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A second-order projection method for the incompressible Navier-Stokes equations. (English) Zbl 0681.76030
We describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. 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.

MSC:
76D05Navier-Stokes equations (fluid dynamics)
65N99Numerical methods for BVP of PDE