Ironically, there is an iterative method called "SIMPLE" (semi-implicit method for pressure-linked equations) designed to resolve the steady state navier-stokes equations based on a predictor-corrector scheme. It works by solving a linearized form of the momentum equation (predictor step) which produces a velocity field which generally does not satisfy continuity. The predicted velocity is used to obtain an equation for pressure which is constrained by the continuity equation. The result of this equation is a pressure field which roughly satisfies the continuity equation (corrector step). Using the updated pressure field, one can easily update the velocity field again so that it satisfies continuity with less error. One repeats this process iteratively until a steady state is reached.

For an example of its implementation, you may want to look at OpenFOAM's simpleFoam solver for an example of such an implementation:

https://github.com/OpenFOAM/OpenFOAM-2.3.x/tree/master/applications/solvers/incompressible/simpleFoam

Admittedly, OpenFOAM is not the easiest software to read and understand, but it shows all of the aforentioned steps (predictor in UEqn.H and corrector in pEqn.H). It may help you as a guide to developing your own implementation.

Ironically, there is an iterative method called "SIMPLE" (semi-implicit method for pressure-linked equations) designed to resolve the steady state navier-stokes equations based on a predictor-corrector scheme. It works by solving a linearized form of the momentum equation (predictor step) which produces a velocity field which generally does not satisfy continuity. The predicted velocity is used to obtain an equation for pressure which is constrained by the continuity equation. The result of this equation is a pressure field which roughly satisfies the continuity equation (corrector step). Using the updated pressure field, one can easily update the velocity field again so that it satisfies continuity with less error. One repeats this process iteratively until a steady state is reached.

For an example of its implementation, you may want to look at OpenFOAM's simpleFoam solver for an example of such an implementation:

https://github.com/OpenFOAM/OpenFOAM-2.3.x/tree/master/applications/solvers/incompressible/simpleFoam

Admittedly, OpenFOAM is not the easiest software to read and understand, but it shows all of the aforentioned steps (predictor in UEqn.H and corrector in pEqn.H). It may help you as a guide to developing your own implementation. For a more detailed explanation of the terms in the software, you may consider looking at some of the tutorials from Chalmers. In particular, I recommend "A look inside icoFoam (and pisoFoam)": http://www.tfd.chalmers.se/~hani/kurser/OS_CFD_2013/aLookInsideIcoFoam.pdf