## 3 Results
### Flow around a Cylinder
Additionally, to check the robustness of the hybrid method, we performed three more simulations: one with a smaller tolerance of \(10^{-4}\), one using a staggered grid for pressure, and one with the pressure Poisson equation solved by the pseudo-spectral method. None of them yielded noticeable differences when compared to the results presented here. Figure 9: Geometry of the flow around a cylinder with periodic lateral boundary conditions. The radius of the cylinder is \(r=0.02\) m, and the side length of the square lattice is \(L=0.1\) m. The three monitoring points 1, 2, and 3 for comparing the time histories of numerical solutions are located at \((-L/2,-L/4)\), \((-r,0)\), and \((r,0)\), respectively. Figure 11: Comparison of the time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the second test case. Figure 10: Comparison of numerical solutions obtained by the reference method **(left)** and the hybrid method **(right)** for the second test case at \(t=300\) s. Shown from top to bottom are the contours of the horizontal velocity component \(U\), the vertical velocity component \(V\), and the pressure \(P\), respectively. On a computer with Intel Broadwell processors running at 2.6 GHz, the total wall clock time of the hybrid solver was about 805 s and the total wall clock time of the reference solver was about 1747 s. It is important to note that the simulation time spent in the first fractional step was not taken into account in the comparison of the computational performance because the two methods share the same algorithm in this part. The hybrid solver was more than 100% faster than the reference solver. Nevertheless, the speedup depends on the tolerance for the AGMG solver and the tolerance for the IB force iteration. Here, the former was set to \(10^{-6}\) (the value recommended in the AGMG application examples) and the latter was set to \(10^{-3}\). Theoretically, this choice can be justified by the fact that the pressure Poisson equation in the hybrid method was solved directly; hence, convergence was not an issue, and the IB force convergence only affected the velocity field locally around the immersed body. Practically, it was observed that further lowering the tolerance for the IB force iteration made no improvement to the results, while further lowering the tolerance for the AGMG solver led to noticeable pressure oscillations. It is worth mentioning that a coarser pressure grid can be used in the hybrid solver to further speed up the simulation. A test using a pressure grid of 256 \(\times\) 256 points showed that the total wall clock time of the hybrid solver was reduced to 496 s to yield a speedup of about three and half times, while the overall simulation accuracy was not considerably worsened, namely there were no obvious differences when comparing the results from this simulation to the standard ones. In this study, both the hybrid and reference solvers were not implemented in parallel. According to the comparative study of state-of-the-art parallel Poisson solvers by [PERSON] et al. (2016) [39], FFT-based parallel solvers can be two orders of magnitude faster than multigrid-based parallel solvers. Moreover, a rectangular computational domain is favourable for parallel computing because straightforward and efficient domain decomposition techniques such as slab or pencil decomposition can be applied. Therefore, the proposed hybrid pressure-correction algorithm has a great potential to result in a high-performance parallel computational fluid dynamics (CFD) solver for incompressible flows.