# Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model
## 5 Model Validation and Case Analysis
### Model Application and GPU Acceleration Performance Evaluation
#### 5.2.1 Comparative Analysis of the Results of Different Simulation Methods  
Figure 8a shows the process of pressure wave curve change at valve C when the end valves C and D are simultaneously closed for the working condition shown in Figure 7. The working condition first runs steadily for 800 s, and then the valve closes linearly within the 60 s. This paper adopts the fifth-order WENO finite-volume method to take 0.01 and 0.005, respectively. As shown in Figure 8a, the calculation results of different schemes are almost the same, and the error of the extreme value point is within 1% when the WENO scheme and the MOC take 0.005. This means that the fifth-order WENO scheme for solving the pipeline water hammer wave can achieve a high accuracy under a very low Courant number. Figure 8b compares the solution results after taking GPU acceleration with the CPU program's results. The GPU is programmed with double-precision floating-point numbers to ensure calculation accuracy. The maximum error between the calculation result and the CPU program calculation result is 0.5%, significantly below the allowable error in actual engineering.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pipe Number** & **Pipe Length/km** & **Pipe Diameter/m** & **Roughness** & **Wave Speed/(m/s)** & **Initial Flow/(m\({}^{3}\)/s)** \\ \hline Pipe 1 & 39.75 & 0.981 & 0.013 & 996 & 0.47 \\ Pipe 2 & 39.75 & 0.981 & 0.013 & 996 & 0.47 \\ Pipe 3 & 99.40 & 1.389 & 0.012 & 1000 & 0.94 \\ Pipe 4 & 59.64 & 0.981 & 0.014 & 994 & 0.47 \\ Pipe 5 & 59.64 & 0.981 & 0.014 & 994 & 0.47 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The pipeline model parameters.  
Figure 7: Schematic diagram of the long-distance water pipeline.  
Figure 9 shows the flow and water hammer wave variations at different locations when the end valves C and D are simultaneously closed. As shown in Figure 9, due to the symmetry of points C and D location and the simultaneous closure of the valve, the pressure wave curves at valves C and D overlap entirely. Simultaneously, after the valve is closed linearly, the water hammer pressure wave reaches its maximum value (131.9 m). The flow rate at A gradually decreases to 0 with the closure of the valve, while the flow rate at B and C still decrease cyclical in the fluctuation changes due to the role of pipe friction, which gradually decreases the amplitude of fluctuations. Figure 10 shows the flow and pressure head variations at different locations when the valve is closed only at C (linear closure within the 60 s). As shown in Figure 10, after closing valve C, the pressure head at C suddenly rises, and its peak reaches 95 m, then oscillates down and stabilizes at 64.3 m. After about 40 min, due to the horizontal arrangement of the pipe, the flow rate in pipe 4 drops to zero, while the pressure head at point C is equal to that at point B. The maximum pressure head generated by the whole process system is still at the end of valve C. It can be seen that the downstream valve is still the most likely place to generate damage, even for a complex bifurcated piping system.  
Figure 8: Results of different simulation methods. (**a**) Comparison of accuracy of different methods. (**b**) Precision comparison of GPU acceleration methods.  
Figure 9: The flow and water hammer wave variations at different locations when the end valves C and D are closed. (**a**) The water hammer changes with time at different positions. (**b**) Flow at different locations varies with time.