# A Numerical Method for the Incompressible Navier-Stokes
Equations Based on an Approximate Projection

###
William G. Szymczak

### Abstract

In this method we present a fractional step discretization of the
time-dependent incompressible Navier-Stokes
equations.
The method is based on a projection formulation in which we
first solve diffusion-convection equations to predict
intermediate velocities which are then projected onto the
space of divergence-free vector fields.
Our treatment of the diffusion-convection step
uses a specialized second-order upwind method for differencing
the nonlinear convective terms that provides a robust treatment of
these terms at high Reynolds number.
In contrast to conventional projection-type discretizations
that impose a discrete form of the divergence-free constraint, we
only approximately impose the constraint; i.e., the velocity field we compute
is not exactly divergence-free.
The approximate projection is computed using a conventional discretization of
the Laplacian and the resulting linear system is solved
using conventional multigrid methods.
Numerical examples are presented to validate the second-order
convergence of the method for Euler, finite Reynolds number, and
Stokes flow. A second example illustrating the behavior of the
algorithm on an unstable shear layer is also presented.