## 3 Immersed Boundary Method
### Diffuse Interface IBM
#### 3.1.4 Velocity Correction IBM
Every element of \(A\) can be computed with \(D(\mathbf{x}-\mathbf{X}(s,t))\):  
\[A_{lk}=\sum_{j}^{M}D_{lj}D_{jk}\mathbf{l}^{3}, \tag{29}\]  
where \(l,k=1,\cdots,N\), \(j=1,\cdots,M\), \(M\) is the number of Eulerian points. By solving \(\mathbf{X}=A^{-1}\mathbf{B}\), \(\Delta\mathbf{U}\Delta s\) can be obtained. However, the coefficient matrix \(A\) is introduced in this method. In the original velocity correction IBM, the procedure for solving \(A^{-1}\) is required. In this way, Equation (28) is difficult to be solved efficiently if the number of Lagrangian points is excessive, especially for 3D problems. This has been considered a drawback of this method. Recently, [PERSON] et al. [83] proposed an explicit technique to handle the matrix \(A\). Using this method, Equation (28) can be solved efficiently. Matrix \(A\) is simplified to a diagonal matrix, where the diagonal elements are computed using  
\[\lambda_{l}=\sum_{k}^{N}A_{lk}, \tag{30}\]where \(l\) represents the row of matrix \(A\). In this way, Equations (27) and (28) can be simplified to each Lagrangian point:  
\[\lambda(s,t)\cdot\Delta\mathbf{U}\Delta s=\mathbf{U}(s,t)-\mathbf{u}^{*}(\mathbf{X}(s,t)). \tag{31}\]  
After obtaining the terms \(\Delta\mathbf{U}\Delta s\), substituting them into Equation (25), the velocity correction \(\Delta\mathbf{u}(\mathbf{x},t)\) is obtained. Finally, Equation (23) can be applied to solve the corrected velocity \(\mathbf{u}(\mathbf{x},t)\) at Eulerian points. Through this procedure, the fluid velocity is enforced to equal the boundary velocity. The no-slip boundary condition can be accurately satisfied, which avoids streamline penetration. This feature is validated by [PERSON] and [PERSON] [31], as shown in Figure 4, where the streamlines penetrate the solid boundary in Figure 4a obtained by the direct forcing IBM [16] while the no-slip boundary condition is satisfied in Figure 4b using velocity correction IBM [31]. In this method, the velocity correction procedure replaces the calculation of the boundary force \(\mathbf{F}(s,t)\) and fluid body force \(\mathbf{f}(\mathbf{x},t)\). The iterative process for the boundary condition in other IBMs [18; 30] is also removed. Obviously, this method is powerful for simulations with both rigid and flexible structures and has successful applications [54; 67; 84; 85; 86]. Some diffuse interface IBMs are presented above. The pros and cons of these methods are also mentioned. For instance, the classic IBM and penalty forcing IBM both have difficulties in simulating problems involving rigid bodies. They are early IBMs. The direct forcing IBM and velocity correction IBM are more prevalent recently. The basic framework of the former is efficient but cannot accurately satisfy the no-slip boundary condition at the solid surface. To overcome this drawback, a multi-direct forcing scheme is proposed. The velocity correction IBM can satisfy the conditions at the immersed boundaries conveniently but introduces a complex matrix in the computations. Some efforts have been put into the aspect of eliminating the complexity. By focusing on the CPU time with different numbers of Lagrangian points, some improved methods for solving Equation (28) are tested in [87], such as the Gaussian elimination method, Cholesky factorization method, and generalized minimal residual (GMRES) method, where the multi-direct forcing IBM is also incorporated into the comparison. The comparison in [87] indicates that with the help of efficient methods for solving the linear equation system, such as the GMRES method, the inefficiency of the velocity correction IBM can be overcome and is more efficient than the direct forcing IBM. Additionally, [PERSON] et al.