Abstract:
The ALcHyMiA project aims at the development of advanced structure preserving numerical methods, with provable mathematical properties, that will convey superior resolution and reliability to the simulation of phenomena described by first order hyperbolic partial differential equations, including the study of extremely challenging astrophysical events.
Indeed, next to fluids and magnetohydrodynamics, key for benchmarks and valuable applications on Earth, we target a new class of first order hyperbolic systems that unifies fluid and solid mechanics and gravity theory. All these unsteady processes may develop features involving a huge disparity of spacetime scales and many different computational difficulties to be managed simultaneously (macro-scale smooth phenomena as waves or vortexes, zero-scale contact and shock discontinuities and multi-scale turbulence features).
Thus, we will employ a particular family of very high order accurate numerical methods (the ADER Finite Volume and Discontinuous Galerkin methods). Here, we plan to incorporate innovative structure preserving techniques, capable in addition of accurately solving the PDE, to also guarantee the exact preservation, even at the discrete level, of both the geometrical and the physical invariants characterizing the studied continuum models.
A crucial ingredient will be the introduction of groundbreaking direct Arbitrary-Lagrangian-Eulerian (ALE) methods on moving polyhedral meshes with changing topology. These are necessary to maintain optimal grid quality even when following rotating compact objects, complex shear flows or metric torsion. They also ensure rotational invariance, entropy stability and Galilean invariance in the Newtonian limit. The breakthrough of our new approach lies in the geometrical understanding and high order PDE integration over 4D spacetime manifolds.
Finally, it is an explicit mission of ALcHyMiA to grow a solid scientific community, sharing know-how by tailored dissemination activities from top-level schools to carefully organized international events revolving around personalized interactions.
