Structure preserving high order Lagrangian schemes for the solution of hyperbolic equations: from fluid-dynamics to astrophysics

Elena Gaburro

Inria center Bordeaux-Sud-Ouest, France

Friday 14th July 2023

14:30:00

Aula Saleri

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 numerical dissipation at contact waves and moving interfaces and to respect the Galilean and rotational invariance of the studied PDEs system. 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. The employed Voronoi 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. On the top of this effective moving mesh framework, we have also made, for the first time in literature, the full ALE ADER DG scheme with a posteriori sub-cell FV limiter well balanced in order to increase our resolution on small perturbations of equilibrium solutions. The presentation is closed by a wide set of numerical results, including simulations of Keplerian disks, which demonstrate all the claimed properties and the increased accuracy and robustness of our novel family of schemes. We will also give an outlook on future works on the Einstein field equations, based on our recent results obtained with Cartesian methods. Acknowledgement: E. Gaburro gratefully acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement SuPerMan (No. 101025563). Contatti: luca.formaggia@polimi.it