Comparison between the PISO algorithm and preconditioning methods for compressible flow

Two widely used family of algorithms, pressure-based and density-based methods, have been developed for computational fluid dynamics (CFD) problems over the years. Pressure-based methods (such as SIMPLE and PISO) use a Poisson-like equation for updating pressure instead of the continuity equation, while density-based methods use the continuity equation to update density (an equation of state is used to provide density in pressure based schemes and pressure in density based schemes). Pressure-based methods were developed originally for incompressible flows at low Reynolds numbers and were then extended to high Reynolds numbers and compressible applications. On the other hand, density based methods were originally developed for transonic flows and have been extended down to low Mach numbers through the use of preconditioning techniques. We compare these two very different approaches to solving the Navier-Stokes equations in order to gain an understanding of their similarities and differences. Specifically, we consider the PISO scheme as a representative pressure-based method and contrast it with a recently developed preconditioning scheme. We also compare the relative performance of the PISO algorithm with a Euler implicit algorithm that is employed to solve the preconditioned equations by means of a vector stability analysis.