## 3 Parallel Computation and Grid-Overlapping Interface
### Grid-Overlapping Interface  
As described in the previous section, the governing equation is solved using the ADI method. The method assumes the absence of variations in the values of dependent variables at the boundary points. Thus, the variables on the points in the interface boundary are excluded from the update that is formed using the ADI method, because the interface boundary is also considered as the boundary for each subdomain. Therefore, it is critical to increase the accuracy of information exchange through the interface boundary to improve the overall accuracy of parallel computation. An overlapping method is proposed to achieve the aforementioned purpose. Figure 2 shows the schematic concept of the overlapping methods. Each subdomain includes dummy points through which the data on the computational points in adjacent subdomains are transferred. The transfer of the data that is computed in adjacent subdomain updates the value on the points in the interface boundary condition with a more reliable value. For example, the matrix equation of the ADI method with eight points is expressed as follows:  
\[\mathrm{S}\Delta\mathrm{Q}=\mathrm{R} \tag{10}\]  
Figure 1: Computational domain for serial and parallel computing. Figure 2: Data exchange at overlapped grids in the interface boundary. where:  
\[\mathrm{S}=\left[ \begin{array}{cccccc}B_{2}&C_{2}&0&0&0&0\\ A_{3}&B_{3}&C_{3}&0&0&0\\ 0&A_{4}&B_{4}&C_{4}&0&0\\ 0&0&A_{5}&B_{5}&C_{5}&0\\ 0&0&0&A_{6}&B_{6}&C_{6}\\ 0&0&0&0&A_{7}&B_{7}\end{array} \right] \tag{11}\]  
With the use of Equation (11), Equation (10) corresponds to a matrix equation formed by applying the ADI method over the single domain. The values of flow variables on points 1 and 8 that are assumed as on the boundary do not exhibit any variation, i.e., \(\Delta\mathrm{Q}_{1}=\Delta\mathrm{Q}_{8}=0\). Figure 3 shows the entire domain decomposed into two subdomains without overlapping points, where the first subdomain includes five points from points 1 to 5, and the second subdomain includes four points, from 5 to 8. If the parallel computation based on the domains is performed with a linear interpolation, the matrix \(\mathrm{S}\) in Equation (11) is modified as follows:  
\[S_{1}=\left[ \begin{array}{cccc}B_{2}&C_{2}&0\\ A_{3}&B_{3}&C_{3}\\ 0&A_{4}&B_{4}\end{array} \right],\;S_{2}=\left[ \begin{array}{cccc}B_{6}&C_{6 }\\ A_{7}&B_{7}\end{array} \right] \tag{12}\]  
In the ADI method, variation in the points of boundary condition in each subdomain is absent. The equation omits C4, A5, B5, C5, and A6, which exist in Equation (11). This implies that it is not possible to obtain the value on point 5 from the solution of Equation (12). The value on point 5 is typically updated using the linear interpolation of the values on the neighboring points, and this is definitely different from that in Equation (11). The overlapped subdomains are employed to reduce the difference. Figure 4 shows the overlapped domains with two points that correspond to the minimum in an overlapping method. If the ADI method is applied in the configuration, and Equation (11) subsequently changes as follows:  
\[S_{1}=\left[ \begin{array}{cccc}B_{2}&C_{2}&0\\ A_{3}&B_{3}&C_{3}\\ 0&A_{4}&B_{4}\end{array} \right],\;S_{2}=\left[ \begin{array}{cccc}B_{5}&C_{5} &0\\ A_{6}&B_{6}&C_{6}\\ 0&A_{7}&B_{7}\end{array} \right] \tag{13}\]  
In the method, the value on points 4 and 5 is computed by solving Equation (12). However, a difference exists between Equations (11) and (13) wherein C4 and A5 are missing in Equation (13) when compared with Equation (11). The effect of the difference is negligible with respect to a simulation for a problem with simple geometry or a small number of grids. However, the four-point overlapped  
Figure 4: Subdomains with two overlapped points. Figure 3: Subdomains with linear interpolation at five points (without overlapped points). method should at least be used to ensure that the numerical solution of parallel simulation is identical to that of the single domain.