A second-order projection method for the incompressible Navier-Stokes equations in arbitrary domains

We present a projection method for the numerical solution of the incompressible Navier-Stokes equations in an arbitrary domain that is second-order accurate in both space and time. The original projection method was developed by Chorin, in which an intermediate velocity field is calculated from the momentum equations which is then projected onto the space of divergence-free vector fields. Our method is based on the projection method developed by Bell and co-workers which is designed for problems in regular domains. We use the continuity equation to derive a pressure equation to compute the gradient part of the vector field. An integral form of the continuity equation is used to give us a natural way to define the discrete divergence operator for cells near the boundary which ensures the diagonal dominance of the resulting pressure equation. We then use the restarted version of the GMRES method to solve the pressure equation.