Type Ia supernovae (SNe Ia) play an important role in astrophysics. They have been successfully used as a cosmological distance indicator because of their high luminosity and remarkable uniformity.This leads to the discovery of the expansion of the Universe, indicating the existence of dark energy. Although SNe Ia have been used as standard candles for accurate distance measurements of faraway galaxies, it is still poorly understood how they form and explod, and there are many uncertainties and controversies about them. This dissertation begins with reviewing the observational history of SNe Ia. It then proceeds to make a brief introduction of recent research progress, in particular theories regarding SNe Ia and some observational characteristics of SNe Ia. Employing the stellar evolution code MESA, we simulate the accretion process of hydrogen-rich or helium-rich material onto CO WDs, and obtain a deeper understanding of the mass accumulation process of CO WDs prior to type Ia supernova explosion. We investigate the accretion process of H-rich material onto a CO WD. The simulation demonstrates the characteristics of the double shell burning on top of the WD, with a hydrogen shell burning on top of a helium burning shell. The results show that helium shell burning is not steady (i.e. it flashes) even though hydrogen shell burning is steady enough (i.e. with a H-rich material accretion rate over the Eddington limit). The CO WD could grow in mass during the flashes and flashes from the helium shell are weaker than those in the case of accretion of helium-rich material onto a CO WD. The carbon to oxygen mass ratio resulting from the helium shell burning is higher than what was previously thought. The flashes become weaker and weaker as the accretion continues. Additionally, we make a description of the helium shell flash in detail. We also simulate the long-term evolution of CO WDs accreting helium-rich material with different initial masses of WDs and mass accretion rates. We find that the C/O ratio varies for different cases. In the steady region for helium shell burning, both a low initial WD mass and a low accretion rate will lead to a higher C/O ratio for the He-accreting CO WDs before its explosion, and this diverse of C/O ratio may be one of the reasons for the diversity of SNe Ia and may be related to the Phillips relation of SNe Ia. Besides, the C/O ratio we get prior to type Ia supernova explosion may influence the research related to the SNe Ia explosion model.
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