Type Ia supernovae (SNe~Ia) are important objects in astronomy and astrophysics. They are one of the brightest and common transients in the universe, a standard candle for a cosmic distance, and a dominant origin of iron group elements. It is widely accepted that SNe~Ia are thermodynamical explosions of carbon-oxygen (CO) white dwarfs (WDs) in binary systems. Nevertheless, it is yet unclear what the companion stars are, and what the masses of the exploding CO~WDs are. Recently, many sub-Chandrasekhar mass explosions in the double-degenerate scenario have been suggested. We have assessed the violent merger and Dynamically-Driven Double-Degenerate Double-Detonation (D$^6$) models by numerical simulations. We have concluded that the violent merger model may not explain normal SNe~Ia, since their circum-binary matter is much larger than the size of SN~2011fe, and their event rate is much smaller than the SN~Ia rate in our Galaxy by an order of magnitude. However, it can be progenitors of peculiar SNe~Ia. The D$^6$ model has roughly consistent features with normal SNe~Ia. However, the model has supernova ejecta with an asymmetric shape and low-velocity carbon-oxygen component due to the presence of the surviving companion CO. If these features can be observed, the D$^6$ model may be inconsistent with normal SNe~Ia. In this case, the D$^6$ model can be also progenitors of peculiar SNe~Ia.