A second-order projection method for the incompressible navier-stokes equations

Abstract

In this paper we describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion-convection step and the projection step we obtain a temporal discretization that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult applications.