  We present evolutionary calculations aimed at describing the born-again
scenario for post-AGB remnant stars of 0.5842 and 0.5885 \msun. Results are
based on a detailed treatment of the physical processes responsible for the
chemical abundance changes. We considered two theories of convection: the
standard mixing length theory (MLT) and the double-diffusive GNA convection
developed by Grossman et al. The latter accounts for the effect of the chemical
gradient ($\nabla\mu$) in the mixing processes and in the transport of energy.
We also explore the dependence of the born-again evolution on some physical
hypothesis, such as the effect of the existence of non-zero chemical gradients,
the prescription for the velocity of the convective elements and the size of
the overshooting zones. Attention is given to the behavior of the born-again
times and to the chemical evolution during the ingestion of protons. We find
that in our calculations born again times are dependent on time resolution. In
particular when the minimum allowed time step is below $5 \times 10^{-5}$ yr we
obtain, with the standard mixing length theory, born again times of 5-10 yr.
This is true without altering the prescription for the efficiency of convective
mixing during the proton ingestion. On the other hand we find that the
inclusion of the chemical gradients in the calculation of the mixing velocity
tend to increase the born again times by about a factor of two. In addition we
find that proton ingestion can be seriously altered if the occurrence of
overshooting is modified by the $\nabla\mu$-barrier at the H-He interface,
strongly altering born again times.
