(2004a) applied the method to multi-dimensional, purely deflagration explosion models of SNe Ia.

In the setup for our hydrodynamic simulations, $80^{2}$ tracer particles are distributed uniformly in mass coordinate, such that each particle represents the same mass of $\sim 2.2 \times 10^{-4} M_{\odot} (=M_{\rm wd}/6400)$. The particles are advected passively, following the velocity field at each time step of the Eulerian hydrodynamic simulations. The thermal history each particle experiences is recorded. The number of the tracer particles is sufficient to accurately follow the nucleosynthesis. Seitenzahl et al. (in preparation) find that in two-dimensional SN Ia simulations with $80^{2}$ tracer particles all isotopes with abundances higher than $\sim 10^{-5}$ are reproduced with an accuracy of better than 5% (except for $^{20}$Ne).

The nuclear postprocessing calculations are then performed for each particle separately. To this end, we recalculate the temperature from the recorded internal energy, rather than directly using the value obtained by the hydrodynamic simulations (see Travaglio et al. 2004a). In deriving the temperature, the electron fraction ($Y_{\rm e}$) is assumed to be 0.5, which introduces some errors when the electron captures are very active. When $T_{9} = T/10^9$ K $ > 6$, we follow the abundance evolution by applying the 