## 2 Materials and Methods
### Design and Implementation of swHPFM  
#### 2.2.1 Algorithm and task mapping scheme  
Large fluid machinery has a physical structure that consists of multiplexers, multiblade rows, multiblade channels, and multi-regions [24]. Correspondingly, it is necessary to communicate the boundary data between blade rows when designing simulations. In addition, blade channel boundary and regional boundary communication are required, as illustrated in Figure 2. In order to illustrate the blade row and blade channel, the geometric model of the axial compressor is shown in Figure 3, in which red blades represent rotor blades, green blades are stator blades, and the polygon stands for the blade channel. The blade channel can be divided into blade regions. The full circle is the blade row. \begin{table}
\begin{tabular}{c c} \hline
**Register Communication** & **Cycles** \\ \hline Point-to-Point delay & 10 \\ Line broadcast delay & 14 \\ Column broadcast delay & 14 \\ \hline \end{tabular}
\end{table} 
Table 1: Delay of the Register Communication. Figure 3: Geometric model of the axial compressor. The red blades represent rotor blades, green blades are stator blades, and the polygon stands for the blade channel. The blade channel can be divided into blade regions. Figure 2: Parallel hierarchy of the axial compressor rotor. Large fluid machinery has a physical structure that consists of multiplexers, multiblade rows, multiblade channels, and multi-regions. Similarly, the Sunway TaihuLight has a multilayer hardware architecture that consists of cabinets, supernodes, processors, and CGs, as shown as in Figure 4. From the bottom up, the main components are the CG of MPE+64 CPEs, the SW26010 processor, which is composed of four CGs, and the super nodes, which are composed of 256 nodes. The four supernodes constitute the cabinet. The multilevel parallelism of the fluid mechanical model and the multilayer hardware architecture of the Sunway TaihuLight naturally correspond to each other, which provides favorable conditions for the task mapping of them. swHPFM adopts two levels of parallelism: coarse-grained task parallelism and fine-grained data parallelism. Before submitting the simulation job to the Sunway TaihuLight supercomputer, the cost of MPI communication among nodes, \(Cost_{communication}=Size_{communication}\times frequency\), was calculated. The MPI tasks are allocated under the principle of minimizing the communication cost. The mapping scheme of swHPFM for the simulation model and the hardware architecture is illustrated in Figure 4.  
Thanks to its powerful logic control and interruption handling capabilities, the MPE on the node is responsible for handling complex scheduling tasks and decomposing the tasks of the blade rows into computing tasks of the kernel functions, shown as Equation (1), to complete the iterative calculation of the flow field. swHPFM distributes the kernel functions to the CPEs. Simultaneously, during the simulation, each CG returns data that are calculated by the CPE cluster in the main process to determine whether the calculation has converged and whether to exit the flow field simulation. The algorithm of the axial compressor rotor designed in this paper is suitable for the finite volume method. The differential form of the fluid control equations is expressed in Equation (1)  
\[\frac{\partial U}{\partial t}+\frac{\partial F_{c}}{\partial x}+\frac{\partial G _{c}}{\partial y}+\frac{\partial H_{c}}{\partial z}=I+\frac{\partial F_{v}}{ \partial x}+\frac{\partial G_{v}}{\partial y}+\frac{\partial H_{v}}{\partial z} \tag{1}\]  
where \(U\) stands for the conservation-type solving vector, \(F_{c}\)\(G_{c}\), and \(H_{c}\) are the convection vectors along the coordinate directions \(x\), \(y\), and \(z\), respectively, \(F_{v}\), \(G_{v}\), and \(H_{v}\) denote the viscous vectors along the three coordinate directions, and \(I\) represents the source term.