# A GPU-Accelerated Shallow-Water Scheme for Surface Runoff Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Numerical Scheme

**n**the outward unit vector normal to C,

**U**the vector of the conserved variables,

**H**= (

**F**,

**G**) the tensor of fluxes in the x and y directions,

**S**and

_{b}_{,}S_{f}**S**the source terms representing the bed slope, the frictional effect and the rainfall rate, respectively. A modified form [59] of SWEs is adopted here, and the vector terms are:

_{r}^{n}is the time step calculated accordingly to the Courant–Friedrichs–Lewy condition [58]. The term

**U**

_{i}

_{,j}

^{n}

^{+1/2}is obtained as:

**D**

_{i}(

**U**

_{i}

_{,j}

^{n}) is defined as:

**F**and

**G**at the intercells i ± ½ and j ± ½ in the x and y direction respectively, are evaluated by using the states at the cell faces as input in the Harten, Lax and van Leer approximate Riemann solver with the contact wave restored (HLLC). The HLLC scheme, which in many situations behaves like the exact Riemann solver, does not require an iterative scheme, is positive definite and generally computationally more efficient than other well-known solvers, is in fact accurate and robust, and preserves the entropy-satisfaction property of the HLL scheme [58,60]. To obtain the Riemann states, a surface reconstruction method (SRM [10]) with hydrostatic reconstruction [61] is used. The flux and the bed and friction slope source term computations are performed in the present approach by adopting the variables obtained from SRM and local bed modification treatment in order to overcome the numerical instabilities induced by the presence of very small depths related to runoff processes. Considering two adjacent cells in the x direction, i and i + 1, the SRM-reconstructed water surface elevations on the left- and right-hand sides of their common interface are given by:

#### 2.2. Evaluation of Rainfall Excess

**S**represents the inflow due to the net rainfall (Equation (1)). This term, variable in both space and time, can be obtained as the difference between rainfall r and infiltration rate f (Equation (2)).

_{r}^{−1}) at each cell or rain time series for predetermined areas (Thiessen polygons). Another way is to exploit the observations at the rain gauges present in the domain by computing the precipitation at each cell (in time) through the inverse distance squared approximation:

_{i}

_{,j}is the rainfall intensity at cell i, j, r

_{g}is the rainfall intensity measured by the gth rainfall gauge, d

_{g}

_{,i,j}is the distance from cell i to the gth rainfall gauge and N is the total number of rainfall gauges. Furthermore, a series of maps (e.g., radar maps) containing the amount of precipitation in time (with constant or variable time spacing) can be provided as input to be transformed to the rainfall rate. In the present model, the rainfall excess is evaluated through the application of the SCS Method, which allows the evaluation of the depth (mm) of the excess precipitation, P

_{e}, through a simple empirical formulation [63]. More details of the numerical implementation of the SCS approach are reported in Appendix A.

#### 2.3. Graphics Processing Unit (GPU) Implementation of Shallow-Water Equation (SWE)

^{TM}GPUs. The main aspects of GPU implementation are briefly recalled in this section, for details see [48,57].

#### 2.4. Block-Uniform Quadtree (BUQ) Grids

_{1}) and the coarsest one (level N, Δ

_{N}= 2

^{N}

^{−1}Δ

_{1}), i.e. each block contains cells with size equal to a power-of-two multiple of the minimum size in the domain. The most important criterion for grid generation is the fact that the variation in the resolution level between neighbor blocks cannot exceed one [51]. This idea is similar to quad-tree partitioning [65], the main difference being the fact that single cells are replaced by blocks of cells in the spatial organization of the quad-tree nodes. As an example, Figure 2 shows a detail of a BUQ grid. Starting from a seeding point where the maximum resolution is requested, four different resolutions are originated during pre-processing. The regular blocks (containing M × M cells each) are shown at the different levels of resolution.

## 3. Experimental and Synthetic Rainfall-Runoff Tests

#### 3.1. Test Case 1: One-Dimensional Three Planes Cascade Rainfall-Runoff Case

_{x1}= 0.02, S

_{x2}= 0.015 and S

_{x3}= 0.01 and length L = 8 m each (Figure 3). The three planes were subject to different rainfall intensities equal to r

_{1}= 3888 mm·h

^{−1}, r

_{2}= 2296.8 mm·h

^{−1}and r

_{3}= 2880 mm·h

^{−1}. The Manning coefficient n = 0.009 m

^{−1/3}s was suggested by the author [67], even though other researchers in the literature [68] considered a slightly different value of n = 0.01 m

^{−1/3}s to achieve better results in the numerical simulations. This accounts for possible uncertainties in the evaluation of the planes roughness coefficient at the time of the experiments. Both values of the Manning coefficient were considered here, and two different runs of the numerical model were performed with n = 0.009 m

^{−1/3}s (Parflood Rain, test I) and n = 0.01 m

^{−1/3}s (Parflood Rain, test II). In addition to the different values of the rainfall intensities above the three planes, different rainfall durations were also considered, namely t = 10 s (case A), t = 20 s (case B) and t = 30 s (case C). The experimental observations for the three test cases considered are compared in Figure 4 with the resulting hydrographs of unit discharge at the downstream boundary, obtained by the complete hydrodynamic model.

^{2}s

^{−1}). However, the investigator had ascribed this strange behavior to the fact that during the experimental runs the lateral inflow rate q achieved at each reach of the flume was not completely uniform. If the three anomalous values are ignored, the comparison between numerical results and observations for case A is quite good. The numerical results related to the adoption of the two different roughness coefficients proposed in the literature (Parflood Rain, test I and test II) show the sensitivity of the numerical computations to the Manning’s parameter adopted. Better results were achieved for case A with n = 0.009 m

^{−1/3}s and for case B with n = 0.01 m

^{−1/3}s.

#### 3.2. Test Case 2: Two-Dimensional V-Shaped Rainfall-Runoff Test Case

^{−1/3}s and 0.15 m

^{−1/3}s, respectively. The whole domain was subject to a uniform rainfall intensity equal to r = 10.8 mm·h

^{−1}for a duration of 1.5 h. A free outflow condition was imposed at the channel outlet.

^{6}to less than 0.3 × 10

^{6}(with a reduction of about 80%). As a result, the BUQ-based simulation performs 6.2 times faster than the Cartesian-based one (Table 1).

_{R}and speed-up S

_{U}defined as:

_{C}and T

_{C}are the same quantities referred to an identical simulation run using a Cartesian grid with size equal to the maximum resolution adopted in the BUQ grid considered.

## 4. Nure Watershed Field Case

^{2}, with a 23 km long main channel, a 943 m mean altitude above sea level, and a 0.248 mean slope.

#### 4.1. Event Description

^{−1}. The precipitation data were acquired every thirty minutes through weather radar and the reflectivity maps obtained were linked to the observations at the rainfall stations present in the drainage area (courtesy of the Regional Agency for Prevention, Environment and Energy of Emilia-Romagna, Italy). The storm center was located South-West of the Nure watershed, yet the total volume of rain on the river basin was over 45 × 10

^{6}m

^{3}(Figure 9).

#### 4.2. Available Field Data

#### 4.3. Bathymetry and Roughness

^{6}cells (4241 rows × 3725 columns) with over 8.35 × 10

^{6}active computational cells was chosen to achieve both acceptable accuracy in the geometrical representation (required by the presence of built up areas or infrastructures), and a limited computational burden. Different DEMs characterized by mesh sizes of 10, 20, 40 and 80 m were then obtained through grid decimation.

#### 4.4. Results for the Nure Watershed Test Case

^{6}to 0.88 × 10

^{6}from BUQ Grid 1 to BUQ Grid 3. As a consequence, a significant speed-up was achieved in the simulations adopting BUQ Grid 3 compared to the reference Cartesian mesh of 5 m × 5 m. The duration of the simulated event was 40 h.

_{C}was equal to 8.49 × 10

^{6}, while T

_{C}was 14.77 h with a simulation to physical time ratio of 0.37. Compression rates and speed-ups achieved with the different BUQ grids adopted are reported in Table 3. The maximum values of C

_{R}and S

_{U}, respectively equal to 9.65 and 7.86, were obtained in the simulation based on BUQ Grid 3, as expected. All simulations were performed using a P100 Tesla

^{®}GPU.

^{6}m

^{3}. The numerical results confirm that the two Sassi Neri check dams were completely submerged and circumvented to the left, as confirmed by the side erosion, and by the damage to the structure and to the wings observed after the event (Figure 11). The flow accelerated and became supercritical with maximum velocities over 10 ms

^{−1}downstream of the valley bottleneck originated by the presence of the landslide. Lower velocities of about 5 ms

^{−1}were predicted downstream of the bridge in Farini (right frame of Figure 18).

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### A.1. Bed Slope Source Term Treatment

_{h}represents the wet/dry threshold (here 10

^{−12}m) and the water surface elevations η

_{i,j}or η

_{i}

_{+1,j}are the original water surface elevations at cell center rather than the reconstructed value (which is used in Equations (A5) and (A6) instead).

#### A.2. Friction Source Term Treatment

**S**and

_{r}**S**. Like Equation (A9), the explicit formulation can be similarly obtained in the y direction.

_{f}#### A.3. Soil Conservation Service SCS Method

_{e}, of the initial abstraction I

_{a}before ponding, for which no runoff occurs, and of the precipitation depth retained in the watershed, F

_{a}:

_{a}can be evaluated through the empirical relation:

_{e}becomes:

_{∞}(mm) is related to the curve number CN by the expression:

_{a}calculated and stored at every time step Δt. The infiltration ΔF

_{a}for the current time interval can be obtained through a difference between the cumulative infiltration at times n and n–1. The related infiltration rate f is then computed as follows:

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**Figure 2.**Particular of a Block-Uniform Quadtree (BUQ) grid, with blocks, cells and a maximum resolution seeding point (yellow cross).

**Figure 4.**Results obtained through the complete hydrodynamic model (tests I and II) compared to the experimental data ([67]) for different rainfall durations: (

**a**) t = 10 s, (

**b**) t = 20 s and (

**c**) t = 30 s.

**Figure 6.**Multiresolution grid for the V-Shaped domain; 1 × 1 m cells in black, 2 × 2 m cells in blue, 4 × 4 m cells in green and 8 × 8 m cells in red.

**Figure 7.**Hydrographs at the end of each hillside and at the channel outlet for different configurations of the Parflood Rain model compared to the kinematic flow wave solution.

**Figure 8.**Nure watershed: (

**a**) administrative map of Italy with Emilia-Romagna Region highlighted and position of the Nure watershed; (

**b**) shaded relief map of the digital elevation model (DEM) of the Nure watershed closed at the Farini outlet.

**Figure 9.**Isohyet map of total rainfall (mm) for the event of interest obtained through the radar reflectivity, linked to rain gauges on the Trebbia and Nure. In red the boundary of the Nure watershed closed at the Farini outlet.

**Figure 11.**The Sassi Neri check dam after the flood event (from downstream). The flow circumvented and damaged the structure on the left.

**Figure 13.**Maps obtained from soil type and land use: (

**a**) Manning’s roughness map; (

**b**) infiltration index map.

**Figure 14.**Discharge hydrographs for uniform grid resolutions (5, 10, 20, 40 and 80 m) at: (

**a**) Cross Section 1; (

**b**) Cross Section 2, (

**c**) Cross Section 5 and (

**d**) Cross Section 6.

**Figure 16.**Water level hydrographs at the Farini bridge for the reference simulation, the multiresolution simulations (BUQ Grid 1, 2 and 3) and maximum water observed level.

**Figure 18.**Computed water surface elevation (left frame) and velocities (right frame) in correspondence with the flood peak; the region of supercritical flow is highlighted. In red are reported the observed maximum water levels.

**Figure 19.**Computed water depths for the whole Nure watershed from 02:00 UTC+1 to 05:00 UTC+1 on 14 September 2015: (

**a**) 02:00 UTC+1; (

**b**) 03:00 UTC+1; (

**c**) 04:00 UTC+1 and (

**d**) 05:00 UTC+1.

**Figure 21.**Detail of the water depth (

**a–d**) and velocity fields (

**e**–

**h**) at the confluence of two right tributaries of the Nure from 02:00 UTC+1 to 05:00 UTC+1 on 14 September 2015: (

**a**,

**e**) 02:00 UTC+1; (

**b**,

**f**) 03:00 UTC+1; (

**c**,

**g**) 04:00 UTC+1 and (

**d**,

**h**) 05:00 UTC+1. (Velocity vectors are plotted only one every five for the sake of graphical representation).

**Table 1.**Characteristics of simulations, compression rates C

_{R}and speed-up S

_{U}for the BUQ grid used in the V-shaped test case.

Grid | N (Millions) | T (Hours) | C_{R}(–) | S_{U}(–) |
---|---|---|---|---|

Cartesian 1 m × 1 m | 1.62 | 2.86 | – | – |

Cartesian 10 m × 10 m | 0.016 | 0.042 | – | – |

BUQ Grid | 0.29 | 0.46 | 5.59 | 6.21 |

Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

CN | 57 | 61 | 72 | 75 | 84 | 86 | 90 |

**Table 3.**Characteristics of simulations, compression rates C

_{R}and speed-ups S

_{U}for the different BUQ grids adopted.

Grid | N (Millions) | T (Hours) | C_{R}(–) | S_{U}(–) |
---|---|---|---|---|

Cartesian 5 m × 5 m | 8.49 | 14.77 | – | – |

BUQ Grid 1 | 4.12 | 7.66 | 2.06 | 1.93 |

BUQ Grid 2 | 1.55 | 2.93 | 5.48 | 5.04 |

BUQ Grid 3 | 0.88 | 1.88 | 9.65 | 7.86 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aureli, F.; Prost, F.; Vacondio, R.; Dazzi, S.; Ferrari, A.
A GPU-Accelerated Shallow-Water Scheme for Surface Runoff Simulations. *Water* **2020**, *12*, 637.
https://doi.org/10.3390/w12030637

**AMA Style**

Aureli F, Prost F, Vacondio R, Dazzi S, Ferrari A.
A GPU-Accelerated Shallow-Water Scheme for Surface Runoff Simulations. *Water*. 2020; 12(3):637.
https://doi.org/10.3390/w12030637

**Chicago/Turabian Style**

Aureli, Francesca, Federico Prost, Renato Vacondio, Susanna Dazzi, and Alessia Ferrari.
2020. "A GPU-Accelerated Shallow-Water Scheme for Surface Runoff Simulations" *Water* 12, no. 3: 637.
https://doi.org/10.3390/w12030637