Resolution

In spite of our ambition to produce accurate results already at low resolution, the reader has been left wondering about the actual resolution dependence of our data. In this subsection, we demonstrate reasonable convergence of our results by separately doubling the resolution in each phase space dimension in otherwise identical simulations. Historically, our standard resolution with adaptive zones, angular bins, and energy groups formed as the minimum resolution where we felt safe about our physical conclusions. In Fig. ()

Figure: Resolution dependence of the results. The standard run (circles) was calculated with adaptive zones, angular bins, and energy groups. We compare with runs with zones (solid line), angular bins (dash-dotted line), and energy groups (dashed line).

we compare the results with a simulation that uses adaptive zones, one that uses angular bins, and one that uses energy groups. We focus on the quantities and instances where we find the largest deviations. We start with two limitations that are well-known. Graph (a) shows the mean flux factor at ms after bounce in the neutrino heating phase. It can clearly be seen, that the run with angular bins determines the flux factor with more accuracy in the free streaming limit Messer_et_al_98,Yamada_Janka_Suzuki_99. The angular resolution determines how close to the radial direction the most forward peaked angular bins are. The closer they are the larger is the asymptotic limit of the flux factor. In graph (b), we show another previously documented effect Mezzacappa_Bruenn_93b. In the last half of a millisecond before bounce, insufficient resolution of the energy phase space leads to a poor representation of the Fermi energy in the degenerate neutrino gas when the trapped neutrinos are compressed to high densities. The consequence are rapid and transient local displacements of neutrinos. At poor energy resolution, the Fermi energies rather increase in steps than continuously, and the neutrino fluxes try to balance the numerical variations in the Fermi energies between adjacent zones. Therefore, the effect is best seen in the electron neutrino luminosities as shown in graph (b). While the conserved lepton number cannot show numerical variations, the electron fraction and neutrino abundances also reflect the transient steps in the Fermi energies. Variations in these quantities are of order . However, because the neutrinos are trapped and because our scheme is conservative, this transient wiggles do not lead to any differences that survive bounce.

This is shown in the two graphs (c) and (d) at bounce where the entropy and luminosity profile, respectively, have again converged with respect to energy resolution. At this stage, we detect an influence of the spatial resolution. During collapse, there is no special region with a high concentration of grid points. At bounce, however, the grid points speed inwards to resolve the newborn shock wave. We detect an effect of this rapid grid displacement in the entropy and electron neutrino luminosity profiles to the extent shown in graphs (c) and (d). Differences in the entropies can also be seen at the interface between the silicon layer and nuclear statistical equilibrium at a radius km. The difference is of no concern because it stems from the granular triggering of zone conversions from silicon to NSE. Each run determines autonomously when output files are dumped. For a given time, we then compare the output with the closest available output files in other runs. Hence, it can happen that the conversion of a zone took already place in one run, but not in another one. The location of the transition to NSE shows an uncertainty of at least one zone width. The first ms after bounce are probably the most dynamical time in the simulations. At this time we find again the most prominent resolution dependencies in the entropy and luminosity profiles. It is still a dependency on the space resolution alone. The entropy in the high energy resolution run carries the slight enhancement from before, overlapped by a slightly narrower and deeper cooling at the launch of the neutrino burst. The higher luminosity peak in graph (f) at higher spatial resolution comes not unexpectedly, as we have seen in subsection that the decay of the outwards propagating neutrino burst depends on the zone width. We can also demonstrate by this resolution study, that the described pulse spreading does only marginally depend on the angular resolution. Note that the apparent difference in the run with higher energy resolution is not a real difference, it stems from an insufficient time match between the output files and shows the rapidly decaying luminosity profile at a slightly earlier time.

Finally, we compare the runs with different resolutions during the important neutrino heating phase, e.g. at ms after bounce. Graphs (g) and (h) show again the entropy and luminosity profiles. We are happy to report that the simulations are converged at this stage. With the exception of the slightly higher entropy in the outer layers of the high space resolution run, none of the previously discussed differences has survived to this time. Moreover, the quality of agreement presented in these graphs is similar to what we find in the velocity, logarithmic density, electron fraction, and rms neutrino energy profiles at any time during the simulations. We conclude that convergence issues are far from affecting our physical conclusions from the simulations. Only if one asks for high precision numbers in specific quantities, we may, with descending importance, recommend an increase of space-, energy-, and angle-resolution.