Third-order projection method for solving the incompressible Navier-Stokes equations

Projection methods are being more widely applied in numerically simulations of incompressible flows, however, their temporal accuracies are no better than second order. This paper shows that the pressure updating formula plays a very important role in higher-order accurate projection methods since it affects not only the accuracy of the pressure solution, but also the stability of the projection scheme. A stable third-order accurate projection method for the incompressible Navier-Stokes equations was developed using a consistent pressure updating equation. A normal mode analysis shows that the velocity and the pressure are both third-order accurate. An inappropriate pressure updating formula will reduce the accuracy of the pressure to at most second-order on the boundary. Finally, numerical results verify that the scheme is stable and both the velocity and the pressure are third-order accurate.