  Particles on Demand formulation of kinetic theory [B. Dorschner, F. B\"{o}sch
and I. V. Karlin, {\it Phys. Rev. Lett.} {\bf 121}, 130602 (2018)] is used to
simulate a variety of compressible flows with strong discontinuities in
density, pressure and velocity. Two modifications are applied to the original
formulation of the Particles on Demand method. First, a regularization by
Grad's projection of particles populations is combined with the reference frame
transformations in order to enhance stability and accuracy. Second, a
finite-volume scheme is implemented which allows tight control of mass,
momentum and energy conservation. The proposed model is validated with an array
of challenging one- and two-dimensional benchmarks of compressible flows,
including hypersonic and near-vacuum situations, Richtmyer-Meshkov instability,
double Mach reflection and astrophysical jet. Excellent performance of the
modified Particles on Demand method is demonstrated beyond the limitations of
other lattice Boltzmann-like approaches to compressible flows.
