  The problem of shock induced ignition by a piston is addressed in the
framework of Fickett's model for reactive compressible flows, i.e., the
reactive form of Burgers' equation. An induction-reaction two-step
chain-branching model is used to study the coupling between the energy release
and the compressible hydrodynamics occurring during the shock ignition
transient leading to a detonation. Owing to the model's simplicity, the
ignition and acceleration mechanism is explained using the two families of
characteristics admitted by the model. The energy release along the particle
paths provides the amplification of forward-travelling pressure waves. These
waves pre-compress the medium in the induction layer ahead of the reaction
zone, therefore changing the induction delays of successive particles. The
variation of the induction delay provides the modulation of the amplification
of the forward travelling pressure waves by controlling the residence time of
the pressure waves in the reaction zone. A closed form analytical solution is
obtained by the method of characteristics and high activation energy
asymptotics. The acceleration of the reaction zone was found to be proportional
to the product of the activation energy, the ratio of the induction to reaction
time and the heat release. This finding provides a theoretical justification
for the previous use of this non-dimensional number to characterize the
ignition regimes observed experimentally in detonations and shock induced
ignition phenomena. Numerical simulations are presented and analysed. Both
subsonic and supersonic internal flame propagation regimes are observed,
consistent with experiment and previous reactive Euler models.
