  We examine the properties of white dwarfs (WDs) accreting hydrogen-rich
matter in and near the stable burning regime of accretion rates as modeled by
time-dependent calculations done with Modules for Experiments in Stellar
Astrophysics (MESA). We report the stability boundary for WDs of masses between
0.51 solar masses and 1.34 solar masses as found via time-dependent
calculations. We also examine recurrent novae that are accreting at rates close
to, but below, the stable burning limit and report their recurrence times and
ignition masses. Our dense grid in accretion rates finds the expected minimum
possible recurrence times as a function of the WD mass. This enables inferences
to be made about the minimum WD mass possible to reach a specific recurrence
time. We compare our computational models of post-outburst novae to the stably
burning WDs and explicitly calculate the duration and effective temperature
(Teff) of the post-novae WD in the supersoft phase. We agree with the measured
turnoff time - Teff relation in M31 by Henze and collaborators, infer WD masses
in the 1.0-1.3 solar masses range, and predict ejection masses consistent with
those observed. We close by commenting on the importance of the hot helium
layer generated by stable or unstable hydrogen burning for the short- and
long-term evolution of accreting white dwarfs.
