Advancing Watershed-Scale Hydrodynamic Modeling: Techniques for Comparative Analysis of Saint-Venant Solvers

Advancing Watershed-Scale Hydrodynamic Modeling: Techniques for Comparative Analysis of Saint-Venant Solvers Edward Tiernan, Cheng-Wei Yu, Saz Sharior, Ben R. Hodges November 2025 Objective This technical report presents the methodological foundation and supplementary analyses associated with a study evaluating the performance of a newly developed Saint-Venant Equation (SVE) solver, SWMM5+, in comparison with two existing hydraulic models: the U.S. EPA Storm Water Management Model (EPA SWMM) and the Simulation Program for River NeTworks (SPRNT). The comparative modeling framework, introduced in the accompanying journal article titled “Advancing Watershed-Scale Hydrodynamic Modeling with the Saint-Venant Equations”, is applied to regional-scale river networks to assess numerical behavior, solution stability, and hydrodynamic accuracy across different solver architectures. This report provides additional technical documentation, implementation details, and supplemental figures that support the findings and discussions presented in the main manuscript. In this study, SWMM5+ is validated through model-to-model comparison rather than direct comparison with observational data, due to the substantial challenges inherent in validating large-scale river network models. Observational flow and stage data are often spatially sparse, temporally inconsistent, and subject to measurement uncertainties, which complicate the assessment of how model errors emerge and propagate throughout the system. Additionally, river bathymetric data—crucial for solving the Saint-Venant Equations—are frequently incomplete or of uncertain accuracy across large domains, making it difficult to isolate algorithmic performance from boundary condition uncertainties. Compounding these challenges is the presence of hydraulic infrastructure, such as dams and weirs, whose operational rules are often proprietary or undocumented. These unmodeled anthropogenic controls can significantly influence flow and stage observations, thereby obscuring the attribution of model-observation discrepancies (Turner, Steyaert, Condon, & Voisin, 2021). Consequently, validation in such settings tends to reflect the degree of parameter calibration rather than the intrinsic accuracy of the numerical schemes. Model-to-model comparison thus provides a more transparent means of evaluating the performance of numerical solvers, independent of the confounding effects of observational and structural uncertainties. In contrast to traditional model-observation comparisons, model-to-model comparison provides a consistent and system-wide framework for evaluating numerical accuracy and stability under identical forcing conditions, boundary specifications, and parameter settings. This approach allows performance to be assessed across a wide spatial domain, overcoming the limitations associated with sparse observational data. Furthermore, it enables detailed examination of hydrodynamic behaviors—such as wave attenuation, numerical dissipation, and solver stability—across complex, watershed-scale networks. By isolating the effects of numerical architecture, model-to-model comparison offers critical insights into the strengths and limitations of each solver, independent of calibration artifacts or observational uncertainties. This technical documentation outlines the methodological components that support such model intercomparison studies and may serve as a reference for similar evaluations. The key topics covered include: 1. Development of regional-scale river network models using publicly available datasets. 2. Criteria and procedures for evaluating SWMM5+ suitability for model-to-model comparison. 3. Initialization procedures for Saint-Venant Equation (SVE) simulations, including model spin-up requirements. 1 4. Post-processing workflows designed to enable robust comparison of spatially and temporally distributed results. 5. Presentation of supplemental simulation results from the SVE-based models. 6. Solver adjustments and configuration settings applied to ensure fair and interpretable comparisons across models. 1 Creation of river network models 1.1 Regional-scale River Basins The regional-scale river basins examined in the main journal article are the Lavaca and San Jacinto River basins, both located within the Texas-Gulf hydrologic region. The Lavaca River originates in central Texas and drains an area of approximately 5,900 km2before discharging into Lavaca Bay. The full river network consists of 3,049 km of channel reaches; however, to streamline the model while preserving hydrodynamic fidelity, first-order tributaries—which have negligible influence on the overall flow dynamics—are represented as nodal inflows rather than explicitly modeled channels. This simplification reduces the effective modeled channel length to 1,291 km. The Lavaca River basin features a relatively mild topographic gradient, with elevations ranging from 0 to 23 m above sea level. Natural channel slopes fall between 0.0001 and 0.002, with an average slope of approximately 0.0003. The network structure and topological configuration of the Lavaca River are illustrated in Figure 1(a). In contrast, the San Jacinto River—which drains an area of approximately 4,600km2, exhibits a more complex branching topology and greater vulnerability to extreme flood events, as evidenced during Hurricane Harvey in 2017. Applying the same channel filtering methodology as used for the Lavaca River, the total modeled channel length is reduced to 1,716 km after excluding first-order tributaries with limited hydraulic influence. Located within the East Texas Plain, the San Jacinto River basin spans elevations from 0 to 120 m above sea level. Excluding localized slope disruptions caused by hydraulic structures, the natural channel slope ranges from 0.0001 to 0.009, with an average slope of 0.00045. The structure and topological configuration of the river network are depicted in Figure 1(b). 2 Fig. 1. The two selected river networks within the Texas-Gulf region in this study. (a) Lavaca River, (b) San Jacinto River. The blue flowlines are extracted from the NHDPlus dataset (USGS, 2020). Gold stars (1) and (2) are hydrograph locations in Fig. 5; gold stars (3) and (4) are hydrograph locations in Fig. 7. 1.2 From Raw Data to SPRNT One-dimensional hydrological models for the Lavaca and San Jacinto Rivers were derived from the USGS National Hydrography Dataset (NHDplus) flowlines for Hydrological Unit Code 12 (USGS, 2020). Definition of channel geometry and roughness followed the approach used in the National Water Model (NWM), which defines trapezoidal cross-sections with constant side-wall slope of 0.5 and a bottom width that depends on the Strahler’s stream order of the reach (David et al., 2011; NOAA, 2016). The bottom width ranges between 7mand 40 m. The roughness from NWM (Strickler-Manning, n) ranges from 0.045 to 0.06. Subcritical flow at the downstream boundary is ensured by fixing the water depth (H) to 3 m. Hydrological flows throughout the river basins were extracted from the NWM archive and introduced as time-varying nodal boundary conditions (NOAA, 2016). Following the approach by Falter et al. (2016) and Paiva, Collischonn, Bonnet, and de Gon¸calves (2012), first order and minor streams (i.e. reaches that are dry a preponderance of the simulation time) were removed. Flow contributions from the excised reaches, when they occur, are incorporated into the hydrological inflows at neighboring nodal boundary conditions. Other network refinement steps were taken, including the omission of 86 “braided” channel reaches, as well as the replacement of 106 reservoirs/dams with simple channels. Such features should be handled by sophisticated river-routing software attempting to make real world predictions. However, for our present scope of comparing different models’ numerical methods for identical simulation parameters, the additional complexity contributed by braided channels or dam-gate operations is not warranted. Generally, favorable behavior from hydraulic models can be expected when channel segments are of roughly equal length. From the NHDplus dataset, the average river reach length for our test case was 3.1 km, with a maximum of 68 km and a minimum of 1 m. The river systems herein were refined such that each river channel segment was ∼150 mby subdividing longer segments and lengthening shorter ones. Table 1 contains details about the discretization of the Lavaca and San Jacinto NHDplus networks into the NWM form. 3 Table 1. Topological information of major river networks, excluding first-order reaches, in the Texas-Gulf watershed (Yu et al., 2021). Network Name Channel Length (km) NHDPlus Flowlines Computational Nodes Inflow Boundaries Lavaca 1291 360 6973 67 San Jacinto 1732 690 9659 116 The Lavaca and San Jacinto River network models were originally built as input files for the Simulation Program for River NeTworks (SPRNT). As SPRNT files, these systems were used by Yu, Liu, and Hodges (2017) to study consistent initial conditions for the Saint-Venant equations in river network modeling, by Yu et al. (2021) to evaluate a new method for detecting instability-causing geometry transitions. 1.3 From SPRNT to SWMM The SPRNT modeling architecture varies from the EPA SWMM modeling architecture in critical ways. To enable a coherent comparison of the results between the modeling paradigms, the models, originally constructed in the SPRNT architecture, must be converted into the EPA SWMM architecture. Fortunately, the SWMM5+ model also used the EPA SWMM architecture, so the Lavaca and San Jacinto models only need to be converted once. All SVE solvers rely on the same physical parameters, e.g., elevation gradients, cross-section geometry, etc., but how these parameters are handled can vary between modeling environments. Converting from one modeling environment to another is an exercise in finding the analogous data structures for each parameter. Some SPRNT objects, such as “segments”, are directly analogous to SWMM “conduits”; properties like length and upstream/downstream connections are simply copied from SPRNT into the SWMM format. Other network objects require data structure transformation. An example of a network object requiring transformation are multi-nodal junctions in SPRNT being collapsed into single node junctions in the SWMM paradigm (shown with a red ∆ in Fig. 2). In SPRNT, a “junction” consists of multiple “nodes” in a complex, where each node can only have one or two connections. Whereas, in SWMM, any node with one or more connections is a considered a junction. Merging junctions from SPRNT into SWMM involves the replacement of SPRNT junction complexes with SWMM junction nodes (illustrated in Fig. 3, below). Other network object parameters (e.g. channel geometry, internal boundary conditions, etc.) are simply stored in different data structures between SPRNT and SWMM. The transformation from SPRNT-compatible input files to SWMM-compatible input files, conserves network topology, geometry, and forcing data (i.e. the network information in Table 1 is preserved). 4 Fig. 2. Schematic showing the data structure shuffling needed to convert from SPRNT-format to SWMMformat. Note the changes in network object jargon between SPRNT and SWMM. Red ∆ indicates Junction complex further illustrated in Fig. 3 Fig. 3. Example schematic of “Junction” complex for SPRNT, SWMM5+, and EPA SWMM. State variables like flow (Q) and head (h) are reported as attributes of different network objects depending on the modeling framework. The SWMM5+ model further transforms the conduit-node EPA SWMM input file into a finite-volume network of elements and faces, while still preserving the network properties. For additional information 5 about the discretization approach used by SWMM5+, please see Hodges et al. (2024). 2 Evaluation of SWMM5+ In the study, river network simulation results from multiple SVE solvers are compared to assess the effects of program architecture on flood wave peak timing and other hydrodynamic factors. The SPRNT program was expressly designed for river network simulation (Liu, 2014), while previous studies have established EPA SWMM as a suitable model for some river routing simulation (Niazi et al., 2020; Swathi, Raju, & Varma, 2020; Xiong & Melching, 2005). The SWMM5+ program, the newest of these SVE-solver models, was designed as an under-the-hood, parallelized, mass-conservative version of EPA SWMM5 (Hodges et al., 2024). However, SWMM5+ has not been explicitly tested on a regional-scale river network prior to its use herein. This section briefly describes some relevant architectural details about SWMM5+, then compares simulation results to observed data at two locations within the Lavaca river basin. 2.1 Parallelization The parallelization strategy in SWMM5+ adopts the single program, multiple data (SPMD) approach. Unlike implicit methods commonly used in SVE solvers, which typically require matrix decomposition, the explicit numerical scheme in SWMM5+ enables direct subdivision and distribution of the computational domain across multiple processors. The network subdivision algorithm from Tiernan and Hodges (2022) is adapted to optimize load balancing across processors. Instead of conventional multiprocessing libraries (e.g., OpenMP), SWMM5+ employs Coarray Fortran for parallelization (Hodges et al., 2021). Although the primary focus of this study is not on parallel computation, we include this information to highlight that SWMM5+ supports efficient parallelization, demonstrating its capability to handle large-scale simulations. 2.2 Numerical Method The numerical framework of SWMM5+ employs an explicit second-order Runge-Kutta (RK2) time-marching scheme to compute flow rate (Q) and volume (V) within finite-volume elements. While the RK2 algorithm exhibits higher temporal discretization discrepancies compared to higher-order time-marching methods, its structure is particularly well-suited for parallel computing architectures. To reduce numerical dependency between adjacent computational elements, SWMM5+ implements a no-neighbor discretization approach within the RK2 scheme (Hodges & Liu, 2019), effectively minimizing communication bandwidth requirements during parallel computation. The time-space discretization structure of the RK2 algorithm is depicted in Figure 4. 6 Fig. 4. The upper panel illustrates the domain of dependency for an RK2 time advancement in a finitevolume scheme with third-order upwind face reconstruction, while the lower panel depicts a no-neighbor compliant scheme with second-order face reconstruction. The thicker arrows in the upper panel highlight the increased spatial dependency introduced by the third-order scheme. Figure adapted from (Hodges & Liu, 2019). As illustrated in Figure 4, the finite-volume element faces are reconstructed dynamically during time integration. Traditional approaches typically rely on a combination of upwind and downwind element values for face reconstruction. In contrast, the no-neighbor method implemented in SWMM5+ ensures that reconstructed faces utilize only adjacent elements, thereby minimizing spatial dependency and reducing communication overhead between processors. This design significantly decreases data exchange between computational cores, improving the efficiency of parallel computation. A detailed explanation can be found in (Hodges & Liu, 2019). The numerical stability of the RK2 scheme is governed by the CFL condition, which necessitates that u∆t /∆x≤Cmax to prevent numerical divergence. In SWMM5+, however, to ensure global numerical stability, the CFL constraint is further restricted to CFL <√2/2, with a dynamically adjusted time step. This limitation is typically imposed by the presence of small elements, such as junctions, which often result in a time step that is disproportionately smaller than the time step in the EPA SWMM model for the same system. While this smaller time step can be a limitation, it can be effectively addressed through the use of parallel computing, which alleviates the computational burden (Hodges et al., 2024). 2.3 Comparison to Observed Data Although the SWMM5+ simulation configuration and forcing in this study are directly adopted from the NWM setup without specific hydrodynamic tailoring (i.e., the model is uncalibrated), it is still valuable to include a preliminary comparison to observed data. Figure 5 shows the discharge timeseries results for SWMM5+ compared against USGS gauge data for two junctions in the Lavaca river network (seen in Fig. 1). The results demonstrate that SWMM5+ can produce reasonable consistency with field observations, capturing comparable peak values and peak arrival times. The simplified configuration of trapezoidal bathymetry inherited from NWM may have lead to overestimates of discharge in the simulations. Nonetheless, as the primary objective of this study is to assess the performance of the SWMM5+ model relative to SPRNT and EPA SWMM, the simplified bathymetry assumption was retained to ensure convergence across all models and maintain consistent comparison standards. 7 2017-01-15 2017-02-01 2017-02-15 2017-03-01 2017-03-15 2017-04-01 2017-04-15 2017-05-01 2017-05-15 Datetime 0 20 40 60 80 100 Discharge ( m 3/ s ) 08164450 Gauge Data SWMM5+ Data 2017-01-15 2017-02-01 2017-02-15 2017-03-01 2017-03-15 2017-04-01 2017-04-15 2017-05-01 2017-05-15 Datetime 08164503 Fig. 5. Comparison of SWMM5+ simulation results with field-observed daily discharge data from two gauges in the Lavaca River network. Left panel: USGS 08164450 (i.e., Star (1) in Fig. 1); right panel: USGS 08164503 (i.e., Star (2) in Fig. 1). 3 Initialize SVE Simulation – “Spin-Up Time” The amount of time before simulated results are free from the impacts of uncertain initial conditions is commonly referred to as the model “spin-up time”. In numerical simulation, the spin-up time is generally unavoidable, as it is rarely possible to obtain precisely accurate spatially distributed initial conditions that are fully consistent with the corresponding spatially distributed boundary conditions at t= 0. As highlighted by Yu et al. (2017), no universal criterion exists for defining spin-up time in river network simulations; rather, it is influenced by the quality of initial condition configurations and the degree of consistency between initial and boundary conditions. Herein the Lavaca and San Jacinto river basins were “spun-up” using slightly variable approaches. For the Lavaca River, relatively stable boundary conditions were imposed; for the San Jacinto River, real forcing data, including a significant runoff event, were utilized to evaluate model responses under dynamic hydrological conditions. Zero initial conditions were applied in the SWMM5+ and EPA SWMM models to assess their convergence to the SPRNT model, with the duration defined as the spin-up time. Convergence was determined based on the percentage difference between simulated hydrographs at the final junction upstream of the outlet, considering them converged when the difference remained within ±1% for a minimum period of six hours. The resulting discharge hydrographs and percentage differences between the models and the benchmark SPRNT model are presented in Figure 6. 8 Lavaca River San Jacinto River Discharge (m3/s)Difference (%) Time (hour) Time (hour) Fig. 6. Spin-up analysis for the Lavaca and San Jacinto River networks using SWMM5+, EPAkin, and EPAdyn. Hydrographs are evaluated at the final junction upstream of the outlet for both river networks. The top panels depict the time required for hydrographs from each model to achieve convergence with the benchmark model, SPRNT. The bottom panels present the percentage differences between the three models and SPRNT over the simulation period. The time a system takes to erase any memory of the initial conditions is largely a physical process, rather than a numerical one. Larger, shallower systems require more spin-up time than smaller, steeper ones. The results of Fig. 6 show that the discharge from the outfall of each comparison model converged to the SPRNT model within ˜ 350 hours for the Lavaca River, ˜ 200 hours for the San Jacinto River. Notably, the SWMM5+ model took the longest to converge for the Lavaca River with consistent boundary conditions, despite being the first to converge for the San Jacinto river with hydrologically realistic boundary conditions. 4 Postprocessing Techniques for Model-to-Model Analysis 4.1 Hydrograph Smoothing Jagged, discontinuous behavior of solution variables presents an impediment to automated processing of hydrograph and stage results. To address this issue, our analysis workflow employs a non-linear median filter (NLMF) for smoothing noisy timeseries data, similar to the approach of White and Hodges (2005) for smoothing bathymetric data. The behavior of the NLMF is similar to a traditional moving-average window, except the use of the median value rather than the arithmetic average reduces the bias from spurious hydrograph spikes. NLMF smoothing can be represented for a filter window of size 2Nf+ 1 as ˜ ϕ[xi, tk] = MEDIAN ϕ[xi, tk−Nf], ϕ[xi, tk−Nf+1], ...ϕ[xi, tk+Nf](1) where ˜ ϕrepresents the NLMF-filtered data set. Values for ϕ[xi, tk< Tstart] and ϕ[xi, tk> Tend] are ignored. The NLMF preserves key hydrograph features such as the slope of the rising limb and timing of peak flows. 9 a similar tendency to produce excessive SDvvalues in downstream channels, attributable to oscillatory numerical results, as we observed in the Lavaca River case. From the author’s observation, the EPAdyn simulations exhibit more pronounced numerical instability in the San Jacinto River compared to the Lavaca River. Despite the application of a moving average filter to mitigate fluctuations, the filter is insufficient to address the severe oscillations, particularly in downstream channels where the numerical instability is most pronounced. For comparison and demonstration purposes, these samples are retained in the scatter plot. The EPAkin continues to exhibit similar behavior to what was observed in the Lavaca case, with a comparable level of SDvto the SWMM5+ model but a broader range of SDt, particularly in channels near the river outlet. 7500100 Marker Size - Catchment Area (km2) Marker Color SWMM5+ EPA kin EPA dyn Marker Shape Jan 20 Event Feb 21 Event May 16 Event SDt (hr) SDv (m3/s) Fig. 11. Scatter plot of timing series difference (SDt) versus magnitude series difference (SDv) for the San Jacinto River, based on three hydrological events (January 20, February 21, and May 16, 2017). Box plots of SDtand SDvare included along the top and right margins, respectively, to illustrate the distribution of timing and magnitude differences for each model. 16 6 Adjustments and configurations made in the solvers to ensure the acquirement of comparable results To preserve the full dynamics of the Saint-Venant governing equations, a specific code block in the Dynamic Wave module (dwflow.c) of the SWMM5 source code was commented out, following guidance from Rossman, Dickinson, and Tiernan (2022). Figure 13 illustrates the simulated flow rate at the outfall of the Lavaca River network over the first 40 days of simulation using the EPAdyn solver, under two configurations: (a) with the normal flow limitation removed to enable full hydrodynamic complexity, and (b) with the default normal flow limitation enabled. The results demonstrate that disabling the normal flow limiter improves numerical stability at the network outlet. Accordingly, all EPAdyn simulations in this study were conducted with the normal flow limitation disabled to ensure consistent application of the complete Saint-Venant equations. Fig. 12. Screenshot of normal flow limiting code block in dwflow.c. When the block is activated, a normal flow condition is enforced. 17 (a) Default normal flow limiter deactivated. (b) Default normal flow limiter activated. Fig. 13. Simulated flowrates at the outfall of the Lavaca River Network from EPAdyn showing stability difference from toggling the normal flow limiting code block in the dwflow.c module. 7 Conclusion This technical report provides supporting documentation for the journal article “Advancing Watershed-Scale Hydrodynamic Modeling with the Saint-Venant Equations”, with a focus on the implementation details, adjustments, and supplemental analyses used in the model-to-model comparison. The material presented here includes practical steps for constructing large-scale river network models, setting up numerical experiments using Saint-Venant solvers, and post-processing methods for evaluating flood wave behavior. This report outlines the procedures used to generate regional-scale river network models for the Lavaca and San Jacinto River basins, including simplifications applied to channel geometry, node inflow handling, and bathymetric assumptions. It also describes modifications made to the SWMM5 source code to preserve full hydrodynamic dynamics, as well as the rationale for including each solver in the comparison. The post-processing methods detailed in this report—such as event-based series distance metrics and spatial error mapping—serve as practical tools for diagnosing numerical behavior across complex river systems. In addition, supplemental results are provided to illustrate model differences in hydrograph shape, flood wave timing, and numerical stability under identical forcing. 18 Overall, this report is intended to enhance the transparency and reproducibility of the modeling workflow described in the main article. It also serves as a reference for researchers seeking to apply or extend the comparison framework to additional river systems or solvers. By documenting implementation-specific choices and solver configurations, we hope this resource supports continued development and evaluation of watershed-scale hydrodynamic models using the Saint-Venant equations. 19 References David, C. H., Maidment, D. R., Niu, G.-Y., Yang, Z.-L., Habets, F., & Eijkhout, V. (2011). River Network Routing on the NHDPlus Dataset. Journal of Hydrometeorology,12 (5), 913–934. doi: 10.1175/2011JHM1345.1 Ehret, U., & Zehe, E. (2011). 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