## 4 Numerical Experiment
### Comparison of Three Kinds of Domain Decomposition Methods  
In this section, a 3D incompressible flow around a sphere with a constant velocity profile, \(u=U_{\infty}=\{0.2 Ma,0,0\}\), was simulated as a numerical test for our various domain decomposition methods. Figure 17 shows the schematic diagram of a three-dimensional flow past a circular cylinder. In order to eliminate the influence of the boundary, the three-dimensional (length \(\times\) width \(\times\) height) size of the calculation domain is \(51.2D\), \(12.8D\) and \(12.8D\), respectively. \(D\) is the radius of the sphere, and its value is 40. The grid scale is \(2048\times 512\times 512\). This comparison experiment was carried out on the third-generation cluster high-performance computer-\"Ziqiang 4000\" of Shanghai University. The cluster includes 140 CPU compute nodes. Each node has two Intel SandyBridge architecture CPUs (Intel E5-2690, 2.9 GHz/8-core), and the cluster has a total of 2240 cores. The shared running memory of each node is 64 GB. The code is compiled under the mpich2 and GCC compilation environment, and the optimization level of the code is \"-O3\". The drag force on the sphere \(F_{d}\) is computed with the momentum-exchange method [56], and the drag force coefficient [57] is obtained by  
\[C_{d}=-\frac{8F_{d}}{\rho u_{\infty}^{2}D^{2}}. \tag{29}\]  
The Figure 18 shows that our experiments are in good agreement with theoretical value [58]. Figures 19 and 20 describe the vortical structure when Re = 1000 and 3700, respectively. In Table 3, the time with 100 iterations to generate the grid; communication time and iterative time, including collision time, streaming time, and communication time; are presented. In 1D DDM, the amount of data needed to be transferred is fixed in accord with Equation (23); thus, the communication time is about same. Using 2D or 3D DDM, the communication time decreases with the increase in \(nPX\), \(nPY\) or \(nPZ\), and the communication time is less under the same cores when adopting the 3D one. The lattice scales are the same in different DDM; thus, the time to generate the grid is relative when cores are same. The values among three kinds of DDMs show that the iterative time of the 3D one is shorter than the 2D one, and the 2D one has a shorter iterative time than the 1D one. Based on the values of Table 3, speedup and efficiency are shown in Figures 21 and 22. It can be observed that 3D DDM has the best scalability among the three divisions.The 3D DDM also has less communication in the design of the parallel algorithmic data exchange. Since in large-scale clusters, the computational resources are very large, to be able to fully utilize the computational resources, a large number of computational sub-regions are needed, and the domain decomposition strategy used in this paper needs to decompose enough sub-computational domains; thus, the 3D division strategy is the most appropriate. The three-dimensional division strategy of LBM given by [PERSON] et al. [59] is similar to our three-dimensional division strategy of RLBM, but we also provide the one-dimensional and two-dimensional division strategies and communication strategies in the paper. Due to the high difficulty of 3D division, large amount of communication and complicated procedures, it is necessary to study the 1D and 2D division strategies. Through experiments, we found that when the number of cores is less than 1000, the speedup ratio of two-dimensional partitioning is close to that of three-dimensional partitioning. When the number of cores is  
Figure 19: Vortical structure, Re = 1000. Figure 20: Vortical structure, Re = 3700.  
greater than 1000, the advantages of three-dimensional partitioning are shown. Therefore, we can choose the division strategy reasonably according to the requirements. Figure 23 shows the parallel efficiency of our 3D partitioning algorithm compared to the results of [PERSON] et al. [59]. It is found that our parallel efficiency is better than [PERSON] et al. [59] when the number of cores is less than 2048.