Derivation of projection methods from formal integration of the Navier-Stokes equations

Projection methods constitute a class of numerical methods for solving the incompressible Navier-Stokes equations. These methods operate using a two-step procedure in which the zero-divergence constraint on the velocity is first relaxed while the velocity evolves, then after a certain period of time the resulting velocity field is projected onto a divergence-free subspace. Although these methods can be quite efficient, there have been certain concerns regarding their formulation. In this paper we show how a formal integration of the Navier-Stokes equations leads to a new and general procedure for the derivation of projection methods. By following this procedure, we show how each of three practical projection methods approximates a system of equations that is equivalent to the Navier-Stokes equations. We also show how the auxiliary boundary conditions required in projection methods are related to the physical boundary conditions. These results should allay the concerns regarding the legitimacy of projection methods, and may assist in their future development.