Invited Lecture
High order space time ADER schemes on unstructured polygonal meshes: direct ALE and multidimensional Riemann solvers
University of Verona, Italy
Abstract
In this talk, we will first introduce the ADER predictor-corrector approach which allows us to make any Finite Volume and Discontinuous Galerkin scheme high order accurate in time with a low dissipative one step fully discrete space time procedure.
We then exploit this framework to build a direct Arbitrary-Lagrangian-Eulerian (ALE) method on moving polytopal meshes with topology changes. Here, we integrate the space time divergence form of the studied hyperbolic PDEs over the space-time control volumes connecting the regenerated tessellations at consecutive time-steps also with unusual shapes, including sliver degenerate elements.
Furthermore, we will present two novel complete genuinely multidimensional Riemann solvers that benefit from the polygonal tessellation and are extended to the high order of accuracy thanks to a CWENO-ADER algorithm. The first is a direct extension of the Osher-Solomon Riemann solver via integration of the dissipation term over manifolds and ii) the second is based on the so-called N-scheme, an optimal upwind flux originally developed for residual distribution methods.
ACKNOWLEDGMENTS
E. Gaburro gratefully acknowledges the support received from the European Union with the ERC Starting Grant ALcHyMiA (grant agreement No. 101114995).