## 5 Application of GPU-MOC to a Long-Distance Water Transmission System  
The transient of a long distance water transmission project is simulated, in which the pumping part is from an example in the book [27] by [PERSON]. We modified it by adding a long trapezoidal channel between the upstream reservoir and the pump station (Figure 7). The parameters of the project are shown in Table 4. The discharge of the two parallel pumps is lumped together, and the suction lines being short are not included in the analysis. Pump trip is considered to be the cause of the transients, and there is no valve action. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Pipe** & **Pump** & **Trapezoidal Channel** \\ \hline Length = 1000 m & \(Q_{\text{R}}\) = 0.25 m\({}^{3}\)/s & Water level of reservoir 1 = 1.5 m \\ \(D\) = 0.75 m & \(H_{\text{R}}\) = 60 m & Length = 200 km, wide = 1 m \\ Wave speed a = 1000 m/s & \(N_{\text{R}}\) = 1100 rpm & Bottom slope = 0.0002 \\ \(Q\) = 0.5 m\({}^{3}\)/s & WR\({}^{2}\) = 16.85 kg\(\cdot\)m\({}^{2}\) & Side slopes 0.5 horizontal to 1 vertical \\ Friction factor = 0.012 & \(\eta_{R}\)= 0.84 & Manning roughness = 0.012 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computation parameters. Figure 6: Performances of GPU-MOC and CPU-MOC in different section numbers. Figure 7: Notation for open channel and pipe combined system. The differential equations for free-surface unsteady flow are solved also by MOC [25]. An interpolation procedure is necessary so that calculations can proceed along characteristic lines. Errors are introduced in the interpolation approximation, and high order interpolation are always adopted to reduce the errors. However, the consumed time and the storage space of the simulation for high order interpolation would increase significantly. To investigate the computing efficiency of the GPU-MOC method in high order interpolation, first-order and second-order Lagrange interpolations [28] are used in simulating the transient processes of free-surface unsteady flow of this system. Figure 8 presents the comparisons of the history curves of pump head, discharge, speed, and head of open channel at downstream by first-order and second-order interpolations. The results of pump parameters for the two interpolations are nearly the same, because the treatments for the pressurized pipe and pump are identical, and the water level change for the open channel is small, which has little impact on pump parameters. After power is lost, the reactive torque of the liquid on the impeller causes the rotational speed to decrease, which, in turn, reduces pressure head and discharge. As a result, the discharge at the channel downstream end that delivers to the pump reduces quickly, causing the water level rises. The water levels at the channel downstream end by the two interpolation methods are different due to interpolation errors. Once the pump reaches its reverse rotating steady-state conditions, the water level at the channel downstream end increases uniformly and slowly. The computing efficiency of the GPU-MOC in high order interpolation is analyzed. The pipe segment length is \(\Delta x=1\) m, time step is \(\Delta t=0.001\) s, and the corresponding section number is 201,000. The transient behavior in the system within 600 s after the pump trip is simulated. The block size is set to 256. GPU-MOC method shows a great advantage in computing intensive problem, especially in high order interpolations (Figure 9). For both first-order and second-order interpolations, the performances of GPU-MOC are much better than those of CPU-MOC. Moreover, for GPU-MOC, the second-order interpolation achieves more significant performance gains compared with the first-order interpolation. For first-order interpolation, the consumed time is 45.726 hours by CPU-MOC, but only 0.217 hour by GPU-MOC, and the achieved speedup ratio is 210.7. For second-order interpolation, the consumed time is 288.9 hours by CPU-MOC, but only 0.281 hour by GPU-MOC, and the achieved speedup ratio is up to 1027.