Numerical Stability with Help from Entropy: Solving a Set of 13 Moment Equations for Shock Tube Problem

Abstract

The shock structures of a 13 moment generalized hydrodynamics system of rarefied gases are simulated. These are first order hyperbolic equations derived from the Boltzmann equation. The investigated moment system stands out due to having an entropy evolution. In addition, a particular interest arises from the fact that the equations not only contain nonconservative products, but also provide the key to solving this mathematical and numerical issue by means of a simple substitution utilizing the physical entropy evolution. The apparent success of this method warrants investigation and provides a new perspective and starting point for finding general approaches to nonconservative products and irreversible processes. Furthermore, the system shows physically accurate results for low Mach numbers and is able to reveal the nonequilibrium entropy profile across a shock wave.

Wave Propagation in Rocks – Investigating the Effect of Rheology

“…This has led our research group to develop, similarly to recent research activity worldwide [24][25][26][27][28][29][30][31][32][33], some own numerical methods, which outperform the commercially existing ones, in each of these aspects. We intend to benefit from the following principles:…”

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

“…and spatial spacetime ones. 2 The results of our ongoing research on 2D and 3D are to be communicated later.…”

“…for simplicity), (2) is the kinematic relationship between the strain field ε 3 and v, and the rheological relationship (3) contains, in addition to Young's modulus E, two positive coefficientsÊ, τ. The Poynting-Thomson-Zener model is a subfamily within the Kluitenberg-Verhás model family, which family can be obtained via a nonequilibrium thermodynamical internal variable approach [16].…”

“…Partly inspired by, and partly based on, the intensive development on symplectic schemes for reversible problems, remarkable research is done in recent years on dissipative systems, more on ones with finite degrees of freedom (including [2][3][4][5]) and less for continua (see, e.g., [6][7][8][9]).…”

Formulation of moment equations for rarefied gases within two frameworks of non-equilibrium thermodynamics: RET and GENERIC

“…Numerical methods that focus on the proper treatment of entropy can have significant advantages, as has been illustrated for the Lattice Boltzmann method [43,44]. In the context of 13 moment equations, it has been shown that the numerical solution of shock wave problems [45] and the formulation of boundary conditions [46] (see also [30]) can benefit enormously if the existence of an entropy is exploited properly. The search for GENERIC integrators, which preserve (as much as possible of) the GENERIC structure under time discretization in the same spirit as symplectic integrators preserve the Hamiltonian structure, is still in its beginning (see [47, and references therein, 48]).…”