# A Parallel Algorithm Based on Regularized Lattice Boltzmann Method for Multi-Layer Grids
###### Abstract  
The regularized lattice Boltzmann method (RLBM) is an improvement of the lattice Boltzmann method (LBM). The advantage of RLBM is improved accuracy without increasing computational overheads. The paper introduces the method of multi-layer grids, the multi-layer grids have different resolutions which can accurately solve problems in computational fluid dynamics (CFD) without destroying the parallelism of RLBM. Simulating fluid flow usually requires a large number of grid simulations. Therefore, it is necessary to design a parallel algorithm for RLBM based on multi-layer grids. In this paper, a load-balancing-based grid dividing algorithm and an MPI-based parallel algorithm for RLBM on multi-layer grids are proposed. The load balancing-based grid dividing algorithm ensures that the workload is evenly distributed across processes, minimizing the discrepancies in computational load. The MPI-based parallel algorithm for RLBM on multi-layer grids ensures accurate and efficient numerical simulation. Numerical simulations have verified that the proposed algorithms exhibit excellent performance in both 2D and 3D experiments, maintaining high stability and accuracy. The multi-layer grids method is significantly better than single-layer grids in terms of CPU runtime and number of grids required. Comparative analysis with the OpenMP multi-threading method on the multi-layer grid RLBM shows that the proposed algorithm in this paper achieves superior speedup and efficiency. regularized lattice Boltzmann method; multi-layer grids; parallel algorithm; numerical simulation +
Footnote †: journal: Applied  
doi:10.3390/0  
surface boundary conditions for compressible flows in the context of regularized models. [PERSON] [15] applied regularized collision models to phase-field multiphase flow models. [PERSON] and [PERSON] [16] showed the similarity between the structure of LBM and RLBM. The RLBM has the properties of easily handling complex boundary conditions and high parallelism [17]. RLBM typically uses a uniform grid for numerical calculations. The fact that the accuracy of the grid cannot be flexibly adjusted limits the application of RLBM in some fields. In numerical simulations, there are regions where flow details need to be finely captured. Refining the entire grid would significantly increase the computational overhead [18]. Therefore, it is necessary to design a method for local grid refinement. [PERSON] and [PERSON] [19] proposed an FH method where the information of the grid is stored in the vertices. The coarse and fine grids in the FH method exchange information in a bi-directional coupled way. An adaptive mesh refinement (AMR) is proposed, which effectively reduces the loss of computational accuracy [20]. [PERSON] et al. [21] used the AMR method in combination with the immersion boundary lattice Boltzmann method (IB-LBM) to validate the effectiveness of the method in both 2D and 3D problems. Since the distribution functions are exchangeable on different grids, [PERSON] and [PERSON] [22] propose a single coupling method. The method is subject to errors in some cases. [PERSON] and [PERSON] [23] proposed a bi-directional coupled local grid refinement method, which solves the error problem in an excellent way. The method can improve the computational efficiency. In addition, a buffer-based interpolation method between coarse and fine grids was proposed by [PERSON] et al. [24]. The method effectively reduces the computation time. A method for storing grid information in the center point was proposed by [PERSON] et al. [25] The grid information at different sizes is stored in the grid intersections. Storing the grid information using this method can accurately and uniformly spread the grid information, which improves the accuracy of numerical calculations [26]. The RLBM with multi-layer grids uses a uniform grid for each layer, and the size of the previous grid is twice the size of the subsequent grid.