As. Almgren et al., A NUMERICAL-METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS BASED ON AN APPROXIMATE PROJECTION, SIAM journal on scientific computing, 17(2), 1996, pp. 358-369
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-convect
ion equations to predict intermediate velocities, which are then proje
cted 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 a high Reynolds number. in contras
t to conventional projection-type discretizations that impose a discre
te form of the divergence-free constraint, we only approximately impos
e the constraint; i.e., the velocity held we compute is not exactly di
vergence-free. The approximate projection is computed using a conventi
onal discretization of the Laplacian and the resulting linear system i
s solved using conventional multigrid methods. Numerical examples are
presented to validate the second-order convergence of the method for E
uler, finite Reynolds number, and Stokes now. A second example illustr
ating the behavior of the algorithm on an unstable shear layer is also
presented.