We realized a time-domain simulator based on the electrodynamics of Cooper pairs and quasiparticles in Josephson junctions. The tool, based on the charge carriers' densities of states described by the Werthamer and Harris formalisms of Bardeen-Cooper-Schrieffer (BCS) theory, allows to analyze the behavior of current-or voltage-controlled Josephson junction-based circuits for any signal waveform, at a physical temperature comprised between the absolute temperature and the critical temperature. The simulator can account for I-V curve hysteresis depending on the McCumber parameter, or Shapiro and photo-assisted steps in the presence of a monochromatic microwave signal. We used the simulator to assess the behavior of MgB2-based Josephson junctions at THz frequencies by taking into account the presence of the two anisotropic gaps of MgB2.