In the framework of the LHC Injector Upgrade project, a new internal dump for the CERN Proton Synchrotron (PS) ring has been designed, installed, and successfully commissioned. This device is designed to move rapidly into the beam and stop charged particles over several thousand turns to provide protection to PS hardware against beam-induced damage. Due to its design, the internal dump absorbs only a fraction of the secondary particles shower produced by the beam particles that impinge on it. The performance of the dump should ensure efficient use throughout the PS energy range, i.e. from injection at 2 GeV (kinetic energy) to flat top at 26 GeV (total energy). This paper presents comprehensive numerical simulations that combine advanced beam dynamics and beam-matter interaction codes to analyse the behaviour of stopped or scattered particles. Based on the impacts computed by multi-turn beam dynamics simulations, detailed shower simulations with FLUKA were performed to assess the impact of the radiation field on downstream equipment, with a particular emphasis on the dose measured by Beam Loss Monitors. The results of these numerical simulations are compared with the data collected during the routine operation of the PS and its internal dump.