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We introduce the LYRA project, a new high resolution galaxy formation modelbuilt within the framework of the cosmological hydro-dynamical moving mesh codeAREPO. The model resolves the multi-phase interstellar medium down to 10 K. Itforms individual stars sampled from the initial mass function (IMF), and trackstheir lifetimes and death pathways individually. Single supernova (SN) blastwaves with variable energy are followed within the hydrodynamic calculation tointeract with the surrounding interstellar medium (ISM). In this paper, wepresent the methods and apply the model to a $10^{10} M_{\odot}$ isolated halo.We demonstrate that the majority of supernovae are Sedov-resolved at ourfiducial gas mass resolution of $4 M_{\odot}$. We show that our SN feedbackprescription self-consistently produces a hot phase within the ISM that drivessignificant outflows, reduces the gas density and suppresses star formation.Clustered SN play a major role in enhancing the effectiveness of feedback,because the majority of explosions occur in low density material. Accountingfor variable SN energy allows the feedback to respond directly to stellarevolution. We show that the ISM is sensitive to the spatially distributedenergy deposition. It strongly affects the outflow behaviour, reducing the massloading by a factor of 2-3, thus allowing the galaxy to retain a higherfraction of mass and metals. LYRA makes it possible to use a comprehensivemulti-physics ISM model directly in cosmological (zoom) simulations of dwarfand higher mass galaxies.

LYRA I: Simulating the multi-phase ISM of a dwarf galaxy with variable energy supernovae from individual stars