5. A parallel Bayesian methodology for estimating upstream hydrographs in

5.8. Concluding remarks

Table 5.10. Parma gamma distributed inflow: Nash-Sutcliffe 𝐸, root mean square error RMSE and error in the peak discharge 𝐸𝑝 values.

𝐸 (-) 𝑅𝑀𝑆𝐸 (m3/s) 𝐸𝑝 (%) No random errors 99.99933 0.4894 -0.0407 Random errors 99.87651 6.6529 0.1451

stable, fast and accurate forward hydraulic model, arose. Moreover, the obtained results have highlighted that the implemented procedure well estimates the unknown inflow hydrographs with different shapes and in presence of corrupted observations: quantitative indicators have proved the accuracy of the methodology. In all the presented tests, the resulted Nash-Sutcliff efficiency criterion exceeded the 99%, the error in the peak discharge was less than 3% and the RMSE error less than 2%.

Future development of the methodology will focus on the possibility of reconstructing the flood waves also in presence of levee breaches and flooding outside the river region.

Conclusions

The thesis focused on the numerical modelling of flooding events involving rural and urban areas in the framework of explicit finite volume schemes based on the Shallow Water Equations (SWEs).

Dealing with the modelling of flooding events, the first goal was the reconstruction of a real event. Since a real flood involves not only the river channel but also the flood prone areas, a fully 2D model was necessary: particularly a second order finite volume scheme solving the SWEs on a Cartesian grid was adopted.

With the aim of capturing the natural and/or artificial elements, which influence the flooding dynamics, a high-resolution mesh was used. Meanwhile, the urban areas were described by means of a different roughness coefficient that was obtained through a calibration procedure.

Even if dealing with a 5 m mesh and a fully 2D-SWE scheme, the parallel implementation on GPUs allowed reaching reasonable computational times (ratio of physical time to runtime greater than 15).

The reconstruction of a real flooding event and the addressed challenges related to the adoption of a stable, accurate and fast numerical scheme, as well as proper input data (bathymetry, breach evolution, urban areas representation, etc.), was considered as a positive starting point for more detailed investigations.

Even if the numerical model was implemented in CUDA language as to take advantage of GPUs capabilities, the goal of reducing the computational times, or of simulating larger domains, still remained.

At this aim, the implementation of the multi-resolution Block Uniform Quadtree grids allowed reproducing small-scale effects while simulating large domains, and exploited the computational capability of GPUs with minimum overheads. As a result, a decrease of the computational costs was achieved (2-10 speed-ups for theoretical tests against Cartesian grids) and the flooding simulation over a 840 km2 domain achieved a ratio of physical to computational time equal to 12, whereas the adoption of Cartesian ones was not enable by the memory requirement of the Tesla K40 GPU.

With reference to urban areas, the BUQ grids allowed modelling these zones with a finer mesh, while describing the rural domain with a coarse one.

Focusing on urban environment description, the thesis investigated the SWEs with porosity, which aim at the global representation of building effects, without requiring for example a detailed information about the velocity field near the obstacles.

The first branch concerned the derivation of a well-balanced 2D model based on isotropic porosity that includes porosity by adding to the classical SWEs additional source terms. Different test cases showed the capability of the porosity model in reproducing the 1D and 2D derived reference solutions.

The second branch concerning the SWEs with isotropic porosity focused on the development of a numerical method capable of handling discontinuities in the porosity field. A 1D and 2D augmented Roe Solver was derived analyzing the physical meaning of a porosity discontinuity and enforcing mass and momentum conservation across it. The test cases showed that the proposed numerical procedure is capable of capturing different Riemann Problems, involving both shocks, rarefactions and transonic rarefactions.

Therefore, in the thesis, the issue related to urban area description in flooding models has been investigated or by modelling those regions with different roughness coefficient or BUQ grid type, or by analyzing dedicated approaches, such as the SWEs with porosity.

The last part of the thesis solved the inverse problem related to the inflow hydrograph estimation in an upstream-ungauged river section, from the knowledge of water level observations in downstream sites, adopting a Bayesian methodology, and some of the results previously assessed (a stable and fast 2D-SWE code, BUQ grids).

The implemented procedure well estimated discharge hydrographs with different shapes and in presence of corrupted observations. Moreover, by taking advantage of a HPC cluster with GPUs the parallel scheme reduced the computational time of a factor 8 against running the 2D-SWE code on a single GPU.

The results outlined in this thesis lead to the possibility of adopting a stable, accurate and fast numerical model, enabled with a proper urban area description, in order to produce hazard maps useful for flood risk definition. This kind of simulation tools opens up new perspectives in the devising and implementing of flood event management strategies for civil protection purposes and with the aim of minimizing the economic loss.

Additionally to these implications, further investigations may be performed, particularly on the porosity approaches and on the inverse problem, in order to model real cases and reconstructing the flood waves also in presence of levee breaches and flooding outside the river region, respectively.

Finally, a brief remark is pointed out regarding the urban area analysis in order to clarify that the thesis did not aim at the definition of the best approach, by testing different techniques (roughness calibration, grid type, classical vs porosity SWEs) on the same benchmark. On the contrary, the main purpose was the development of independently branches motivated by different challenges, which were interconnected one another, but each of one presented a theoretical derivation and/or implementation and required a proper validation procedure.

In document 2D Shallow Water modelling of flood propagation: GPU parallelization, non-uniform grids, porosity, reverse flow routing (Page 194-200)