Energy conservation

After having put much effort into the consistent finite differencing of the transport equation in favor of an accurate evolution of expectation values, we investigate in this section the energy conservation properties in the most challenging run started from the massive M progenitor star. Our scheme preserves lepton numbers to machine precision by construction. In Fig. ()

Figure: The different components of the energy budget are shown. The internal energy of the fluid is represented with a thin solid line in the upper half of the figure. We set the free constant in the internal energy such that we start with zero total energy (thick solid line). Most of the internal energy is balanced by the gravitational energy (thin solid line in the lower half of the figure). We show also the kinetic energy of the fluid (thin dashed line) and the surface work exerted at the border of the computational domain (dotted line). The dash-dotted line represents the energy of the neutrinos in the computational domain and the dashed thick line represents the accumulated energy of the escaped neutrinos. The total energy is nicely conserved during core collapse and exhibits a systematic increase of order of an explosion energy during the crucial phase around ms after bounce.

we show how the total energy in the simulation divides up into different energy forms during the simulation. The gravitational energy (thin solid line in the lower part of the figure) and the internal energy of the fluid (thin solid line in the upper part of the figure) form the largest contributions. For this figure, we have set the undetermined constant in the internal energy such that the total energy vanishes at the start of the simulation. The energy stored in neutrinos (dash-dotted line) grows with respect to the internal fluid energy until bounce. Afterwards, it evolves almost proportionally to the internal fluid energy. The kinetic energy (thin dashed line) also grows during collapse. It peaks at bounce and settles on a lower level afterwards, slightly decaying during the stationary postbounce phase because of the decreasing density of infalling material. The dotted line at the center of the figure is the work exerted on the surface of the computational domain. The thick dashed line is the accumulated energy emitted by neutrinos. Its steep increase around bounce is delayed with respect to the other energy contributions because of the delay in the neutrino burst and the propagation time to the surface of the computational domain at km radius. The thick solid line represents the evolution of the total energy in our simulation. It is very accurately conserved during the core collapse phase. It shows a perturbation of order erg in the most dynamic phase around bounce when the grid points rush to the center to resolve the shock front. It systematically increases afterwards and reaches the order of an explosion energy at the end of the simulation when the neutron star collapses to a black hole. We further investigate this energy violation in the next two figures.

We would obtain energy conservation to machine precision if we could enforce perfect cancellation in the six chains discussed after Eq. (), i.e. , , , , , and . The hydrodynamics scheme is designed to absorb the energy exchange by the collision integral to machine precision and conserve energy perfectly, even on the adaptive grid. By selecting only the positive contributions in the canceling terms we obtain a measure of the importance for an accurate cancellation. After an integration over the computational domain, Fig. ()

Figure: For the six chains of terms that are supposed to cancel in Eq. () we show the size of all positive contributions to measure the importance of accurate cancellations. The implementation of the O cancellations and are most important (thick solid line and thick dashed line, respectively). Of similar importance is a matching in the general relativistic term (thin solid line). About one order of magnitude less important is the matching (thin dashed line). Of lowest importance is the matching among observer corrections themselves in and (dash-dotted and dotted lines, respectively). The figure illustrates the steep increase in the challenge to conserve energy after bounce.

illustrates this measure for the six above-mentioned expressions with a thick dashed line, a thick solid line, a dash-dotted line, a dotted line, a thin dashed line, and a thin solid line, respectively. After bounce, the maximum individual contribution to the energy conservation equation in Eq. (27) reaches a typical level of erg/s. Fig. () makes immediately evident that maintaining accurate energy conservation is more challenging after bounce than before bounce. Moreover, we note that the O terms and are much larger than the higher order terms and .

Nevertheless, these terms are perfectly matched in our implementation. Violation of energy conservation therefore stems from the terms (the matching is tuned for an isotropic neutrino distribution) and (the matching is based on an energy flux averaging) as discussed in the context of Eqs. () and () respectively. In section we have discussed energy conservation violations by the adaptive grid corrections when they are applied to the radiation quantities. Where do they enter the conservation check? If we evaluate Eq. (27) on the adaptive grid according to the recipe in Eq. (), we note that the integration of the energy over the whole star reduces the adaptive grid corrections to surface terms. These surface terms vanish because the grid velocity is zero at the center and the surface of the star. If we compare with the more detailed Eq. () we find that this time derivative corresponds exactly to the term . The energy violations by the adaptive grid show only up in the terms and . If these terms are evaluated with all the grid corrections in the time evolution of , , , and , they will numerically differ from the terms and we have used in Eqs. () and () for the approximate matching. We check the influence of the adaptive grid corrections by a comparison with a run using a pure Lagrangian grid, where no adaptive grid corrections can compromise energy conservation. However, in order to get enough resolution in the run with the fixed grid, we had to run with spatial zones instead of the we used with the adaptive grid. This run is extremely slow. On the one hand, the solution vector is four times larger. But much more important is that every single zone has to change its value from the preshock conditions to the postshock conditions in the allowed -change steps. This requires an almost times smaller time step than with the adaptive grid, where a zone can follow the shock. In the latter case, the conditions in the zone changes on a much longer time scale determined by the drift between the zone speed and the shock propagation. In order to let a run reach the interesting phase around ms after bounce in reasonable time, we had to reduce the angular resolution to only two angular bins. Both measures, the increase of spatial and time resolution and the decrease of angular resolution can in principle affect energy conservation. Nevertheless, we hope to get the correct impression of the influence of the adaptive grid on the energy conservation. The two terms and of the Lagrangian run are also shown in Fig. () (thin lines). We find that the order of magnitude of energy violation in the cancellation does not significantly change on the fixed grid. However, the cancellation in the term is greatly reduced on the fixed grid. This suggests that the adaptive grid corrections of the radiation quantities are the dominant remaining sources of energy violation, about five times larger than the mismatch in .

Figure: The thick solid line and thick dashed line represent the error in the matching of the terms and respectively. All other matches are to machine precision. Therefore, the time integration of these energy rates must reproduce the drift in the total energy represented by a thick solid line in Fig. (). The two monitored expressions do not allow a distinction of errors induced by approximations in the matching procedure in Eqs. () and () from errors induced by the application of the adaptive grid to radiation quantities in the comoving frame (see discussion following Eq. ()). In an attempt to disentangle these two contributions to the violation of energy conservation, we compare the cancellation in the terms and with a Lagrangian run with zones and angular bins (thin solid line and thin dashed line, respectively). The energy violation of the former term is greatly reduced on a fixed grid.

The absolute necessity of accurate energy conservation may be discussed. But certainly, it provided an invaluable check for the congruence between the programmers intention and the actual implementation of the many intricate finite difference expressions in our code.