On the stability of Godunov-projection methods for incompressible flow

Abstract

An analysis of the stability of certain numerical methods for the linear advection-diffusion equation in two dimensions is performed. The advection-diffusion equation is studied because it is a linearized version of the Navier-Stokes equations, the evolution equation for density in Boussinesq flows, and a simplified form of the equations for bulk thermodynamic temperature and mass fraction in reacting flows. It is found that various methods currently in use which are based on a Crank-Nicholson type temporal discretization utilizing second-order Godunov methods for explicitly calculating advective terms suffer from a time-step restriction which depends on the coefficients of diffusive terms. A simple modification in the computation of the advective derivatives results in a method with a stability condition that is independent of the magnitude of the coefficients of the diffusive terms. © 1996 Academic Press, Inc.