  Hyperbolic partial differential equations (PDEs) cover a wide range of
interesting phenomena, from human and hearth-sciences up to astrophysics: this
unavoidably requires the treatment of many space and time scales in order to
describe at the same time observer-size macrostructures, multi-scale turbulent
features, and also zero-scale shocks. Moreover, numerical methods for solving
hyperbolic PDEs must reliably handle different families of waves: smooth
rarefactions, and discontinuities of shock and contact type. In order to
achieve these goals, an effective approach consists in the combination of
space-time-based high-order schemes, very accurate on smooth features even on
coarse grids, with Lagrangian methods, which, by moving the mesh with the fluid
flow, yield highly resolved and minimally dissipative results on both shocks
and contacts. However, ensuring the high quality of moving meshes is a huge
challenge that needs the development of innovative and unconventional
techniques. The scheme proposed here falls into the family of
Arbitrary-Lagrangian-Eulerian (ALE) methods, with the unique additional freedom
of evolving the shape of the mesh elements through connectivity changes. We aim
here at showing, by simple and very salient examples, the capabilities of
high-order ALE schemes, and of our novel technique, based on the high-order
space-time treatment of topology changes.
