  In this exploratory study, we apply shock-capturing schemes within the
framework of the Particles on Demand kinetic model to simulate compressible
flows with mild and strong shock waves and discontinuities. The model is based
on the semi-Lagrangian method where the information propagates along the
characteristics while a set of shock-capturing concepts such as the total
variation diminishing and weighted essentially non-oscillatory schemes are
employed to capture the discontinuities and the shock-waves. The results show
that the reconstruction schemes are able to remove the oscillations at the
location of the shock waves and together with the Galilean invariance nature of
the Particles on Demand model, stable simulations of mild to extreme
compressible benchmarks can be carried out. Moreover, the essential numerical
properties of the reconstruction schemes such as their spectral analysis and
order of accuracy are discussed.
