On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory

Abstract

AbstractEver since the time of Chorin's classic 1968 paper on projection methods, there have been lingering and poorly understood issues related to the best—or even proper or appropriate—boundary conditions (BCs) that should be (or could be) applied to the ‘intermediate’ velocity when the viscous terms in the incompressible Navier–Stokes equations are treated with an implicit time integration method and a Poisson equation is solved as part of a ‘time step’. These issues also pervade all related methods that uncouple the equations by ‘splitting’ the pressure computation from that of the velocity—at least in the presence of solid boundaries and (again) when implicit treatment of the viscous terms is employed. This paper is intended to clarify these issues by showing which intermediate BCs are ‘best’ and why some that are not work well anyway. In particular we show that all intermediate BCs must cause problems related to the regularity of the solution near boundaries, but that a near‐miraculous recovery occurs such that accurate results are nevertheless achieved beyond the spurious boundary layer introduced by such methods. The mechanism for this ‘miracle’ is related to the existence of a higher‐order equation that is actually satisfied by the pressure. All that is required then for projection (splitting, fractional step, etc.) methods to work well is that the spurious boundary layer be thin—as has been largely observed in practice.