Stability of approximate projection methods on cell-centered grids

Projection methods are a popular class of methods for solving the incompressible Navier-Stokes equations. If a cell-centered grid is chosen, in order to use high-resolution methods for the advection terms, performing the projection exactly is problematic. An attractive alternative is to use an approximate projection, in which the velocity is required to be only approximately discretely divergence-free. The stability of the cell-centered, approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient. This is demonstrated by analyzing a model problem and conducting numerical simulations of the Navier-Stokes equations.