On the penalty-projection method for the Navier–Stokes equations with the MAC mesh

We deal with the time-dependent Navier–Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear’s scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method.