Hamiltonian Information Geometric Regularization of the Compressible Euler Equations

Abstract

The recently proposed information geometric regularization (IGR) was the first inviscid regularization of the multi-dimensional compressible Euler equations, which enabled the simulation of realistic compressible fluid models at an unprecedented scale. However, the thermodynamic effects of this regularization have not yet been understood in a principled manner. To achieve a proper understanding of the thermodynamic aspects of the IGR, we decompose the regularization into its conservative dynamics, framed as a Hamiltonian subsystem, and its dissipative dynamics. In so doing, we further introduce two more models to compare to IGR, the Hamiltonian regularized Euler (HRE) model, which is the first multi-dimensional, non-dispersive Hamiltonian regularization of the compressible Euler equations with energy, as well as the Hamiltonian IGR (HIGR) model, which modifies the dissipation used in the IGR model to instead utilize a metriplectic dissipative force. Despite having many attractive features, the HRE and HIGR models exhibit notable defects in numerical tests on colliding shock problems, which preclude their use as computational tools without further study of dissipative weak solutions to these models. Additionally, our analysis presents new results on the IGR model itself, including its ability to conserve acoustic waves, as well as local energy transport laws and entropy production rates. By separating the conservative and dissipative dynamics of the IGR, our hope is that subsequent of analysis of the IGR model can benefit from this natural decomposition, such as, for example, rigorous proofs of strong solutions for multi-dimensional IGR for the compressible Euler system with thermodynamics.