Second order fractional step schemes for the incompressible Navier-Stokes equations. Inherent pressure stability and pressure stabilization

摘要

Fractional step methods for the time integration of the incompressible Navier-Stokes equations are very popular for their low computational cost. Moreover, very often it has been considered that they allow us to use equal velocity-pressure interpolation, thus avoiding the need to satisfy the classical inf-sup condition. In this paper we analyze this point for two families of second order fractional step methods. The first one is the classical pressure splitting (PS), in which the pressure is treated explicitly in the momentum equation and then it is updated, whereas the second one is the more recent velocity splitting (VS), which consists of treating explicitly the velocity in the momentum equation and updating it once the pressure and a fractional velocity are computed. We show that these two methods provide some pressure stability independent of the choice of the velocity and pressure interpolation. For the PS scheme, whose analysis is fully detailed, this stability is enough for first order schemes, but very poor for the second order scheme we consider. Contrary, it is possible to design VS methods that provide higher pressure stability. In fact, if the time step is properly chosen (equal to the critical time step of the forward Euler time integration), the pressure stability obtained is optimal, although in this case the time accuracy is strongly coupled with the space accuracy. We discuss all these details and also present a pressure stabilization technique that allows us to obtain the proper pressure stability without relying on the time step size.