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

Introduction

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.

Innovative Numerical Methods

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
• 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.

Applications in Astrophysics

This allows us to study:

1. Gravitational waves
2. Binary neutron stars
3. Accretion disks around black holes

These phenomena 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.

Grid Quality and Invariance

These methods are necessary to maintain optimal grid quality even when following:

• Rotating compact objects
• Complex shear flows
• Metric torsion

They also ensure:

• Rotational invariance
• Entropy stability
• Galilean invariance in the Newtonian limit

Breakthrough Schemes

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.

High-Risk High-Gain Challenge

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.

Community Building

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.