On the Stability of Godunov-Projection Methods for Incompressible Flow

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.

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