D-Math

In this bachelor thesis we present a proof of Kirszbraun's theorem for Hilbert spaces and then we prove a gluing result of Makarychev and Makarychev. Namely we prove that if X = A ∪ B is a metric space such that A embeds into l a 2 with distortion D A and B into l b 2 with distortion D B , then X embeds into l a+b+1 2 with distortion at most 7D A D B + 5(D A + D B).

We investigate properties which remain invariant under the action of quasi-Möbius maps of quasi-metric spaces. A metric space is called doubling with constant D if every ball of nite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting complex wave ...

ABSTRACT We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multidimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multidimensional triangular system.

ABSTRACT We consider a system of nonlinear partial differential equations that arises in the modeling of two-phase flows in a porous medium. The phase velocities are modeled using a Brinkman regularization of the classical Darcy's law. We propose a notion of weak solution for these equations and prove existence of these solutions. An efficient finite difference scheme is proposed and is shown to converge to the weak solutions of this system. The Darcy limit of the Brinkman regularization is studied numerically using the convergent finite difference scheme in two space dimensions as well as using both analytical and numerical tools in one space dimension. The results suggest that the Brinkman regularization may not approximate the accepted entropy solutions of the Darcy model and raise fundamental questions about the use of Brinkman type models in two-phase flows.

ABSTRACT This book introduces novel design techniques developed to increase the safety of aircraft engines. The authors demonstrate how the application of uncertainty methods can overcome problems in the accurate prediction of engine lift, caused by manufacturing error. This in turn ameliorates the difficulty of achieving required safety margins imposed by limits in current design and manufacturing methods. This text shows that even state-of-the-art computational fluid dynamics (CFD) is not able to predict the same performance measured in experiments; CFD methods assume idealised geometries but ideal geometries do not exist, cannot be manufactured and their performance differs from real-world ones. By applying geometrical variations of a few microns, the agreement with experiments improves dramatically, but unfortunately the manufacturing errors in engines or in experiments are unknown. In order to overcome this limitation, uncertainty quantification considers the probability density functions of manufacturing errors. It is then possible to predict the overall variation of the jet engine performance using stochastic techniques. Uncertainty Quantification in Computational Fluid Dynamics and Aircraft Engines demonstrates that some geometries are not affected by manufacturing errors, meaning that it is possible to design safer engines. Instead of trying to improve the manufacturing accuracy, uncertainty quantification when applied to CFD, is able to indicate an improved design direction.

Abstract. We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the ...