On the hyperbolicity of evolution equations for relativistic fluids
In this dissertation, systems of partial differential equations describing neutron stars in numerical relativity are investigated concerning their hyperbolicity properties. First, fundamental definitions are given and well-posedness of the initial value problem is explained. The main tool for the hyperbolicity analysis of fluid systems in this thesis is the so-called dual frame formalism in which two different frames can be related to each other. With the help of the formalism, it is shown that strong hyperbolicity is independent of the chosen frame if the appearing speeds are subluminal. Second, with regard to the numerical modeling of neutron stars, the partial differential equation systems of ideal hydrodynamics, ideal magnetohydrodynamics, and resistive magnetohydrodynamics are investigated in general relativity. The system of ideal hydrodynamics serves as a test system for the application of the dual frame formalism. The main focus of this work lies on the investigation of ideal magnetohydrodynamics used in numerical relativity. Two formulations of the system of equations are distinguished, determined by the presence of parametrized combinations of the magnetic field constraint in the evolution equations. The first formulation is strongly hyperbolic. In contrast, the second so-called flux-balance law formulation, which is used in numerical relativity, turns out to be only weakly hyperbolic and hence, no well-posed initial value problem can be found. Finally, the two numerically used systems of equations for resistive magnetohydrodynamics are investigated, and again both systems turn out to be only weakly hyperbolic. The flux-balance law formulation of classical magnetohydrodynamics, as well as the systems of dust and charged dust also turn out to be weakly hyperbolic, for the latter at least in the minimally coupled case. Thus, the results have great impact on the current numerical modeling of neutron stars.