Entropy and weak solutions in the thermal model for the compressible Euler equations

TL;DR

This paper investigates the entropy conditions in the lattice Boltzmann method for the KT model of compressible fluids, revealing potential causes of numerical instability and non-physical oscillations near shocks.

Contribution

It derives entropy functions aligned with Euler-like equations in the lattice Boltzmann framework, clarifying the entropy condition's role in numerical stability.

Findings

Negative entropy correlates with non-physical oscillations near shocks

Entropy condition is not fully satisfied in the KT model

Explicit subsidiary entropy condition can be derived from macroscopic equations

Abstract

Among the existing models for compressible fluids, the one by Kataoka and Tsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigorous theoretical background. The drawback of this KT model is that it can cause numerical instability if the local Mach number exceeds 1. The precise mechanism of this instability has not yet been clarified. In this paper, we derive entropy functions whose local equilibria are suitable to recover the Euler-like equations in the framework of the lattice Boltzmann method for the KT model. Numerical examples are also given, which are consistent with the above theoretical arguments, and show that the entropy condition is not fully guaranteed in KT model. The negative entropy may be the inherent cause for the non-physical oscillations in the vicinity of the shock. In contrast to these Karlin's microscopic entropy approach, the corresponding subsidiary entropy condition in the LBM calculation could also be deduced explicitly from the macroscopic version, which provides some insights on the numerical instability of the lattice Boltzmann model for shock calculation.