10.2.3 Navier-Stokes Solver

We have implemented a sophisticated incompressible flow solver using DIME and DIMEFEM. The algorithm is described more completely in [Bristeau:87a]. The evolution equation for an incompressible Newtonian fluid of viscosity n is

We use a three-stage operator-split scheme whereby for each time step of length dt, the equation is integrated

• from t to with incompressibility and no convection, then
• from to with convection and no incompressibility condition, then
• to t + dt as in stage one with incompressibility and no convection.

The parameter is .

Each of these three implicit steps involves the solution of either a Stokes problem: 

or the nonlinear problem:\

where is a parameter inversely proportional to the time step. We solve the Navier-Stokes equation,  and consequently also these subsidiary problems, with given velocity at the boundary (Dirichlet boundary conditions).