Well posed problems and boundary conditions in computational fluid dynamics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact by discussing well-posedness of the most important set of governing equations in computational fluid dynamics, namely the time-dependent compressible Navier-Stokes equations. In particular, for a simplified model problem of the compressible Navier-Stokes equations, we will discuss i) how many boundary conditions are required, ii) where to impose them and iii) which form they should have. The procedure is based on the energy method and generalizes the characteristic boundary procedure for the Euler equations to the compressible Navier-Stokes equations. Once the boundary conditions in terms of i-iii) are known, one issue remains; they can be imposed weakly or strongly. The differences and similarities between a weak and strong imposition is discussed for the continuous case. It will be shown that the weak and strong boundary procedures produce identical solutions and that the boundary conditions are satisfied exactly also in the weak procedure. We conclude by relating the well-posedness results to energy-stability of the numerical approximation.