Fakultätskolloquium

In the first part of the talk we present a family of arbitrary high order accurate (ADER) finite volume and discontinous Galerkin finite element schemes for the numerical solution of a new unified first oder symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of Newtonian continuum physics, including a general description of fluid and solid mechanics as well as electro-magnetic fields in one single system of governing partial differential equations (PDE). The model is based on previous work of Godunov, Peshkov and Romenski (so-called GPR model) on symmetric hyperbolic and thermodynamically compatible systems.

In the second part of the talk, we show a successful extension of the GPR model to general relativity, leading to a novel and unified first order hyperbolic formulation of general relativistic continuum mechanics. The model is able to describe nonlinear elasto-plastic solids, as well as ideal and non-ideal (viscous) fluids in full general relativity. Formal asymptotic expansion of the governing PDE reveals the structure of the viscous stress tensor in the asymptotic relaxation limit. The key features of the new model are its symmetric hyperbolicity and thermodynamical compatibility. The proposed PDE system is causal, covariant and has bounded signal speeds for all involved processes, including disspative ones. Since the new model also contains elastic solids as a special case, it should be understood as an alternative to existing models for vicous relativistic fluids that are usually derived from kinetic theory and extended irreversible thermodynamics. We present numerical results obtained with high oder ADER schemes for inviscid and viscous relativistic flows obtained in the stiff relaxation limit of the system, as well as results for solid mechanics.

In the last part of the talk we introduce a new, provably strongly hyperbolic first order reduction of the CCZ4 formalism of the Einstein field equations of general relativity and its solution with high order ADER discontinuous Galerkin finite element schemes.

The Newtonian three body problem has undergone a Renaissance in recent years. I will present an overview of old and new results on periodic solutions, symbolic dynamics, and chaos in this problem. Then I will describe new results about the symplectic symmetry reduction and dynamics of relative equilibria when the spatial dimension is at least four. In particular we will show that there are families of relative equilibria that are minima of the reduced Hamiltonian, and hence are Lyapunov stable. This establishes the first proof of Lyapunov stable periodic orbits in the three body problem, albeit in dimension four.