How To Model Supernovae in Simulations of Star and Galaxy Formation
  We study the implementation of mechanical feedback from supernovae (SNe) and
stellar mass loss in galaxy simulations, within the Feedback In Realistic
Environments (FIRE) project. We present the FIRE-2 algorithm for coupling
mechanical feedback, which can be applied to any hydrodynamics method (e.g.
fixed-grid, moving-mesh, and mesh-less methods), and black hole as well as
stellar feedback. This algorithm ensures manifest conservation of mass, energy,
and momentum, and avoids imprinting 'preferred directions' on the ejecta. We
show that it is critical to incorporate both momentum and thermal energy of
mechanical ejecta in a self-consistent manner, accounting for SNe cooling radii
when they are not resolved. Using idealized simulations of single SN
explosions, we show that the FIRE-2 algorithm, independent of resolution,
reproduces converged solutions in both energy and momentum. In contrast, common
'fully-thermal' (energy-dump) or 'fully-kinetic' (particle-kicking) schemes in
the literature depend strongly on resolution: when applied at mass resolution
>100 solar masses, they diverge by orders-of-magnitude from the converged
solution. In galaxy-formation simulations, this divergence leads to
orders-of-magnitude differences in galaxy properties, unless those models are
adjusted in a resolution-dependent way. We show that all models that
individually time-resolve SNe converge to the FIRE-2 solution at sufficiently
high resolution. However, in both idealized single-SN simulations and
cosmological galaxy-formation simulations, the FIRE-2 algorithm converges much
faster than other sub-grid models without re-tuning parameters.
