# Comparative Analyses between the Zero-Inertia and Fully Dynamic Models of the Shallow Water Equations for Unsteady Overland Flow Propagation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Governing Equations

_{b}is the topographic (or ground) level, h is the water depth, g is the gravitational acceleration, p is a source term (e.g., due to the applied rain) and n is the Manning friction coefficient. The water level (or piezometric level), H, is the sum of the water depth and the topographic level, H = h + z

_{b}. The momentum equations represent a balance of forces due to (1) the local inertia (or local acceleration) li, (2) the convective inertia (or convective acceleration) ci, (3) the sum of the gradients of the hydrostatic pressure and ground level gr, and (4) the bottom friction term bfr. In Equations (2) and (3) we neglect the viscous effects. System (1)–(2) is written in conservative form in the unknowns h, uh, and vh.

## 3. Brief Overview of the Applied Numerical Procedures

_{T}triangles k

_{T,e}(e = 1, ...., N

_{T}) and N nodes (or vertices) P

_{i}(i = 1, ..., N), satisfying the Generalized Delaunay (GD) property [9] (see A.2 in Supplementary Materials).

_{T}) is not a drawback in the comparative analysis of the two solvers.

## 4. Presentation of the Models Applications

_{j}have been introduced in Equation (4), indices k and k + 1 mark the beginning and the end of the time step, respectively, index e marks the variables of the eth triangle, and $\left|{k}_{{T}_{e}}\right|$ is its area. h

_{e}, q

_{j,e}, and ∇

_{j}H

_{e}in Equation (6) are the water depth, the flow rate per unit width, and the component of the gradient of H in the same element in the j direction, respectively, computed at the end of the time step. F

_{l,e}is the flux across side l of cell e, computed as in Equations (A.22) and (A.23) in Part A in Supplementary Materials. A positive (negative) value of ci

_{j}

_{,e}means that the absolute value of the leaving momentum flux is greater (lower) than the one of the incoming momentum flux in the cell. The way we compute the gr

_{j,e}term in Equation (6) is consistent with the numerical procedure, where, in both the prediction and correction steps, the gradients of h and z

_{b}are merged in the gradient of H.

_{j,e}terms of the 0ISWEsM, given as in Equation (6), using the values of h and q

_{j}components of the triangular element e.

## 5. Test 1. Steady Flow in a 1D Channel with Undulating Bottom Profile

_{b}(x) can be easily obtained by integrating Equation (8).

^{1/3}, uh = 2 m

^{2}/s, and two values of L, 500 m and 2500 m. According to the exact and computed (by the FDSWEsM) values of Fr (see Table 1 and Table 2), the flow is subcritical in the entire domain (Fr ranges from 0.44 to 0.78). The boundary conditions are the same for both the numerical solvers, i.e., the known flow rate at the upstream end and h from Equation (7) at the downstream end.

_{b}, h and H are obtained for the shortest channel (shortening L from 2500 to 500 m, we reduce by the same factor both the channel and the wave lengths in Equation (7)). This test was presented in [47], where the authors apply their zero-convective inertia model, which becomes a 0ISWEsM for steady flow. They analyze separately the effects of L and Fr on the scatters between the solution of their 0ISWEsM model and the fully dynamic one (Figures 7 and 8 and related comments in [47]). They do not investigate the role of the terms in the momentum equations and do not quantify the value of the inertial term compared to the other terms in the momentum equation of the fully dynamic model.

_{0}and S

_{f}the bottom and the friction slope,

_{FD}–(dh/dx)

_{0I}. We observe that (1) the maximum absolute values of the ci term correspond to the maximum scatters between the dh/dx computed by the two solvers, and (2) similar values of dh/dx are computed by the two solvers also for significant values of Fr.

_{b}in case 2 are smoother than in case 1, and it is reasonable to expect smaller changes of the longitudinal flow velocity, flux, and momentum flux along the channel. Even if the range of Fr is the same as in case 1, due to reduced absolute values of the ci term (approximately 1/4 smaller than in case 1), the solution of the 0ISWEsM is closer to the exact one, compared to case 1.

_{1}and L

_{2}norms of the relative errors of h and uh with respect to the exact solution, computed by the two present solvers over the unstructured meshes. The convergence order r

_{c}shown in Table 1 and Table 2 is computed as in Equations (49) and (50) in [9] and Equation (70) in [40]. As expected, the norms of h computed by the 0ISWEsM for case 1 are much greater than the ones given by the FDSWEsM.

^{1/3}and uh = 0.5 m

^{2}/s. These runs are referred to as cases 3 and 4, for the 500 m and the 2500 m long channel, respectively. For simplicity we show the L

_{1}and L

_{2}norms of the relative errors and r

_{c}in Figure S1 in Supplementary Materials. According to the numerical computed results, the absolute value of the ci term is less than 1/6 of the other terms in the momentum equations of the FDSWEsM. This is due to the assigned values of (uh,n), respectively smaller and greater than in the previous series of numerical runs. The small effect of the inertial term is the reason why the values of the norms of the 0ISWEsM are very similar to the ones in the FDSWEsM.

_{c}approaches one, due to the spatial approximation order of the unknown variables inside the computational cells [9,22,39,40].

## 6. Test 2. Rain in a 1D Channel

_{out}at the downstream end of the flume. The value of n suggested by the author is 0.009 s/m

^{1/3}, consistent with the material of the bottom, and referred to as n

_{0}. More details on the experiments can be found in [49]. We simulated the experiments with rain duration 10 s and 30 s, referred to as case 1 and case 2, respectively, whose details are listed in Table 3. The flow rate per unit width corresponding to the rain applied to each reach of the channel is computed by multiplying the rain intensity of each reach by the length of the reach (see Table 3). This is the p source term in Equations (1) and (5).

^{2}). The time step size is 0.1 s. We assumed all the boundaries to be impervious walls, except the downstream side, where we assumed the boundary conditions specified as follows. In the FDSWEsM, if the flow is supercritical in the cells at the downstream side, no boundary conditions have to be imposed, otherwise we assume the critical water depth value (which depends on the value of the specific flow rate in the downstream cells). The latter condition is assumed in the 0ISWEsM, where the flow is always subcritical. Further details on the boundary conditions can be found in [9,40] for the FDSWEsM and 0ISWEsM, respectively.

_{opt}, i.e., that value which minimizes the L

_{1}, L

_{2}, and L

_{inf}norms of the relative errors of q

_{out}between the computed and measured data. This is quite similar to the concept proposed in [50], where the authors change the Manning coefficient in order to compute “at large scale point of view”, a flow field equivalent to the one obtained assuming the effects of the convective accelerations. For each of these simulations, we assumed a spatial uniform value of the n coefficient, selected in the range 0.007–0.019 s/m

^{1/3}, and, in successive runs, changed by steps of 0.000005 s/m

^{1/3}. The functions L

_{1(2)}(n) are shown in Figure S3 (in Part B in Supplementary Materials). The n

_{opt}value of the FDSWEsM is essentially equal to n

_{0}, whereas the n

_{opt}values of the 0ISWEsM range over a wider interval (see Figure S3). The n

_{opt}values are listed in Table 4. Generally, we observe that the values of the norms of the FDSWEsM are smaller than the ones of the 0ISWEsM, and the results of the 0ISWEsM are more affected by the value of n compared to the ones of the FDSWEsM. In the range of physically feasible values of n, indeed, the curvature of the functions L

_{1(2)}(n) of the 0ISWEsM is greater than the one of the functions of the FDSWEsM.

_{opt}value (these results are marked with “FDSWEsM” and “0ISWEsM”, respectively). We also provide the results of the 0ISWEsM obtained with n

_{0}(results marked with “0ISWEsM 1”).

_{0I}”, respectively. Before the end of the rain, in the upstream portion of the domain, $\sum}_{term}^{FD$ is significantly higher than bfr

_{0I}. We could regard the FDSWEsM as a 0ISWEsM with an “augmented” bottom resistance, due to the effects of the two inertias which decelerate the flow. After the end of the rain (e.g., at 20 s), $\sum}_{term}^{FD$ and bfr

_{0I}have similar values, and this is consistent with the balancing of the inertial terms, which assume opposite sign and similar absolute values, observed as above. When the effects of the inertial terms become negligible, the two solvers have similar behavior irrespective of the value of Fr.

_{0I}term computed by the 0ISWEsM 1 has trend and values very similar to the ones plotted in Figure 3, and for simplicity are not shown.

_{opt}and of the 0ISWEsM 1 are 2.49 and 2.42, respectively.

_{out}. The FDSWEsM matches reasonably well the measures for both tests. The 0ISWEsM fails in simulating the profile of q

_{out}for the shortest rain duration, and anticipates the discharge for the rain duration of 30 s.

_{1}and L

_{2}norms computed by the FDSWEsM for case 1 differ significantly from each other (see Figure S3), and the reason could be the sharp profile of q

_{out}due to the arrival of the contribution of the upstream reach. On the contrary, probably due to the smoother computed profiles with respect to the measures, the two norms of the 0ISWEsM have similar values (see Figure S4).

^{1/3}, respectively. The time of the peak of q

_{out}simulated by the two literature models in [51,52] is slightly higher than the experimental one, whereas it is anticipated by our 0ISWEsM. The authors in [52] motivate the difference between their computed and measured q

_{out}as “the maximum values are not accurately reproduced due to the diffusive nature of the approximation”. We motivate the reason why the present 0ISWEsM fails in simulating the measures as follows. From the analysis of the values of u and q

_{out}predicted by the two present solvers for case 1, it seems that the 0ISWEsM does not properly reproduce the superposition of the effects of the discharge from the three reaches. According to the measures, the contribution of the upstream reach arrives at the downstream end of the channel approximately after 25 s (see Figure 4a), much later than the contributions of the other two reaches. This implies that the contributions of the three reaches do not add up and arrive separately at the end of the channel. The present 0ISWEsM, which predicts, during the rain, a faster flow field compared to the FDSWEsM (described as above), anticipates the arrival of the flow discharge of the upstream reach and this is added to the contributions of the other two reaches.

_{out}in case 2, with higher rain duration, suggests that the contributions of the three portions of the channel reach the downstream end during the rain event. Even if the present 0ISWEsM predicts higher flow velocity before the end of the rain, we observe smaller differences between the measured q

_{out}and the one computed by this solver, compared to case 1.

_{out}computed by the FDSWEsM for case 1 (see Figure 4b), whereas in the other scenario the results simulated over the refined meshes are very close to the ones obtained over the starting mesh.

## 7. Test 3. Rainfall in a 2D Catchment

^{1/3}, consistent with the material of the bottom, referred to as n

_{0}. The configuration of the bottom slope generates one impluvium between zones 2 and 3 and two impluviums, between zones 1 and 2 and zones 1 and 3, respectively. This simple geometry was complicated by the authors in [53] placing two walls inside the basin (in Figure 5a). We discretize the domain with a GD mesh with 14,735 triangles and 7647 nodes (the mean area of the triangles is 3.1d–04 m

^{2}). The gauges P1, ..., P6 in Figure 5a, have been used for the comparison of the results predicted by the two models.

^{3}/s. The authors in [53] measured only the outflow discharge q

_{out}. More details on the experiments can be found in [53]. In our numerical simulations we excluded the two walls from the computational domain, and their contours were regarded as impervious boundaries with imposed free slip condition. We compensate the rain falling over the two walls by increasing the rain intensity over the other portions of the basin by the ratio between the area of the walls and the total area of the basin. At the downstream external side of the domain (bottom side), we applied the boundary conditions for the two solvers as described for test 2, and the other three external sides were considered as impervious walls, with assigned free slip condition. We computed the n

_{opt}values of the two models in the same way as in test 2, investigating the range of n values 0.007–0.03 s/m

^{1/3}. The n

_{opt}value of the FDSWEsM essentially corresponds to n

_{0}, whereas the n

_{opt}of the 0ISWEsM is 0.017 s/m

^{1/3}, much higher than the value consistent with the material of the bed surface. In Figure S4 (in Part B in Supplementary Materials) we plot the functions L

_{1(2)}(n) of the relative error of q

_{out}.

_{out}. The nomenclature is the same as for test 2. Due to the uncertainties of the measurements listed by the authors in [53], we assume that both models with their n

_{opt}values satisfactorily match the registrations and properly reproduce the rising and falling limb, as well as the peak values. They accurately simulate the stop and restart of the rain. The 0ISWEsM 1 badly reproduces the measures, anticipates the outflow discharge, overestimates the peak values, and does not properly reproduce the stop/restart of the rain.

_{opt}value for the simulations of the 0ISWEsM.

_{0I}terms of the 0ISWEsM and 0ISWEsM 1 at 52 s. As for test 2, we could regard the FDSWEsM as a zero-inertia solver with a different bottom friction coefficient, due to the effects of the inertial terms. Generally, in some portions of the domain, e.g., as along the central impluvium, the absolute value of bfr

_{0I}of the 0ISWEsM 1 is smaller than the absolute value of ${\sum}_{term}^{FD}$. This could be the reason for the higher values of the flow velocity computed by the 0ISWEsM 1, compared to the values of the fully dynamic model. The alternating sign of ${\sum}_{term}^{FD}$ in both x and y directions, downstream of the walls and in the central portion of the domain could be motivated as follows. Because of the bottom slope, the values of h in contiguous cells are different, and this affects both the values of the fluxes and the momentum fluxes across the sides of the cells, and, consequently, also the sign of the ci terms (see the last of Equation (6)). This is confirmed by the numerical computed values of the ci terms.

_{out}provided by the two solvers, and, assuming as ‘reference solution’ q

_{out}

_{,ex}the results of the FDSWEsM, we computed the Nash-Sutcliffe efficiency coefficient of the q

_{out}of the 0ISWEsM, E

_{0I}[54]. In Figure 9a, we plot the threshold curves r vs. n, whose meaning is as follows. For fixed n, we expect that E

_{0I}is not smaller than 0.85, if the parameter r of the applied hydrograph does not exceed the value on the curve corresponding to the n coefficient. The parameter of the curves is the maximum value of Fr computed for each series of simulations. Despite the simplicity of this approach limited to geometry of the lab flume in [53], the importance of the above study is that the 0ISWEsM can be regarded as a ‘reliable substitute’ of the FDSWEsM only in cases with rough enough bottom surfaces and hydrological events with a very smooth rising limb. As expected, for fixed r, the value of n for which the two models compute similar solutions, increases with Fr. For n less than 0.015 s/m

^{1/3}, E

_{0I}is smaller than 0.85, regardless of the values of Fr and r.

## 8. Test 4. Laboratory Scaled Physical Model of the Toce River

^{3}/s, (corresponding, at the real scale, to 35,000 m

^{3}/s), and a total duration of 180 s (in Figure 11 (left)) [55]. This event caused the overtopping of the levee of the central reservoir [55]. The authors in [36,38] assumed a single value for the Manning coefficient, 0.016 s/m

^{1/3}, suggested in [55] according to the material of the bottom of the lab flume, and corresponding to 0.035 s/m

^{1/3}at the real scale. This value of n is referred to as n

_{0}. The authors in [36,38] assert that, (1) except for a few local effects not properly simulated, the zero-inertia model correctly predicts the overall propagation of the impulsive wave; and (2) the computed arrival times of the wavefront at the gauges, the water level peaks and the spatial distribution of the water level are similar to the ones provided by other literature fully dynamic SWEs models.

^{2}for the coarse mesh, and from 0.00236 to 0.0019 m

^{2}for the fine mesh. The time step sizes ∆t are 0.03 and 0.02 s, for the simulations performed over the coarse and the fine mesh, respectively.

_{0}. The maximum values of the CFL number in the simulations of the FDSWEsM are 2.15 and 3.78, for the coarse and fine mesh, respectively. The corresponding values obtained with the 0ISWEsM are 4.8 and 6.417, respectively.

_{1}, L

_{2}, and L

_{inf}norms of the relative errors of the two solvers, computed with respect to the measures. As expected, the norms of the FDSWEsM have generally smaller values compared to the ones of the 0ISWEsM.

_{1}and L

_{2}norms of the 0ISWEsM, generally being twice or more compared to the corresponding ones of the FDSWesM. The values of the L

_{inf}norms of the 0ISWEsM, greater than the ones of the FDSWEsM for several gauges, confirm that the 0ISWEsM generally underestimates the measures.

_{0I}at 25 and 50 s. In several parts of the domain the absolute value of ${\sum}_{term}^{FD}$ is greater than bfr

_{0I}, and, in the light of the argumentations in tests 2 and 3, we could explain the higher values of the flow velocity predicted by the 0ISWEsM. The authors in [36] did not investigate the effects of the inertial terms of their proposed FDSWEsM.

^{1/3}. We performed several simulations, and, for each simulation we used a single value of n and, for each gauge, we computed the optimal value n

_{opt}which minimizes the L

_{1}, L

_{2}, and L

_{inf}norms of the relative error of H for that gauge. In Table 5 (right) and Table S1 we show, for all the gauges, the n

_{opt}values and the corresponding computed norms, respectively. The n

_{opt}values of the FDSWEsM range between 0.0155 and 0.017 s/m

^{1/3}, and for most of the gauges they are very close to n

_{0}, whereas for the 0ISWEsM they range in the much wider interval 0.008–0.05 s/m

^{1/3}(see Table 5 (right)). The water levels computed by the FDSWEsM at the gauges, setting n

_{opt}, if different from n

_{0}, are very close to the ones plotted in Figure 16 and for simplicity are not shown.

_{opt}at the gauges P9 and P12 are very similar (Table 5 (right)). The maximum water level simulated at P9 with n

_{opt}reaches the top of the levee (see Figure 16), but the overtopping occurs in a downstream portion of the levee, close to the intake of the reservoir, where the topographic level is smaller. The different position of the overtopping is also confirmed by the time delay of the arrival of the front at gauge P12, compared to the measured one (see Figure 16). The overtopping of the reservoir does not occur for values of the Manning coefficient smaller than the n

_{opt}value of gauge P9. The values of n

_{opt}at gauges P23–P26 are significantly smaller than the one at gauge P9. This confirms that the overtopping/not overtopping is not the only one factor determining the dynamic of the wave propagation in the downstream portion of the domain.

_{1}, L

_{2}, and L

_{inf}norms of the FDSWEsM computed setting n equal to n

_{0}, are 2.32%, 1.92%, and 2.7%, respectively, and occur for gauge P12. The more refined discretization of the topography could slightly modify the prediction of the volume of incoming water to the central reservoir. The maximum differences in the norms of the 0ISWEsM computed for n equal to n

_{0}, are 1.06%, 0.84%, and 1.14%, respectively. The n

_{opt}values of the two present solvers remain essentially the same as the ones obtained over the coarse mesh.

_{ex}the one computed by the FDSWEsM, we computed for all the gauges the Nash-Sutcliffe efficiency coefficient E

_{0I}of the 0ISWEsM. In Figure 18 we plot the threshold curves r vs. n, whose parameter, as for Figure 9b, is the maximum value of Fr computed for each series of simulations. The meaning of the curves is as follows. For fixed n, we expect that the maximum value of the E

_{0I}coefficients of all the gauges is not smaller than 0.85, if the parameter r of the applied hydrograph does not exceed the value on the curve corresponding to the n value. The 0ISWEsM can effectively substitute the FDSWEsM only for events with very smooth rising limb and over very rough bottom surfaces, whose n coefficients, at the real scale, could correspond to unrealistic values. For instance, in Figure 19 we plot the flooded areas and the water depths computed by the two solvers at 100 and 180 s for an event with r = 0.00088 m

^{3}/s

^{2}(peak of the flow rate 0.0892 m

^{3}/s and time to peak 90 s) and n = 0.03 s/m

^{1/3}(0.065 s/m

^{1/3}at the real scale).

_{0I}, (see Figure 20, where for brevity we only plot the x components of these terms) and, as explained as above, we could explain the higher values of the flow velocity predicted by the 0ISWEsM. For this scenario we computed the values of the n

_{opt}for the 0ISWEsM which minimize the scatters (in terms of the L

_{2}norms of the relative errors) with respect to the results of the FDSWEsM. For some of the gauges we obtain n

_{opt}values quite different from the imposed one (0.03 s/m

^{1/3}). For instance, the n

_{opt}values for gauges S6S, P9, P24, P25, and P26 are 0.019, 0.033, 0.035, 0.039, and 0.034 s/m

^{1/3}, respectively.

## 9. Investigation of the Computational Costs

_{cc}is the number of computational cells (triangles or nodes in FDSWEsM or 0ISWEsM, respectively). In Figure 21 we plot the laws in Equation (11) required for the solution of the FDSWEsM and 0ISWEsM for test 1, case 1, and the grow with N

_{cc}is only a bit more than linearly in the 0ISWEsM (β = 1.019) and a bit less than linearly in the FDSWEsM (β = 0.987). The ${\overline{CPU}}_{u}$ is almost constant with the growth of N

_{cc}, for both solvers (see the values in the table in Figure 21). The reason is motivated in the reference papers [9,40,41] for the 0ISWEsM and FDSWEsM, respectively.

^{−4}s. This computational time is much greater than the ones required by the present 0ISWEsM (approximately 380 s and 864 s, over the coarse and the refined mesh, respectively), even if the number of triangles in the computational mesh adopted in [38] is approximately half of ours.

## 10. Conclusions

^{3}/s

^{2}and n = 0.03 s/m

^{1/3}).

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Test 1. (

**a**) exact and computed water depth h, case 1, (

**b**) the same for case 2, (

**c**) terms in the momentum equations of FDSWEsM and Fr (on secondary axis), case 1, (

**d**) the same for case 2, (

**e**) difference of dh/dx computed by the two solvers, case 1, (

**f**) the same for case 2 (h’ = dh/dx).

**Figure 2.**Test 2, case 1. (

**a**,

**d**,

**g**) terms in the momentum equations and Fr (on the secondary scale) at 6 s, 10 s and 20 s, respectively). (

**b**,

**e**,

**h**) computed profiles of h at 6 s, 10 s and 20 s. (

**c**,

**f**,

**i**) computed profiles of u at T = 6 s, T = 10 s and T = 20 s.

**Figure 3.**Test 2, case 1. Comparison between $\sum}_{term}^{FD$ and bfr

_{0I}(bfr

_{0I}is computed with the n

_{opt}of the 0ISWEsM) (

**a**) T = 6 s, (

**b**) T = 10 s, (

**c**) T = 20 s.

**Figure 4.**Test 2. (

**a**) measured and computed q

_{out}; (

**b**) effects of the mesh refinement on the computed solution of the FDSWEsM for case 1. Measures in [49].

**Figure 5.**Test 3. (

**a**) geometry of the experimental basin (dimensions in meters); (

**b**) measured and computed q

_{out}. Measures in [53].

**Figure 6.**Test 3. (

**a**) computed h and vectors of unitary flow rate, FDSWEsM; (

**b**,

**c**) the same for the 0ISWEsM and 0ISWEsM 1, respectively; (

**d**) zoom of the vectors of the flow rate, FDSWEsM; (

**e**,

**f**) the same for the 0ISWEsM and 0ISWEsM 1, respectively. T = 52 s.

**Figure 7.**Test 3. Computed values of h (

**a**) and modulus of the flow velocity (

**b**) at some of the gauges.

**Figure 8.**Test 3. Comparison of the bfr

_{0I}and $\sum}_{terms}^{FD$, T = 52 s. (

**a**) bfr

_{0I,x}of 0ISWEsM 1; (

**b**) bfr

_{0I,x}of 0ISWEsM; (

**c**) $\sum}_{terms,x}^{FD$ of FDSWEsM; (

**d**) bfr

_{0I,y}of 0ISWEsM 1; (

**e**) bfr

_{0I,y}of 0ISWEsM; (

**f**) $\sum}_{terms,y}^{FD$ of FDSWEsM.

**Figure 10.**Test 4. Digital Elevation Model of the Toce valley [55].

**Figure 11.**Test 4. (

**a**) The experimental hydrograph [55]. (

**b**) Digital Elevation Model and coarse mesh near the central reservoir.

**Figure 14.**Test 4. Computed vectors of the unitary flow rate upstream of the central reservoir. (

**a**) FDSWEsM, (

**b**) 0ISWEsM. T = 40 s.

**Figure 15.**Test 4. Computed vectors of the unitary flow rate close to gauges P24–P25. (

**a**) FDSWEsM, T = 70 s; (

**b**) FDSWEsM T = 95 s; (

**c**) 0ISWEsM T = 70 s.

**Figure 16.**Test 4. Measured and computed water levels at the gauges. Measures from [55].

**Figure 19.**Test 4. Event with r = 0.00088 m

^{3}/s

^{2}and n = 0.03 s/m

^{1/3}. Computed maps of inundations.

**Figure 20.**Test 4. Event with r = 0.00088 m

^{3}/s

^{2}and n = 0.03 s/m

^{1/3}. Σ

_{terms}

_{,x}and bfr

_{x}

_{,0I}at 100 s.

**Figure 21.**(

**a**) ${\overline{CPU}}_{}$ and ${\overline{CPU}}_{u}$ for test 1, case 1 and test 4. (

**b**) Grow rate laws in Equation (11) (test 1, case 1).

**Table 1.**Test 1. L

_{1}and L

_{2}norms of the relative errors for h and uh of the 0ISWEsM. Unstructured mesh. (Abbreviations. L

_{1(2),h}are the L

_{1(2)}norms of h. L

_{1(2)}, uh are the L

_{1(2)}norms of uh. r

_{c}is the convergence order).

r_{c} | L_{1,h} | r_{c} | L_{1,uh} | r_{c} | L_{2,h} | r_{c} | L_{2,uh} |

case 1 | |||||||

0.323 | 0.0311 | 0.350 | 0.0031 | 0.384 | 0.0526 | 0.348 | 0.0053 |

0.520 | 0.0248 | 0.464 | 0.0024 | 0.481 | 0.0403 | 0.496 | 0.0042 |

0.615 | 0.0173 | 0.660 | 0.0017 | 0.597 | 0.0289 | 0.577 | 0.0030 |

0.0113 | 0.0011 | 0.019 | 0.0020 | ||||

case 2 | |||||||

0.712 | 0.0070 | 0.502 | 0.0028 | 0.799 | 0.0126 | 0.476 | 0.0041 |

0.817 | 0.0043 | 0.634 | 0.0020 | 0.844 | 0.0073 | 0.694 | 0.0029 |

0.991 | 0.0024 | 0.876 | 0.0013 | 1.083 | 0.0040 | 0.807 | 0.0018 |

0.0012 | 0.0007 | 0.0019 | 0.0010 |

**Table 2.**Test 1. L

_{1}and L

_{2}norms of the relative errors for h and uh of the FDSWEsM. Unstructured mesh. (Abbreviations as for Table 1).

r_{c} | L_{1,h} | r_{c} | L_{1,uh} | r_{c} | L_{2,h} | r_{c} | L_{2,uh} |

case 1 | |||||||

0.949 | 0.0036 | 0.987 | 0.0023 | 0.996 | 0.0059 | 0.907 | 0.0049 |

1.055 | 0.0019 | 1.059 | 0.0011 | 1.030 | 0.0030 | 0.927 | 0.0026 |

1.065 | 0.0009 | 1.026 | 0.0006 | 1.012 | 0.0015 | 1.077 | 0.0014 |

0.0004 | 0.0003 | 0.0007 | 0.0006 | ||||

case 2 | |||||||

0.971 | 0.0036 | 0.994 | 0.0023 | 0.948 | 0.0060 | 0.963 | 0.0049 |

1.047 | 0.0019 | 1.089 | 0.0011 | 1.050 | 0.0031 | 1.048 | 0.0025 |

1.103 | 0.0009 | 1.030 | 0.0005 | 1.001 | 0.0015 | 1.007 | 0.0012 |

0.0004 | 0.0003 | 0.0007 | 0.0006 |

Case | Rain Duration (s) | Geometrical and Hydrological Data | Reach 1 (0–8 m) | Reach 2 (8–16 m) | Reach 3 (16–24 m) |
---|---|---|---|---|---|

1 | 10 | slope channel (–) | 0.02 | 0.015 | 0.01 |

2 | 30 | rain intensity (mm/h) | 3890 | 2300 | 2880 |

flow rate (m^{2}/s) | 0.025933 | 0.015333 | 0.0192 |

Case | FDSWEsM | 0ISWEsM | ||||
---|---|---|---|---|---|---|

L_{1} | L_{2} | L_{inf} | L_{1} | L_{2} | L_{inf} | |

1 | 0.0090512 | 0.00905122 | 0.0090725 | 0.0112 | 0.01125 | 0.01125 |

2 | 0.0091203 | 0.009124 | 0.009214 | 0.01325 | 0.01326 | 0.01326 |

**Table 5.**Test 4. Differences of the computed and measured arrival time of the wavefront at the gauges (left). n

_{opt}values for the two solvers which minimize the L

_{1}, L

_{2}L

_{inf}norms of the relative errors at each gauge (right).

Gauge | ∆t of the Arrival Time of the Front [s] | n_{opt} Values [s/m^{1/3}] Which Minimize the Norms | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

FDSWEsM | 0ISWEsM | FDSWEsM | 0ISWEsM | |||||||

n0 | n_{opt} | n0 | n_{opt} | L_{1} | L_{2} | L_{inf} | L_{1} | L_{2} | L_{inf} | |

P1 | 0.01 | 0.01 | 0.01 | 0.01 | 0.016 | 0.0155 | 0.0142 | 0.016 | 0.015 | 0.014 |

P2 | 0.01 | 0.01 | 0.02 | 0.02 | 0.0162 | 0.0161 | 0.0168 | 0.022 | 0.022 | 0.022 |

P3 | 0.9 | 0.9 | 1.2 | 1.2 | 0.0161 | 0.0161 | 0.017 | 0.018 | 0.016 | 0.022 |

S4 | 0.1 | 0.1 | 0.1 | 0.1 | 0.0164 | 0.0166 | 0.016 | 0.014 | 0.014 | 0.014 |

P4 | 0.9 | 0.9 | 0.8 | 0.8 | 0.016 | 0.0162 | 0.016 | 0.016 | 0.016 | 0.016 |

S6S | 0.01 | 0.01 | 1.4 | 2 | 0.016 | 0.0162 | 0.017 | 0.008 | 0.008 | 0.008 |

S6D | 0.04 | 0.04 | 0.04 | 0.02 | 0.016 | 0.0161 | 0.0164 | 0.022 | 0.023 | 0.019 |

P5 | 0.45 | 0.1 | 0.7 | 0.1 | 0.016 | 0.016 | 0.0163 | 0.028 | 0.029 | 0.032 |

P8 | 0.2 | 0.1 | 0.5 | −0.8 | 0.0158 | 0.0165 | 0.0165 | 0.027 | 0.028 | 0.027 |

S8D | −0.6 | −0.6 | 0.9 | −0.9 | 0.0159 | 0.0161 | 0.0164 | 0.035 | 0.035 | 0.035 |

P9 | 0.01 | 0.01 | 0.01 | −16 | 0.016 | 0.0162 | 0.0167 | 0.04 | 0.039 | 0.017 |

P10 | −1 | −1.1 | 0.5 | −7 | 0.016 | 0.0164 | 0.0163 | 0.038 | 0.039 | 0.03 |

P12 | −1 | −1.1 | no | −22 | 0.016 | 0.0162 | 0.0165 | 0.04 | 0.04 | 0.038 |

P13 | 0.6 | 0.4 | 0.8 | −5 | 0.01607 | 0.0162 | 0.0164 | 0.028 | 0.028 | 0.025 |

P18 | −0.8 | −0.78 | 0.2 | −4 | 0.01602 | 0.0162 | 0.0161 | 0.025 | 0.025 | 0.022 |

P19 | −0.9 | −0.92 | 0.8 | −2 | 0.01602 | 0.016 | 0.0161 | 0.022 | 0.022 | 0.022 |

P21 | −6 | −6 | 0.02 | −14 | 0.016 | 0.0161 | 0.0162 | 0.031 | 0.031 | 0.028 |

P23 | 0.1 | 0.11 | 3 | −10 | 0.0161 | 0.0162 | 0.0163 | 0.025 | 0.03 | 0.022 |

P24 | −0.6 | −0.9 | 4 | −15 | 0.0165 | 0.0171 | 0.0163 | 0.027 | 0.031 | 0.025 |

P25 | −2.3 | −3 | 14 | −21 | 0.0164 | 0.0168 | 0.0161 | 0.038 | 0.05 | 0.05 |

P26 | −0.7 | −0.7 | 8 | −10 | 0.016 | 0.0161 | 0.016 | 0.033 | 0.032 | 0.035 |

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**MDPI and ACS Style**

Aricò, C.; Nasello, C.
Comparative Analyses between the Zero-Inertia and Fully Dynamic Models of the Shallow Water Equations for Unsteady Overland Flow Propagation. *Water* **2018**, *10*, 44.
https://doi.org/10.3390/w10010044

**AMA Style**

Aricò C, Nasello C.
Comparative Analyses between the Zero-Inertia and Fully Dynamic Models of the Shallow Water Equations for Unsteady Overland Flow Propagation. *Water*. 2018; 10(1):44.
https://doi.org/10.3390/w10010044

**Chicago/Turabian Style**

Aricò, Costanza, and Carmelo Nasello.
2018. "Comparative Analyses between the Zero-Inertia and Fully Dynamic Models of the Shallow Water Equations for Unsteady Overland Flow Propagation" *Water* 10, no. 1: 44.
https://doi.org/10.3390/w10010044