On Discrete Projection and Numerical Boundary Conditions for the Numerical Solution of the Unsteady Incompressible Navier-Stokes Equations

Abstract

The unsteady incompressible Navier-Stokes equations are discretized in space and studied on the fixed mesh as a system of differential algebraic equations. With discrete projection defined, the local errors of Crank Nicholson schemes with three projection methods are derived in a straightforward manner. Then the approximate factorization of relevant matrices are used to study the time accuracy with more detail, especially at points adjacent to the boundary. The effects of numerical boundary conditions for the auxiliary velocity and the discrete pressure Poisson equation on the time accuracy are also investigated. Results of numerical experiments with an analytic example confirm the conclusions of our analysis.