The double-degenerate (DD) model, involving the merging of massive double carbon-oxygen white dwarfs (CO WDs) driven by gravitational wave radiation, is one of the classical pathways for the formation of type Ia supernovae (SNe Ia). Recently, it has been proposed that the WD+He subgiant channel has a significant contribution to the production of massive double WDs, in which the primary WD accumulates mass by accreting He-rich matter from a He subgiant. We evolved about 1800 CO WD+He star systems and obtained a large and dense grid for producing SNe Ia through the DD model. We then performed a series of binary population synthesis simulations for the DD model, in which the WD+He subgiant channel is calculated by interpolations in this grid. According to our standard model, the Galactic birthrate of SNe Ia is about 2.4*10^{-3} yr^{-1} for the WD+He subgiant channel of the DD model; the total birthrate is about 3.7*10^{-3} yr^{-1} for all channels, reproducing that of observations. Previous theoretical models still have deficit with the observed SNe Ia with delay times <1 Gyr and >8 Gyr. After considering the WD+He subgiant channel, we found that the delay time distributions is comparable with the observed results. Additionally, some recent studies proposed that the violent WD mergers are more likely to produce SNe Ia based on the DD model. We estimated that the violent mergers through the DD model may only contribute to about 16% of all SNe Ia.
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