Type Ia supernovae (SNe Ia) are the thermonuclear explosion of a white dwarf star. At peak brightness, their light output rivals that of their host galaxy, enabling them to be seen across great distances in the Universe. Their application to cosmology has led to the discovery that the expansion rate of the Universe is accelerating. Despite their observational successes, our theoretical understanding of SNe Ia is far from complete. Simulations of these events need to be concerned with the large scale motions on the scale of the star (1.e8 cm) down to the thermal thickness of the burning front (1.e-5 to 1 cm), and timescales including the 100 years of convection preceding the 1 second duration of the explosion. This is extremely challenging for modern computers and hydrodynamic algorithms. In this talk, I will discuss the current state of the field, outline the outstanding problems, and motivate the next generation of algorithms that will be used to simulate the lead up to the explosion.