A general algorithm for compressible and incompressible flows. Part III: The semi-implicit form

In this paper we consider some particular aspects related to the semi-implicit version of a fractional step finite element method for compressible flows that we have developed recently. The first is the imposition of boundary conditions, We show that no boundary conditions at all need to be imposed in the first step where an intermediate momentum is computed. This allows us to impose the real boundary conditions for the pressure, a point that turns out to be very important for compressible flows. The main difficulty of the semi-implicit form of the scheme arises in the solution of the continuity equation, since it involves both the density and the pressure. These two variables can be related through the equation of state, which in turn introduces the temperature as a variable in many cases. We discuss here the choice of variables (pressure or density) and some strategies to solve the continuity equation. The final point that we study is the behaviour of the scheme in the incompressible limit. It is shown that the method has an inherent pressure dissipation that allows us to reach this limit without having to satisfy the classical compatibility conditions for the interpolation of the velocity and the pressure. (C) 1998 John Wiley & Sons, Ltd.