"MISSING " BOUNDARY CONDITIONS? DISCRETIZE FIRST, SUBSTITUTE NEXT, AND COMBINE LATER*

Abstract. A simple approach exists to prevent the need for constructing boundary conditions in situations where they are not explicitly supplied by the original analytical formulation of the problem. An example is the Poisson equation for the pressure in calculations of incompressible flow. Other examples are the streamfunction-vorticity formulation where no condition for the vorticity is present, and ADI methods where boundary conditions for the intermediate timesteps must be provided. In short, this approach can be described as follows: first discretize the equations of motion, next substitute the original boundary conditions (for the velocity), and finally combine the discrete equations (e.g., to a modified Poisson equation). Key words, boundary conditions, discretization methods, incompressible Navier-Stokes equations AMS(MOS) subject classifications. 65N05, 76D05 1. Introduction. Whe