(1) A staggered mesh compact difference scheme is presented for solving the unsteady viscous incompressible Navier-Stokes equations. It is fourth order accurate both in the spatial direction and in the time direction, at least third order accurate on the boundary; (2) Describe a pressure-Poisson-equation that is equivalent to the discrete continuity equation provided the discrete momentum equations remain. The discrete continuity equation may have derivative boundary conditions,e.g., the compact difference scheme; (3) A new ADI iterative method is proposed.The pressure-Poisson-equation is in the discrete form. It is difficult to be solved with a usual ADI method. We translate it to be tridiagonal in each spatial direction of each step of the ADI iterations, then add a pseudo time term to get a tridiagonal equation which is easily to be solved; (4) The driven flow in a square cavity with Re = 10000 is simulated numerically.
We present iterative pressure Poisson equation method for solving the viscous incompressible Navier-Stokes equations. Comparing with the pressure Poisson equation method, it uses the increment of the pressure, the difference of two successive pressures (pkn+1 and pk+1n+1), instead of the pressure p（n+1） as an unknown variable. We apply it to a fourth order accurate staggered mesh compact difference scheme, simulate the driven flow in a square cavity with Re = 5000 and 10000.
