A 2D compact fourth-order projection decomposition method

A 2D fourth-order compact direct scheme projection decomposition method for solving incompressible viscous flows in multi-connected rectangular domains is devised. In each subdomain, the governing Navier–Stokes equations are discretized by using fourth-order compact schemes in space and second-order scheme in time. The coupling between subdomains is based on a direct non-overlapping multidomain method: it allows to solve each Helmholtz/Poisson problem resulting of a projection method in complex geometries. The major difficulty of the Poisson–Neumann problem solvability is addressed and correctly treated. The present numerical method is checked through some classical numerical experiments. First, the second-order accuracy in time and the fourth-order accuracy in space are shown by matching with the analytical solution of the Taylor problem. The method is also tested by simulating the flow in a 2D lid-driven cavity. The utility of the compact scheme projection decomposition method approach is further illustrated by two other benchmark problems, viz., the flow over a backward-facing step and the laminar flow past a square prism. The present results are in good agreement with the experimental data and other numerical solutions available in the literature.