GENERIC, structure preserving integrators and hyperbolic partial differential equations of hydrodynamic type

Siccha, Nikolas; Torrilhon, Manuel (Thesis advisor); Westdickenberg, Michael (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

Both hyperbolic partial differential equations (PDEs) and the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework are ubiquitous in continuum mechanics, being applicable to very similar problem areas. In particular first-order systems of conservation laws have always been intimately related to gas dynamics, which served as a starting point for the GENERIC framework. However, while a lot of mathematical theory has been developed for hyperbolic PDEs, the GENERIC framework is notable mainly for its application in the modeling of thermodynamic systems in non-equilibrium. There it facilitates the development of thermodynamically compatible evolution equations, while also imposing the severe restrictions of Poisson structures on the reversible part of the evolution equation. The original goal my thesis was to investigate the exact connection between these two disparate theories.For this there were initially two main questions:* When does a system being GENERIC imply hyperbolicity?* When does hyperbolicity imply the existence of a GENERIC formulation? Furthermore, inspired by current research, another question arose:* How can structure-preserving integrators facilitate the numerical solution of GENERIC systems? Consequently, the results factor into two distinct, but related groups.It is well-known that in the smooth regime many hyperbolic PDEs can be regarded as time-reversible infinite-dimensional Hamiltonian systems, while this property is generally lost in the presence of discontinuities. This immediately suggests to investigate the connection between hyperbolic PDEs and the reversible part of GENERIC systems, and indeed we were able to show that the irreversible part of GENERIC systems can generally not contribute first-order terms to the evolution equation. Consequently, we were able to exploit the extensive body of work on hydrodynamic Poisson systems, including recent classification results for low numbers of spatial dimensions and low numbers of dependent variables to classify combinations of differential operators and Hamiltonian functionals, yielding conditions on the Hamiltonian that lead to or preclude hyperbolicity of the resulting PDEs.Second, we developed a family of superconvergent, arbitrary order reversible-irreversible numerical integrators based on the discontinuous Galerkin (dG) timestepping method and tested it both for GENERIC ODEs and PDEs. The goal was to obtain a method which combines the advantages of reversible integrators for reversible problems and the favorable properties of the dG-methods for irreversible and stiff problems such as the heat equation. Based on a simple and easily computed heuristic (the local free energy dissipation rate) our method is able to recover the characteristic L-stability and superconvergence of the dG-method for purely irreversible problems, while also retaining A-stability, time-reversibility and gaining an increased-by-one order of convergence for purely reversible problems. For intermediate problems, the qualitative behavior of the solution (as measured in the free energy dissipation rate) is generally better captured for large timesteps by our method than by either the purely reversible or standard (purely irreversible) dG-method, while the quantitative behavior of the solution converges towards the higher-order purely reversible dG-method for decreasing timestep sizes.

Institutions

• Department of Mathematics [110000]
• Chair of Applied and Computational Mathematics [115010]