# Accelerating FVM-Based Parallel Fluid Simulations with Better Grid Renumbering Methods
## 5 Results
### Impact of the Renumbering Algorithm  
This section focuses on the specific impact that the renumbering algorithm described above has on the numerical simulation calculations. In this section, the \(Valgrind\) third-party tool is used to simulate and test the access process in CFD calculations. \(Valgrind\) will simulate a large access space during the test; it will greatly increase the numerical simulation time by a factor of ten or more. Here, we select 270,000 grid, 800,000 grid and 2,240,000 grid (with parallelism of 12 cores, 16 cores and 40 cores, respectively) from \(FuelRodBundleFlow\) case for testing. The above three cases were iterated by \(Valgrind\) in 10 steps. The six standard test cases in YHACT are also used as test subjects, with grid sizes ranging from 2000 to 70,000. All examples can be found in the ANSYS FLUID DYNAMICS VERIFICATION MANUAL. The above six classic test cases were iterated by Valgrind in the same number of iteration steps. The feasibility of MDMP is further validated. In YHACT, the original grid numbering of the six test cases is optimal. The three \(FuelRodBundleFlow\) cases are large real-world engineering cases whose original grid arrangement is chaotic. Therefore, by renumbering these two types of cases, this section further verifies whether the MDMP metrics can correctly reflect the relationship between different numbering algorithms and the overall numerical simulation efficiency. Table 7 shows that the MDMP increases for six test cases and decreases for three \(FuelRodBundleFlow\) cases after processing by Greedy and CQ algorithms. The corresponding CFD numerical simulation time and cache miss rate also change. Figure 8 visualizes the correspondence between the elapsed time and the cache miss rate for each case iteration of 100 steps. The iteration time and cache miss rate of the original mesh for the six YHACT test cases are the smallest. The iteration times and cache miss rates of the meshes of the three A cases processed by [PERSON]'s algorithm are the smallest. The cache hit rate of CFD computation for all cases after CQ renumbering is lower than the grid after Greedy renumbering, and the computation time is longer. The above is consistent with the judgment of MDMP, that is, the smaller the MDMP of the cases, then the better the renumbering algorithm used. It can be concluded that, in the CFD numerical simulation by finite volume method, the MDMP of the sparse matrix can correctly reflect the cache hit rate during the computation, and thus judge the impact of different renumbering algorithms on the overall CFD numerical simu \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Grid Size**} & \multirow{2}{*}{**Core**} & \multicolumn{3}{c}{**Time**} \\ \cline{3-5}  & & **Original** & **Greedy** & **CQ** \\ \hline \multirow{3}{*}{270,000} & 12 & 367.24 s & 302.724 s & 322.315 s \\  & & (17.57\%) & (12.23\%) \\ \cline{2-5}  & 5 & 436.46 s & 397.684 s & 401.355 s \\  & & (8.88\%) & (8.04\%) \\ \hline \multirow{3}{*}{800,000} & 40 & 389.29 s & 306.8 s & 320.838 s \\  & & (21.19\%) & (17.65\%) \\ \cline{2-5}  & 16 & 974.57 s & 668.019 s & 708.43 s \\  & & (31.46\%) & (27.31\%) \\ \hline \multirow{3}{*}{2,240,000} & 100 & 1229.56 s & 1020.05 s & 1146.17 s \\  & & (17.04\%) & (6.78\%) \\ \cline{2-5}  & 40 & 1943.6 s & 1201.31 s & 1597.39 s \\  & & (38.19\%) & (17.81\%) \\ \hline \multirow{3}{*}{4,000,000} & 192 & 1147.78 s & 1020.49 s & 1092.3 s \\  & & (11.1\%) & (4.83\%) \\ \cline{2-5}  & 72 & 1159.9 s & 875.68 s & 932.46 s \\  & & (24.5\%) & (19.61\%) \\ \hline \multirow{3}{*}{8,648,000} & 384 & 1997.66 s & 1662.41 s & 1757.01 s \\  & & (16.78\%) & (12.08\%) \\ \cline{2-5}  & 160 & 5012.54 s & 4264.56 s & 4611.35 s \\ \cline{1-1} \cline{2-5}  & & (14.92\%) & (8.0\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scalability of the grids renumbering algorithms on different processors and grid sizes of \(FuelRodBundleFlow\) case. Column 1 indicates the grid size. Column 2 indicates the number of parallel processes.