  In this work, we present a novel family of high order accurate numerical
schemes for the solution of hyperbolic partial differential equations (PDEs)
which combines several geometrical and physical structure preserving
properties. First, we settle our methods in the Lagrangian framework, where
each element of the mesh evolves following as close as possible the local fluid
flow, so to reduce the numerical dissipation at contact waves and moving
interfaces and to satisfy the Galilean and rotational invariance properties of
the studied PDEs system. In particular, we choose the direct
Arbitrary-Lagrangian-Eulerian (ALE) approach which, in order to always
guarantee the high quality of the moving mesh, allows to combine the Lagrangian
motion with mesh optimization techniques. The employed polygonal tessellation
is thus regenerated at each time step, the previous one is connected with the
new one by space-time control volumes, including hole-like sliver elements in
correspondence of topology changes, over which we integrate a space-time
divergence form of the original PDEs through a high order accurate ADER
discontinuous Galerkin (DG) scheme. Mass conservation and adherence to the GCL
condition are guaranteed by construction thanks to the integration over closed
control volumes, and robustness over shock discontinuities is ensured by the
use of an a posteriori subcell finite volume (FV) limiting technique.
