$\Delta t=t_1-t_0$. This quantity effectively controls the consistency between the instantaneous energy generation and the subsequent chemical evolution based on it. Whenever $|1-\delta L |$ exceeds a certain limit (presently set to about 0.7), the chosen time-step is immediately reduced and the chemical evolution is recalculated. The time-step is decreased until $|1-\delta L | $ becomes sufficiently small. With this and additional criteria for the determination of time steps and grid resolution, it is possible to follow numerically the He-flash and HEFM evolution of low-mass metal-free stars both accurately and efficiently.

In the lower panel of Fig. \[fig1\] $\delta L$ during the approach of the HECZ to the H-rich layers is shown. Already at $t-t_0 \approx 10\,{\rm yr}$, i.e. before $L_{\rm H}$ exceeds $L_{\rm He}$, hydrogen is mixed into the He-burning convective zone, causing an increase in $\log (L_{\rm
H}/L_\odot)$ from initially about -0.5 to 2.5 (thick solid line in upper panel of Fig. \[fig1\]). This energy release is, however, negligible compared to the energy produced by the fusion of helium. Hence, even with relatively large time-steps, $|1-\delta L|$ remains below 0.5. When the amount of ingested hydrogen is sufficient to affect the subsequent evolution ($t-t_0
\approx29 \,{\rm yr}$), $\delta 