## 4 Conclusions and Future Work  
A new hybrid pressure-correction algorithm was developed for applying projection methods to simulate unsteady incompressible flows with the presence of obstacles. The hybrid algorithm combines different numerical methods and associated spatial discretisations in the pressure-correction step of projection methods, and it was demonstrated/implemented in a classical second-order time-marching projection scheme together with a meshless method with the irregular distribution of points for solving the momentum equations. The key of the hybrid algorithm relies on adopting a double-grid system and combining the weighted least-squares approximation with the idea inspired by immersed boundary methods. Such a design makes it feasible to solve the pressure Poisson equation on a regular grid for flow problems of obstacles. Hence, the solution process can be accelerated by using the FFT while keeping the implementation of the boundary conditions as easy as body-conforming numerical methods do with structured or unstructured grids. The numerical simulations of five test cases with different boundary conditions, geometries, and Reynolds numbers were performed to verify and validate the proposed methodology. The results showed that the new hybrid algorithm works and is robust, and as expected, it had no influence on the temporal and spatial accuracy of the underlying projection method. In terms of computational performance, the hybrid method outperformed the reference method, which adopts the same temporal and spatial discretisation schemes in the first sub-step as the hybrid method does, but solves the pressure Poisson equation in the second sub-step in the conventional way with a multigrid-based method. It was shown that, without deteriorating the overall simulation accuracy, a three-times speedup can be achieved by the hybrid method by setting the tolerance value to \(10^{-3}\) for the IB force iteration and using a grid for pressure coarser than that for velocity. Although the proposed hybrid algorithm was here demonstrated/implemented in the classical second-order Adams-Bashforth-based projection method, it can be readily implemented in other projection schemes such as high-order Runge-Kutta-based projection methods. Furthermore, the numerical method in the first fractional step is not necessarily of the meshless type, and it can be a mesh-based method such as finite-volume. Considering the facts that domain decomposition techniques/libraries for rectangular domains are mature and efficient and with which FFT-based parallel Poisson solvers can be two orders of magnitude faster than multigrid-based parallel Poisson solvers, the hybrid algorithm provides a promising alternative for developing next-generation high-performance parallel CFD solvers for incompressible flows. In future work, the hybrid framework will be tested in high-order projection methods, implemented with parallel computing, and extended for three-dimensional turbulent flows and large-eddy simulation of atmospheric boundary layers over complex terrain. Figure 21: Comparison of the radial and azimuthal velocity components along four radial lines for the lid-driven polar cavity flow at different Reynolds numbers. Ultimately, this work will serve as the basis for a high-performance parallel incompressible flow solver. This research received no external funding. The data presented in this study are available upon request from the corresponding author. This work was facilitated though the use of advanced computational infrastructure provided by the Scientific IT and Application Support (SCITAS) platform at the Swiss Federal Institute of Technology in Lausanne (EPFL). The authors declare no conflict of interest.