Approximate projection method for the incompressible Navier-Stokes equations

An approximate projection method has been developed for the incompressible Navier–Stokes equations. The algorithm was based on a generic family of Runge–Kutta time-stepping schemes, and it was particularly efficient from a computational point of view since the pressure projection step was operated only at the final stage. In contrast to previously developed methods, the new procedure employed a higher-order approximation for pressure within the substages, and hence allowed achievement of third-order temporal accuracy for the velocity field. Also, by improving the pressure approximation, the divergence error at each substep could be significantly reduced, leading to enhanced robustness. Preliminary turbulent simulations demonstrated the effectiveness of the method. The formulation was general and could, in principle, accommodate Runge–Kutta methods of any class and type.