# A Parallel Algorithm Based on Regularized Lattice Boltzmann Method for Multi-Layer Grids
## 5 Conclusions  
The regularized lattice Boltzmann method is an improvement based on the lattice Boltzmann method. It can improve accuracy without significantly increasing computational overheads. The multi-layer grids method can be effectively combined with RLBM. Focusing the fine grid on the parts that require more accurate calculations. The aim is to improve the computational accuracy and stability. A large number of grids is usually required for numerical calculations. RLBM has a natural parallelism; therefore, designing a parallel algorithm is necessary. In this paper, a load-balancing grid division algorithm is designed. It can achieve computational load balancing among the processes and minimize the communication overhead. Based on the above algorithm, an MPI-based parallel algorithm for RLBM on multi-layer grids is proposed. It can obtain the numerical results in a faster way. The accuracy and feasibility of the proposed method are verified by numerical experiments on flow over two circular cylinders in the 2D case and flow around a sphere in the 3D case. The experimental results show that the computation time is significantly reduced as the number of processes increases. Parallel algorithms have excellent performance in terms of speedup. In this paper, the proposed algorithm is compared with the use of OpenMP in a 3D flow around a sphere simulation scenario. The speed-up is close to 60 percent when using 16 cores. Using MPI is 28% more efficient than the parallel method using OpenMP. The algorithm performs well in large-scale computing tasks. Meanwhile, the load balancing strategy effectively reduces communication overhead, maintaining high parallel efficiency. In conclusion, the parallel algorithm on a regularized lattice Boltzmann method for multi-layer grids proposed in this paper offers an effective solution for massively parallel computing. The accuracy and stability of the algorithm will be tested in more CFD scenarios in the future, such as multiple cylinder arrangement, moving boundary cases, etc.  
Figure 26: Comparison of the efficiency between the parallel algorithm in this paper and the parallel algorithm using OpenMP.  
**Author Contributions:** Conceptualization, W.Z. and Y.W.; Data curation, Z.L. and Y.Z.; Formal analysis, Z.L. and Y.Z.; Methodology, Z.L. and Y.Z.; Resources, Z.L.; Software, Z.L. and Y.Z.; Supervision, Z.L., W.Z. and Y.W.; Validation, Z.L. and Y.Z.; Writing--original draft, Z.L. and Y.Z.; Writing--review and editing, Z.L., W.Z. and Y.W. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 42376194).  
**Data Availability Statement:** 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].  
**Acknowledgments:** The authors would like to express their gratitude for the support of Fishery Engineering and Equipment Innovation Team of Shanghai High-level Local University.  
**Conflicts of Interest:** The authors declare no conflicts of interest.