  For simulating incompressible flows by projection methods. it is generally
accepted that the pressure-correction stage is the most time-consuming part of
the flow solver. The objective of the present work is to develop a fast hybrid
pressure-correction algorithm for numerical simulation of incompressible flows
around obstacles in the context of projection methods. The key idea is to adopt
different numerical methods/discretizations in the sub-steps of projection
methods. Here, a classical second-order time-marching projection method which
consists of two sub-steps is chosen for the purpose of demonstration. In the
first sub-step, the momentum equations are discretized on unstructured grids
and solved by conventional numerical methods, here, a meshless method. In the
second sub-step (pressure-correction), the proposed algorithm adopts a double
discretization system and combines the weighted least squares approximation
with the essence of immersed boundary methods. Such a design allows us to
develop a FFT-based solver to speed up the solution of the pressure Poisson
equation for flow cases with obstacles, while keeping the implementation of
boundary conditions for the momentum equations as easy as conventional
numerical methods do with unstructured grids. Numerical experiments of five
test cases have been performed to verify and validate the proposed hybrid
algorithm and evaluate its computational performance. The results show that the
new FFT-based hybrid algorithm is working and robust, and it is significantly
faster than the multigrid-based reference method. The hybrid algorithm opens an
avenue for the development of next-generation high-performance parallel
computational fluid dynamics solvers for incompressible flows.
