Analysis and Numerical Treatment of Nonlinear Hyperbolic Conservation Laws on Graphs

Abstract

Hyperbolic conservation laws are used to model various important applications such as gas flow or traffic flow. Those phenomena are interesting to study in a one-dimensional setting, it would be even more relevant for real world applications, to study those equations on networks. Whilst the theory for hyperbolic conservation laws in 1D is fairly extensive, many questions are still open for the network case. My thesis addresses and solves several of these open questions. In particular, my thesis addresses the question of well-posedness of hyperbolic conservation laws on networks. The question of well-posedness consists of three sub-questions, which are existence, uniqueness, and stability of a solution. In my thesis I present a fairly general well-posedness theory for a large class of equations that include models of gas flow and traffic flow on networks. Furthermore, I developed a computer program that allows to compute approximate solutions to said equations and showed that this algorithm converges towards the actual solution. In addition to showing convergence of the algorithm, I also show results on how fast the algorithm converges towards the actual solution. This is important to know for computations of actual use cases.