A new perspective on the stability of unsteady stream-function vorticity calculations

Abstract

The stability of a numerical solution of the Navier-Stokes equations is usually approached by considering the stability of an advection-diffusion equation for either a velocity component, or in the case of two-dimensional flow, the vorticity. Stability restrictions for discretised advection-diffusion equations are a very serious constraint, particularly when a mesh is refined, so an accurate understanding of the stability of a numerical procedure is often of equal or greater importance than concerns with accuracy. The stream-function vorticity formulation provides two equations, one an advection-diffusion equation for vorticity and the other a Poisson equation between the vorticity and the stream-function. These two equations are usually not coupled in stability considerations, commonly only the stability of time marching of the advection diffusion equation is taken into account. In this work, we derive a global time-iteration matrix for the full system and show that this iteration matrix is far more complicated than that for just the advection-diffusion equation. We show how for a model system, the complete equations have much tighter stability constraints than would be predicted from the advection-diffusion equation alone