## 5 Conclusions  
In this paper, we analyze the serial computing details of RLBM with LES in detail. Considering the architecture characteristics of the large-scale computing cluster, the highly scalable parallel algorithm with computing load balancing is proposed. This paper used the domain decomposition method to design three grid partition strategies, and introduced the buffer technology to provide the corresponding parallel data exchange strategy. MPI is used  
Figure 24: Speedup of 3D domain decomposition on hundreds of thousands of cores. Figure 25: Efficiency of 3D domain decomposition on hundreds of thousands of cores. for parallel data exchange between computing nodes. The proposed parallel algorithm has the properties of computational load balancing and high scalability. We used this parallel algorithm to simulate the flow of the 3D lid-driven cavity flow under the high Reynolds number, and compare with the results of [PERSON] et al. The experimental results show that RLBM with LES has a good stability under the high Reynolds number. When computing resources are relatively small, DDM decomposition along one direction is feasible. However, when the operating environment becomes a large cluster with hundreds of thousands of cores, the three-dimensional decomposition is reasonable and efficient. By analyzing the experimental results calculated on Shanghai University's \"Ziqiang 4000\" and Sunway BlueLight MPP in NSCC-JN, we prove that the effect of 3D decomposition is the best and the presented algorithms are efficient and scalable. It is foreseeable that even on one million CPU cores, the speedup ratio will not drop significantly. In the field of HPC, integrating parallel algorithms with Multigrid techniques is crucial, especially in LES turbulence modeling. The multigrid method, a sophisticated hierarchical approach, excels in solving LES equations on multiple grid levels, ensuring rapid convergence in turbulent flow simulations. Its adaptability is vital in understanding complex turbulence phenomena within LES. Moreover, the parallel algorithms of RLBM and LES under multi-layer mesh will be studied in depth in the future work. The challenge in using multiple GPUs lies in efficient communication. Within a single node, communication is easier and more efficient. However, when GPUs span multiple nodes, factors such as grid partitioning and communication strategies complicate matters. Achieving a high efficiency in cross-node GPU communication is difficult and requires careful planning and strategic implementation of communication methods and computational coordination. Despite challenges in adapting our proposed parallel algorithm to heterogeneous CPU-GPU platforms, future efforts will refine our approach for multi-GPU or multi-MIC configurations in LES simulations. These endeavors will optimize the synergy between parallel algorithms and the Multigrid framework, not only in LES but also exploring integrations with other turbulence models, such as Reynolds stress models or detached eddy simulation. This pursuit promises advanced solutions for large-scale LES simulations, marking a significant step in this evolving computational field. Conceptualization, W.X. and W.S. ; Data curation, Z.L. and Y.C. ; Formal analysis, Z.L. and Y.C. ; Methodology, Z.L. and Y.C. ; Resources, Z.L. ; Software, Z.L., Y.C. and Y.L. ; Supervision, Z.L. and W.S. ; Validation, Z.L., Y.C. and Y.L. ; Writing--original draft, Z.L., Y.C. and Y.L. ; Writing--review and editing, Z.L., W.X. and W.S. All authors have read and agreed to the published version of the manuscript. This research was funded by National Key Research and Development Program of China (Grant No. 2021 YFC3101601); National Natural Science Foundation of China (Grant No. 61972240) and the Program for the Capacity Development of Shanghai Local Colleges (Grant No. 20050501900). Experimental data related to this paper can be requested from the authors by email if any researcher in need of the data, email: [EMAIL_ADDRESS].