Projection methods for the incompressible Navier-Stokes equations

The incompressible Navier-Stokes equations (INSE) are the basic governing equations of fluid dynamics, and their numerical solutions are of great significance. In this review paper, we first recollect some classical projection methods and their relatives in the past 50 years and then fully explain the recent fourth-order projection method called GePUP [Zhang Q 2016 J. Sci. Comput. 67 1134]. Based on a generic projection operator and the UPPE formulation of the INSE [Liu J G, Liu J, Pego R L 2007 Comm. Pure Appl. Math. 60 1443], we derive the GePUP equations, which retain the advantage of UPPE that the velocity divergence is governed by a heat equation and is thus well under control. In comparison with UPPE, the GePUP formulation is advantageous in three aspects: (1) its derivation depends on none of the properties of the Leray-Helmholtz projection; (2) the evolutionary velocity can be divergent, thus it is directly applicable to numerical calculations with nonzero velocity divergence; (3) the Leray-Helmholtz projection does appear on the right-hand sides of the governing equations, thus making it transparent to analyze the accuracy and stability issues raised by numerically approximating the Leray-Helmholtz projection. As the most appealing feature of GePUP, temporal integration and spatial discretization are completely decoupled and can be treated as black boxes, so that the user can choose his favorite methods for the two parts to form his own GePUP method. In particular, high-order accuracy in time can be easily obtained since no internal details of the ODE solver are needed. The flexibility in time makes the GePUP method applicable to both low-Reynolds-number flow and high-Reynold-snumber flow. The flexibility in space makes the GePUP method applicable to both rectangular boxes and irregular domains. The numerical results and elementary analysis show that the fourth-order GePUP method may be much more accurate and efficient than classical second-order projection methods by many orders of magnitude.