  Thermal-diffusional pulsation behaviors in planar as well as outwardly and
inwardly propagating white dwarf carbon flames are systematically studied. In
the 1D numerical simulation, the asymptotic degenerate equation of state and
simplified one-step reaction rates for nuclear reactions are used to study the
flame propagation and pulsation in white dwarfs. The numerical critical
Zel'dovich numbers of planar flames at different densities ($\rho=2$, 3 and
4$\times 10^7$~g/cm$^3$) and of spherical flames (with curvature $c=$-0.01, 0,
0.01 and 0.05) at a particular density ($\rho=2\times 10^7$~g/cm$^3$) are
presented. Flame front pulsation in different environmental densities and
temperatures are obtained to form the regime diagram of pulsation, showing that
carbon flames pulsate in the typical density of $2\times10^7~{\rm g/cm^3}$ and
temperature of $0.6\times 10^9~{\rm K}$. While being stable at higher
temperatures, at relatively lower temperatures the amplitude of the flame
pulsation becomes larger. In outwardly propagating spherical flames the
pulsation instability is enhanced and flames are also easier to quench due to
pulsation at small radius, while the inwardly propagating flames are more
stable.
