In this paper, we review the present state of theoretical models of thermonuclear supernovae, and compare their predicitions with the constraints derived from observations of Type Ia supernovae. The diversity of explosion mechanisms usually found in one-dimensional simulations is a direct consequence of the impossibility to resolve the flame structure under the assumption of spherical symmetry. Spherically symmetric models have been successful in explaining many of the observational features of Type Ia supernovae, but they rely on two kinds of empirical models: one that describes the behaviour of the flame on the scales unresolved by the code, and another that takes account of the evolution of the flame shape. In contrast, three-dimensional simulations are able to compute the flame shape in a self-consistent way, but they still need a model for the propagation of the flame in the scales unresolved by the code. Furthermore, in three dimensions the number of degrees of freedom of the initial configuration of the white dwarf at runaway is much larger than in one dimension. Recent simulations have shown that the sensitivity of the explosion output to the initial conditions can be extremely large. New paradigms of thermonuclear supernovae have emerged from this situation, as the Pulsating Reverse Detonation. The resolution of all these issues must rely on the predictions of observational properties of the models, and their comparison with current Type Ia supernova data, including X-ray spectra of Type Ia supernova remnants.
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