Strong gravity and relativistic plasma flows are among the fundamental ingredients powering high-energy astrophysical phenomena such as short and long gamma ray bursts, core-collapse supernovae and relativistic outflows from black-hole accreting systems. General-relativistic hydrodynamics is also essential in modelling the merger of neutron stars binaries and black-hole neutron- star binaries that are among the best sources for future gravitational-wave detectors such as LIGO, Virgo or KAGRA. Over the past decade, the understanding of these phenomena has benefited significantly from the results obtained through non-linear numerical calculations. Key factors in this progress have been the switch to more advanced numerical schemes that are able to properly treat relativistic shock waves, and the progressive inclusion of more “physics”, such as magnetic fields or realistic equations of state. Following this trend, even better numerical tools and more accurate physical description will be be essential to understand these phenomena. This thesis aims at contributing to both of these aspects.