Implementation of feedback in SPH: towards concordance of methods
  We perform simulations of feedback from supernovae and black holes with
smoothed particle hydrodynamics (SPH). Such strong perturbations are
inaccurately handled with standard time integration schemes, leading to poor
energy conservation, a problem that is commonly overlooked. We show for the
first time that, in the absence of radiative cooling, concordance of thermal
and kinetic feedback are achieved when using an accurate time integration. In
order to preserve the concordance of feedback methods when using a more
efficient time integration scheme - as for instance the hierarchical time-step
scheme - we implement a modified version of the time-step limiter proposed by
Saitoh & Makino (2009). We apply the limiter to general test cases, and first
show that this scheme violates energy conservation up to almost four orders of
magnitude when energy is injected at random times. To tackle this issue, we
find necessary, not only to ensure a fast information propagation, but also to
enforce a prompt response of the system to the energy perturbation. The method
proposed here to handle strong feedback events enables us to achieve energy
conservation at percent level in all tests, even if all the available energy is
injected into only one particle. We argue that concordance of feedback methods
can be achieved in numerical simulations only if the time integration scheme
preserve a high energy conservation level.
