Convergent numerical approximations and analysis of viscous compressible flow

Partial differential equations modeling the flow of fluids such as air or water are widely used in both academics and industry. Typically, they are used to predict the behavior of a fluid prior to construction and experiments. By the aid of computers it is possible to construct approximations of these equations with accurate results. Today, such computer approximations constitutes an indispensable tool in all parts of engineering. Compared to the enormous practical success enjoyed by fluid modeling, it is remarkable how little is known about the equations from a purely theoretical point of view. Indeed, the basic questions of existence, uniqueness, and regularity of solutions are still open for most realistic models. This is also reflected in the theore tical foundation for numerical approximations of viscous gas flow. In fact, it is not known whether or not any numerical approximation of viscous compressible flow converges as discretization parameters go to zero. Hence, even though experiments indicate that a numerical method is appropriate, there are no theoretical guarantees that this is actually the case. The proposed project aims at contributing to the mathematical and numerical theory of viscous compressible flow in two directions: Primarily, the project will focus on the development of convergent numerical methods for the approximation of viscous gas flow. Using innovative discretization techniques, the goal is to develop the first convergent numerical methods for the flow of gas in various situa tions. Secondly, the project will conduct analytical studies on the equations governing viscous gas flow. Through these studies, the project aims at extending the current existence theory to include more flow situations and more types of gases. Both dire ctions of the project will be conducted in close collaboration with some of the worlds leading experts.