  The dynamics of flame propagation in systems with infinite Lewis number and
spatially discretized sources of heat release is examined, which is applicable
to the combustion of suspensions of fuel particles in air. The system is
analyzed numerically using a one-dimensional heat equation with a source term
for the reaction progress variable, which is specified to have zero
diffusivity, and the model reveals a spectrum of flame-propagation regimes. For
the case of a switch-type reaction rate and homogeneous media (continuous
regime), the flame propagates steadily at a velocity in agreement with
analytical solutions. As the sources are spatially concentrated into
{\delta}-function-like sources, propagation approaches the discrete regime with
a fixed period between ignition of the sources, for which an analytic solution
is also available for validation. When the source term is governed by an
Arrhenius rate and the activation energy is increased beyond the stability
boundary, the flame begins to exhibit a long-wavelength (4-5 times the thermal
flame thickness) oscillation characteristic of the thermo-diffusive
instability, in good agreement with prior stability analysis. When spatial
discreteness is introduced, a competition is observed between the long-period
oscillations of the thermo-diffusive instability and the pulsations associated
with the rapid heat release of the concentrated sources. Interestingly, the
presence of spatial discreteness is able to excite higher modes (period
doubling and chaotic solutions) of the thermo-diffusive instability, suggesting
that the introduction of discreteness may have an influence qualitatively
similar to that of increasing activation energy. Relevance of the model
parameters to experimental systems is then discussed.
