Recent experiments indicate that cavitation has a much bigger impact on injection processes than previously assumed. However, cavitation is still neglected in most numerical simulations and models. One reason for this is its numerical complexity due to compressible and turbulent flow conditions as well as resulting shocks and discontinuities. Currently, models predicting the impact of cavitation on spray processes without resolving very small length scales do not exist. A simulation framework enabling highly accurate simulations of cavitating fuel injection processes was established for this work and relies on three components: First, adequate numerical methods for Large-Eddy Simulations (LES) of compressible nozzle flows including an equation-of-state based cavitation model were adopted. Second, accurate methods for computing primary breakup in the vicinity of the nozzle exit were incorporated. These are based on a 3D unsplit forward/backward Volume-of-Fluid (VOF) approach that is coupled to a Level Set (LS) method (3DU-CLSVOF) and a hybrid discretization of the convective transport term and of the pressure-projection. Third, the framework couples the highly accurate results of the primary breakup simulations to Lagrangian particle-based spray simulations in order to achieve results for the spray evolution further downstream of the nozzle with currently available computing resources. In the current work, simulations of a gasoline direct injection (GDI) system are performed and the results are verified with Xray measurements, spray penetration length data and Laser Diffraction Spray Analyzer (LDSA) data.