an assessment of numerical accuracy (
For instance, there is a wealth of accumulated experience on the numerical
properties of the well-known Preissmann scheme, wherein stability and convergence are
determined by the spatial resolution An excellent example of the use of Fourier analysis in numerical modeling of flood flows is that of the Muskingum-Cunge model, a diffusion wave model that is based on the matching of physical and numerical diffusivities (Cunge, 1969). A review of the amplitude and phase portraits of the Muskingum-Cunge model, including an online calculator, has recently been accomplished by Vuppalapati and Ponce (2016).
In its newest implementation, the RSM model uses the Total Variation Diminishing Lax-Friedrichs method (TVDLF), which is shown to be accurate and stable for both kinematic and diffusion flows such as those prevalent in Southern Florida (Lal and Gabor, 2013). The method uses a linearized conservative implicit formulation of the simplified St. Venant equations, thereby avoiding the iterative formulations that would normally be necessary when solving a nonlinear scheme. SFWMD scientists have extensively tested the method, with favorable results in terms of numerical accuracy and runtime.
The success of the method in simulating a wide array of problems, including dry channel bed and steep bottom slopes, must be attributed to
its use of weighting factors to incorporate numerical diffusion as needed to
control the instabilities
that would normally appear in connection with sharp (
An established approach to modeling flood flows used in RSM
is that of Hayami (1951), who combined the governing
equations of continuity and motion (the Saint Venant equations)
into a second-order partial
differential equation with discharge
The hydraulic diffusivity used in RSM follows the original Hayami formulation
of a diffusion wave, wherein the inertia terms (in the equation of motion)
are neglected. This approximation works well for low Froude number flows.
However, for high Froude number flows, the neglect of inertia proves to be increasingly
unjustified. As shown in Ponce (1991), the true hydraulic diffusivity
of the convection-diffusion model of flood flows
is the i.e., the so-called roll waves.
This fits admirably with physical reality, confirming the theoretical basis
of the Vedernikov-dependent diffusivity, i.e., the dynamic hydraulic diffusivity.
We recommend that a dynamic hydraulic diffusivity be incorporated into all instances where surface-water convection-diffusion is
being modeled in RSM. This extension provides
The determination of the proper spatial resolution
lies at the crux of good modeling practice, as the experience with RSM clearly shows.
No amount of time spent on this effort is wasted.
Our recommendation is to start with a
target spatial resolution
Fig. 1 Dimensionless relative wave celerity vs dimensionless wavenumber in open-channel flow.
The choice between explicit and implicit schemes continues to haunt numerical modelers. While implicit schemes are unconditionally stable, a similar statement may not follow for explicit schemes. This is certainly the case for both surface and groundwater flows. On this basis, implicit schemes are generally preferred over explicit schemes, but the complete story remains to be told.
It may be true that implicit schemes are not subject to
The tradeoffs between explicit and implicit schemes are, therefore, clear:
While implicit schemes are more stable,
they require matrix inversion and the actual
time step is effectively limited in size by accuracy considerations.
Explicit schemes, on the other hand, are simpler to develop and execute,
requiring no matrix inversion and no downstream boundary
(Ponce
In the typical case, the forces acting on a 1-D formulation of
unsteady open-channel flow are: (1) gravity,
(2) friction, (3) pressure gradient, and (4) inertia.
Significantly, in a horizontal channel the gravitational force vanishes,
while the three other forces remain.
This renders the Manning equation inapproriate, since the driving force
in this equation is the gravitational force. Thus,
for modeling unsteady flows in
horizontal channels (the case of south Florida) there appears to be no other choice
than to use
the full Saint-Venant equations. The need to include lateral contributions (seepage in and out of the control volume) in the analysis of wave propagation in south Florida applications remains to be fully clarified. Great strides along these lines have already been made by SFWMD scientists. The additional terms in the mass and momentum balance equations need to be carefully identified. Their relative importance may be determined following the work of Ponce (1982).
The term "RSM model" is being currently used to describe any and all
activities under the RSM modeling framework. This explains
the District's (SFWMD) reluctance
to engage in explicit model version numbers to describe what amounts to
activities of varied scope and in many areas.
We recommend that SFWMD consider a thorough and full documentation of the RSM model via a technically edited User Manual, accompanied by a Reference Manual, as a way to ensure that potential users of the model will be able to use it in the future. Background material would consist of relevant published papers listed in the bibliography and included therein with hot links to online pdf files.
As an alternative, one certainly requiring fewer resources, the District could sponsor a publications series to be entitled, for example,
We offer the following miscellaneous recommendations: The 2-D momentum equations originate in the 3-D Navier-Stokes equations, and, as such, are technically *not closed*(Flokstra, 1976; 1977). Some sort of surrogate for the missing effective stresses appears in order (Kuipers and Vreugdenhil, 1973). This is an obscure subject, perhaps deserving of more attention than that given so far.Caution is recommended when using a 2-D formulation of a diffusion wave, wherein the inertia terms are neglected. Neglecting inertia is bound to eliminate physical circulation (Ponce and Yabusaki, 1981). However, it may be a reasonable assumption in the largely convective 2-D flows that prevail in south Florida. The Muskingum-Cunge model of 1-D flood flows, effectively a diffusion wave model, has been analytically verified by Ponce *et al.*(1996). We suggest that the District consider including this verification test in their set of cases for model verification.
Cunge, J. A.. 1969.
Flokstra, C. 1976. Generation of two-dimensional horizontal secondary currents. Report S 163-II, Delft Hydraulics Laboratory, Delft, The Netherlands, July.
Flokstra, C. 1977. The closure problem for depth-averaged two-dimensional
flow.
Hayami, S. 1951.
Kuipers, J. and C. B. Vreugdenhil. 1973. Calculations of two-dimensional
horizontal flows.
Lal, A. M. and G. Toth. 2013.
Leendertse, J. J. 1967. Aspects of a computational
model for long-period water wave propagation.
Lighthill, M. J. and Whitham, G. B. 1955.
Nuccitelli, N. R. and V. M. Ponce. 2014.
O'Brien, G. G., M. A. Hyman and S. Kaplan. 1950.
Ponce, V. M. and D. B. Simons. 1977.
Ponce, V. M., H. Indlekofer and D. B. Simons. 1978.
Ponce, V. M. and S. B. Yabusaki. 1981.
Ponce, V. M. 1982.
Ponce, V. M. 1991.
Ponce, V. M., A. K. Lohani and C. Scheyhing. 1996.
Ponce, V. M., A. V. Shetty and S. Ercan. 2001.
Ponce, V. M. 2014.
Powell, R. W. 1948.
Vuppalapati, B. and V. M. Ponce. 2016. |

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