Structure and Operator Preserving High Order Schemes for Thermodynamically Compatible Hyperbolic PDE Systems

Objetivo

In SOPHOS we develop new numerical schemes with mathematically provable properties for some of the most challenging hyperbolic PDE systems in continuum physics, namely geometric formulations of continuum mechanics, compressible turbulence and the Einstein-Euler equations of general relativity. The overarching framework are overdetermined, hyperbolic and thermodynamically compatible (HTC) systems. Our new schemes will mimic their elegant mathematical structure for the first time entirely also on the discrete level. We devise nonlinearly stable algorithms that are consistent with the asymptotic limits of the PDE, preserve stationary equilibria and satisfy all differential and algebraic constraints. Nonlinear hyperbolic systems admit discontinuities, which are notoriously difficult. We will treat them via new genuinely nonlinear space-time discontinuous Galerkin schemes that employ data dependent function spaces to represent the discrete solution.
Recent theoretical results on nonlinear hyperbolic PDE indicate that the non-uniqueness of weak solutions of the Euler equations in multiple space dimensions may be linked to turbulence, a long-standing and still unresolved problem in mathematics and physics. We will design new schemes that follow the theory of dissipative weak solutions at the discrete level. We will also shed light on the question if the recent theoretical findings can be linked to existing turbulence models. At the macro scale turbulence can be described by the Reynolds stress tensor, which in SOPHOS will be obtained from a novel hierarchy of HTC systems that mimic the energy cascade down to the dissipative scale.
Up to now there are no schemes for the Einstein field equations of general relativity that are provably nonlinearly stable and which preserve all their nonlinear differential constraints exactly at the discrete level. In SOPHOS we will design these missing algorithms to allow provably stable long-time simulations in numerical general relativity.