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In this work we study compact stars, i.e. neutron stars, as cosmic laboratories for the nuclear matter. With a mass of around 1 - 3 solar masses and a radius of around 10km, compact stars are very dense and, besides nucleons, can contain exotic matter such as hyperons or quark matter. The KaoS collaboration studied nuclear matter for densities up to 2-3 times saturation density by analysing kaon multiplicities from Au+Au and C+C collisions. The results show that nuclear matter in the corresponding density region is very compressible, with a compressibility of <200MeV. For such soft nuclear equations of state the maximum masses of neutron stars are ca. 1.8 - 1.9 solar masses, whereas the central densities are higher than 5 times nuclear saturation density and therefore point towards a possible phase transition to quark matter. If quark matter would be present in the interior of neutron stars, so-called hybrid stars, it could be produced already during their birth in supernova explosions. To study this we implement a quark matter phase transition in a hadronic equation of state which is used in supernova simulations. Supernova simulations of low and intermediate mass progenitors and two different bag constants show a collapse of the proto neutron star due to the softening of the equations of state in the quark-hadron mixed phase. The stiffening of the equation of state for pure quark matter halts the collapse and leads to the production of a second shock wave. The second shock wave is energetic enough to lead to an explosion of the star and produces a neutrino burst when passing the neutrinospheres. Furthermore, first studies of the longtime cooling of hybrid stars show, that colour superconductivity can significantly influence the cooling behaviour of hybrid stars, if all quarks form Cooper Pairs. For the so-called CSL phase (colour-spin locking) with pairing energies of several MeV, the cooling of the quark phase is suppressed and the hybrid star appears as a pure hadronic star.
High-energy astrophysics plays an increasingly important role in the understanding of our universe. On one hand, this is due to ground-breaking observations, like the gravitational-wave detections of the LIGO and Virgo network or the black-hole shadow observations of the EHT collaboration. On the other hand, the field of numerical relativity has reached a level of sophistication that allows for realistic simulations that include all four fundamental forces of nature. A prime example of how observations and theory complement each other can be seen in the studies following GW170817, the first detection of gravitational waves from a binary neutron-star merger. The same detection is also the chronological starting point of this Thesis. The plethora of information and constraints on nuclear physics derived from GW170817 in conjunction with theoretical computations will be presented in the first part of this Thesis. The second part goes beyond this detection and prepares for future observations when also the high-frequency postmerger signal will become detectable. Specifically, signatures of a quark-hadron phase transition are discussed and the specific case of a delayed phase transition is analyzed in detail. Finally, the third part of this Thesis focuses on the inclusion of radiative transport in numerical astrophysics. In the context of binary neutron-star mergers, radiation in the form of neutrinos is crucial for realistic long-term simulations. Two methods are introduced for treating radiation: the approximate state-of-the-art two-moment method (M1) and the recently developed radiative Lattice-Boltzmann method. The latter promises
to be more accurate than M1 at a comparable computational cost. Given that most methods for radiative transport or either inaccurate or unfeasible, the derivation of this new method represents a novel and possibly paradigm-changing contribution to an accurate inclusion of radiation in numerical astrophysics.