In this talk, 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.

Indeed, first, we settle in the Lagrangian framework, where each element of the mesh evolves following as close as possible the local fluid flow, so to reduce the dissipation at contact waves and moving interfaces and to respect Galilean and rotational invariance.

In particular, we choose the direct Arbitrary-Lagrangian-Eulerian setting which, in order to always guarantee the high quality of the moving mesh, allows to combine the Lagrangian motion with mesh optimization techniques.

Our 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 the respect of 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 sub-cell finite volume (FV) limiter.

In addition, our schemes are able to guarantee the exact preservation, up to machine precision, of equilibria and involution constraints: this allows to obtain stable and robust simulations of complex equations as the Einstein field equations of general relativity.

Acknowledgement

E. Gaburro gratefully acknowledges the support received from the European Union with the ERC Starting Grant ALcHyMiA (No. 101114995).