## 4 Application of GPU-MOC to a Water Distribution Network System  
The transient process of an actual water distribution network system (Figure 4) in [PERSON]'s work [26] is simulated. The length of the original system (Table 2), namely Case 1, is about 9 km, with seven nodes along the pipeline, a surge tank at node 3, a relief valve at node 6, and a control valve at node 7 (partially closed with \(\tau\) = 0.6 initially). The transient phenomena are caused by closing the control valve from opening \(\tau\) = 0.6 to \(\tau\) = 0.2 within 10 s. The relief valve is initially closed, but it is set to open linearly within 3 s once the pressure exceeds 210 m and then it is closed linearly within 60 s. Details of other parameters can be found in [26]. As shown in Figure 5, the results of GPU-MOC and [PERSON]'s work at nodes 3, 6, and 7 show quite minor differences. Thus, the GPU-MOC method is reliable. The high pressure waves initiated at node 7 by the closing valve are propagated throughout the system. At \(t\) = 3 s water starts to flow into the surge tank and makes the tank water level rises gradually, and pressure at junction 3 fluctuates in a relative narrow range. The pressure relief valve exceeds 210 m at \(t\) = 6.4 s. When the valve opens, the release of fluid at that location reduces the pressure. The water through the relief valve at node 6 and the water level change in the surge tank at node 3 attenuate the transient pressure and allow the system to reach steady state quickly. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Section Numbers**} & \multicolumn{2}{c}{**GPU-MOC with Different Block Sizes**} \\ \cline{2-4}  & **64** & **128** & **256** \\ \hline \(2^{14}\) & 36 & 36.2 & 36.3 \\ \(2^{16}\) & 109.2 & 110.1 & 111.9 \\ \(2^{18}\) & 234.6 & 235.5 & 236.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Speedup ratios of GPU-MOC to CPU-MOC. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pipe Number** & **L1 (1 to 2)** & **L2 (2 to 3)** & **L3 (3 to 4)** & **L4 (3 to 5)** & **L5 (6 to 5)** & **L6 (2 to 6)** & **L7 (6 to 7)** \\ \hline Case 1 & 1001.2 & 2000 & 2000 & 502.5 & 502.5 & 1001.2 & 2000.2 \\ Case 2 & 10,012 & 20,000 & 20,000 & 5025 & 5025 & 10,012 & 20,002 \\ Case 3 & 100,120 & 200,000 & 200,000 & 50,250 & 50,250 & 100,120 & 200,020 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The length of pipe for different Cases. Figure 3: Performances of CPU-MOC in different section numbers and GPU-MOC in different section numbers with different block sizes. To demonstrate the strong computing capability of the GPU-MOC in computing intensive problems, the pipe length of the above Case 1 is increased tenfold and hundredfold, which are corresponding to Case 2 and Case 3 (Table 2), respectively. The pipe segment lengths \(\Delta x\) of Case 1, Case 2 and Case 3 are all set to 1 m, and the corresponding section numbers are 9014, 90,077, 900,767, respectively. The first 60 s transient behavior of the system caused by operating the control valve at node 7 is simulated. The block size is set to 256. The performances of GPU-MOC are much better than those of the CPU-MOC (Figure 6). The maximum performance for GPU-MOC is up to 568 MSUPS, but for CPU-MOC is only 3.6 MSUPS. In addition, the performances of GPU-MOC improve significantly as the scale of the system becomes larger. For Case 3, it updates 568 MSUPS, but for Case 1 is only 55 MSUPS. That is to say, the computing efficiency of Case 3 is more than 10 times to that of Case 1. Figure 4: Sketch map of the simple hydraulic network. Figure 5: Comparison of head and discharge at surge tank (Node 3), pressure relief valve (Node 6), and control valve (Node 7) with data from [PERSON] [26]. The consumed time of GPU-MOC is far less than that of CPU-MOC (Table 3). Especially in Case 3, the consumed time of GPU-MOC is less than two minutes, but for CPU-MOC is more than four hours, the achieved speedup ratio is up to 158. Therefore, finer calculation sections can be used to simulate the large scale problems by GPU-MOC method.