Advanced Structure Preserving Lagrangian schemes for novel first order Hyperbolic Models: towards General Relativistic Astrophysics (ALcHyMiA)

Framework: Horizon Europe — Excellent Science
Programme: European Research Council (ERC)
Theme: European research council
Destination: Starting grants
Action type: ERC actions
Panel: Mathematics
Panel description: All areas of mathematics, pure and applied, plus mathematical foundations of computer science, mathematical physics and statistics
Countries involved: Italy
Call: ERC STARTING GRANTS (ERC-2023-STG, 2023)
Start date: 2024-04-01
End date: 2029-03-31
Duration (months): 60.0
Total project cost (EUR): 1500000.0
EU funding (EUR): 1500000.0
Participants: 1
Project status: SIGNED

ALcHyMiA will make substantial progress in applied mathematics, targeting long-time stable and self-consistent simulations in general relativity and high energy density problems, via the development of new and effective structure preserving numerical methods with provable mathematical properties. We will devise innovative schemes for hyperbolic partial differential equations (PDE) which at the discrete level exactly preserve all the invariants of the continuous problem, such as equilibria, involutions and asymptotic limits. 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. This allows to study gravitational waves, binary neutron stars and accretion disks around black holes that require the coupled evolution of matter and spacetime. Here, high resolution and minimal dissipation at shocks and moving interfaces are crucial and will be achieved by groundbreaking direct Arbitrary-Lagrangian-Eulerian (ALE) methods on moving Voronoi 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 Finite Volume and Discontinuous Galerkin ALE schemes lies in the geometrical understanding and high order PDE integration over 4D spacetime manifolds. The high-risk high-gain challenge is the design of smart DG schemes with virtual, bound-preserving, genuinely nonlinear data-dependent function spaces, taking advantage of the Voronoi properties. 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.

Keywords: first order hyperbolic partial differential equations pde with involutions,high order methods on manifolds,arbitrary lagrangian eulerian ale methods on moving voronoi meshes with topology changes,finite volume fv and discontinuous galerkin dg schemes,virtual finite element methods vem,structure preserving sp schemes,well balanced wb methods,bound preserving schemes,mesh optimization,polytopes,einstein field equations,unified model of continuum mechanics,teleparallel gravity theory,numerical general relativity,computational astrophysics,binary neutron stars,gravitational waves