High-order Arbitrary-Lagrangian-Eulerian schemes on crazy moving Voronoi meshes

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.