  We present a numerical investigation of the cellular burning regime in Type
Ia supernova explosions. This regime holds at small scales (i.e. below the
Gibson scale), which are unresolved in large-scale Type Ia supernova
simulations. The fundamental effects that dominate the flame evolution here are
the Landau-Darrieus instability and its nonlinear stabilization, leading to a
stabilization of the flame in a cellular shape. The flame propagation into
quiescent fuel is investigated addressing the dependence of the simulation
results on the specific parameters of the numerical setup. Furthermore, we
investigate the flame stability at a range of fuel densities. This is directly
connected to the questions of active turbulent combustion (a mechanism of flame
destabilization and subsequent self-turbulization) and a
deflagration-to-detonation transition of the flame. In our simulations we find
no substantial destabilization of the flame when propagating into quiescent
fuels of densities down to ~10^7 g/cm^3, corroborating fundamental assumptions
of large-scale SN Ia explosion models. For these models, however, we suggest an
increased lower cutoff for the flame propagation velocity to take the cellular
burning regime into account.
