  Supernovae of type Ia are thought to arise from the thermonuclear
incineration of a carbon-oxygen white dwarf stellar remnant. However, the
detailed explosion scenario and stellar evolutionary origin scenario -- or
scenarios -- which lead to observed supernovae are still quite uncertain. One
of the principal tests of proposed scenarios is comparison with the explosion
products inferred, for example, from the spectrum of the supernovae. Making
this comparison requires computation of the combustion dynamics and products
through simulation of proposed scenarios. Here we discuss two specific proposed
explosion scenarios, the deflagration-detonation transition and the helium
shell double detonation, With these two examples in mind, we proceed to discuss
challenges to computational modeling of the combustion taking place in these
explosions. Both subsonically and supersonically propagating reaction fronts
are discussed, called deflagrations and detonations respectively. Several major
stages of the combustion occur on length and time scales that are many orders
of magnitude smaller than those accessible in simulations of the explosion.
Models which attempt to capture this sub-grid behavior and the verification of
those models is briefly discussed.
