  Type Ia supernovae (SNe Ia) are among the most energetic events in the
Universe. They are excellent cosmological distance indicators due to the
remarkable homogeneity of their light curves. However, the nature of the
progenitors of SNe Ia is still not well understood. In the single-degenerate
model, a carbon-oxygen white dwarf (CO WD) could grow its mass by accreting
material from an asymptotic giant branch (AGB) star, leading to the formation
of SNe Ia when the mass of the WD approaches to the Chandrasekhar-mass limit,
known as the AGB donor channel. In this channel, previous studies mainly
concentrate on the wind-accretion pathway for the mass-increase of the WDs. In
the present work, we employed an integrated mass-transfer prescription for the
semidetached WD+AGB systems, and evolved a number of WD+AGB systems for the
formation of SNe Ia through the Roche-lobe overflow process or the
wind-accretion process. We provided the initial and final parameter spaces of
WD+AGB systems for producing SNe Ia. We also obtained the density distribution
of circumstellar matter at the moment when the WD mass reaches the
Chandrasekhar-mass limit. Moreover, we found that the massive WD+AGB sample AT
2019qyl can be covered by the final parameter space for producing SNe Ia,
indicating that AT 2019qyl is a strong progenitor candidate of SNe Ia with AGB
donors.
