Fabrice Durier & Claudio Dalla Vecchia; arXiv:1105.3729v1

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.

Sedov's explosion test for the reference simulations, when the system is evolved on global time-steps. Results are given in the natural system of units at t=0.02 after the explosion. We show in the upper row the projected gas density in a slice of thickness &Delta z=0.1 centred on the box origin. The dashed circle corresponds to the position of the spherical shock-front given by the similarity solution at that time. The white dots mark the position of all particles that initially received the energy. In the bottom row, the average radial density profiles are compared to the profiles given by Sedov's analytic solution (red lines). We see that both methods agree remarkably well in reproducing the similarity solution for both the location and the width of the blast shell.

Sedov's explosion test when the individual time-step integration is used. Images are the same as previously. Results are given soon after the explosion at time t=0.004. Given the large energy jump introduced by the explosion, it is clear from these plots that none of the feedback scheme are able to describe properly Sedov's test.

Sedov's explosion test when the limiter is applied alone. Animations are covering the entire simulation. We see that for the thermal implementation of feedback a stable shell develops too quickly ahead from the similarity solution because of the initial large violation of energy conservation. Given the periodic property of the simulated volume, the shock re-enter the box from each other sides and creates interferences. In the kinetic feedback case inter-particle crossing is evident. No density contrast has yet developed given that kicked particles have traveled far from the explosion site. These results illustrate how the lack of a prompt response of the medium leads to two distinct effects from the two feedback schemes.

Sedov's explosion test when both the limiter and the time-step update are applied. We show here that, for our recommended set of parameters, the two feedback schemes are able to reproduce precisely the similarity solution and hence provide again concordant results as for the global time integration scheme. This demonstrate that both a fast information transport and a prompt response to the explosion are needed in order to resolve strong feedback events with individual time-step integration.

The off-centre explosion in a self-gravitating gas sphere for thermal (left column) and kinetic (right column) energy injection. In all movies we show the gas density in a slice of thickness &Delta z=0.1 centred on the box origin. The white dots mark the position of all particles that initially received thermal or kinetic energy. Top row: standard individual time-step scheme. The behaviour is similarly wrong in both cases. The halo atmosphere is disrupted and no expanding bubble forms. Bottom row: time-step limiter scheme only is applied using the conservative value of f_step=2. The energy violation is severe in the thermal case and the bubble have blown away a large fraction of the gas halo.

The off-centre explosion in a self-gravitating gas sphere when time-step limiter and update are applied. All movies are as the previous ones. Top row: the energy is injected on 32 particles in thermal or kinetic form. Impacted particles are made active and the time-step of surrounding particles has been corrected. The results are qualitatively identical, showing concordance of the two feedback methods. Bottom row: all available energy is injected into one particle. The results excellently agree with each other and with the cases in which the energy is injected into 32 particles.