Analysis and Methods in Fluid Mechanics

When constructing numerical methods for partial differential equations, it is important to have a thorough understanding of the continuous model and the characteristic properties of its solutions. We shall present methods of analysis for determining well-posedness of hyperbolic and mixed hyperbolic-parabolic equations which are applicable to the time-dependent Euler and Navier-Stokes equations. We shall then discuss difference- and finite volume methods and the construction of grids. The geometry of realistic problems is usually such that it is almost impossible to construct one structured grid. One way to overcome this difficulty is to use overlapping grids, where each domain has a structured grid. We discuss stability and accuracy of difference methods applied on such grids. Many problems in physics and engineering are defined in boundary domains, and artificial boundaries are introduced for computational reasons. In some cases one can construct accurate boundary conditions at these open boundaries. We shall indicate how this can be achieved, but we will also point out certain cases where accurate solutions are impossible to be obtained on limited domains. Finally some comments will be given on the difficulties arising when almost incompressible flow is computed. This corresponds to small Mach-numbers, and extra care must be taken when designing numerical methods. The theory will be complemented by numerical experiments for various flow problems in two space dimensions.