In this paper we report some specific features of the numerical technique used to study the dynamic evolution of massive white dwarfs following the explosive ignition of nuclear fuel under degenerate conditions. We focus on three important points: (1) how to construct a stable initial model for white dwarfs with a central density ρc > 109 g cm-3 in the context of smoothed particle hydrodynamics (SPH); (2) the procedure devised in the numerical handling of combustion fronts and thermal discontinuities; and (3) a proposed method based on techniques of analysis of dynamic sets of points to characterize the flame front structure. As we will show, the combination of these methods along with the standard SPH technique makes the study of deflagrations in massive white dwarfs feasible even in three dimensions. After explaining in detail the numerical scheme, we show the results of several calculations, in three dimensions, addressed to checking the ability of the hydrocode to handle deflagrations in massive white dwarfs in two density regimes. First, several tests were carried out under the physical conditions that characterize Chandrasekhar-mass models for Type Ia supernovae, and some of the results were compared with standard one-dimensional calculations. We also explored the consequences of deflagrations at very high densities, where electron captures play a fundamental role in the further evolution of the white dwarf, and where collapse to a neutron star instead of an explosion is expected. Our calculations support the idea that the SPH method and various fractal analysis techniques can successfully be used to model the gross features of deflagrations in white dwarfs provided that the nuclear energy injected at the first stages of the explosion is sufficient to dominate the numerical noise. An extensive number of calculations for both Type Ia supernovae explosions and accretion-induced collapse of a white dwarf to a neutron star are in progress and will be reported in future publications.

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