Implementation of feedback in smoothed particle hydrodynamics: towards concordance of methods

We perform simulations of feedback from supernovae and black holes with smoothed particle hydrodynamics. 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. We apply the limiter to general test cases, and first show that this scheme violates energy conservation up to almost 4 orders of magnitude when energy is injected at random times. To tackle this issue, we find it 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 per cent 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.