Linear stability of lattice Boltzmann models with non-ideal equation of state

TL;DR

This paper analyzes the linear stability of lattice Boltzmann models with non-ideal equations of state, focusing on how density-dependent pressures alter sound speeds and stability. Through a comprehensive spectral and multiscale (Chapman–Enskog) analysis of D1Q3 and D2Q9 LBGK systems, it derives necessary stability conditions on normal-mode speeds and demonstrates that, for 1D, these conditions are also sufficient for all wave-numbers, while in 2D they are necessary but not sufficient. The study shows that typical non-ideal EOS (e.g., shallow water and van der Waals) yield bounded stable velocity ranges and that standard equilibria do not guarantee unconditional stability; however, entropic equilibria can achieve unconditional linear stability. The results provide a principled pathway to construct unconditionally stable LBGK models for non-ideal fluids by designing velocity-consistent, velocity-dependent pressure laws and identifying how non-equilibrium corrections influence dissipation of higher-order modes.

Abstract

Detailed study of spectral properties and of linear stability is presented for a class of lattice Boltzmann models with a non-ideal equation of state. Examples include the van der Waals and the shallow water models. Both analytical and numerical approaches demonstrate that linear stability requires boundedness of propagation speeds of normal eigen-modes. The study provides a basis for the construction of unconditionally stable lattice Boltzmann models.

Figures (16)

• Figure 1: Clapeyron diagram of van der Waals equation of state. Here $\rho_c$ and $P_c$ are the critical density and pressure.
• Figure 2: Positivity domain of attenuation rates $\mathcal{R}^{\pm}$\ref{['eq:attenuation_rates_1D']} as a function of eigen-modes $c^{\pm}$. Red: Positivity domain of $\mathcal{R}^{-}$. Blue: Positivity domain of $\mathcal{R}^{+}$. Purple: Positivity domain of both attenuation rates $\mathcal{R}^{\pm}$ simultaneously. The positive square root convention restricts the stability domain to the top-left quadrant, shown with solid black lines.
• Figure 3: Attenuation rates of normal modes \ref{['eq:attenuation_rates_1D']}, $\mathcal{R}^+$ in red and $\mathcal{R}^-$ in blue, for shallow water equation of state \ref{['eq:swe']} for ${\rho}=1$ and $g=2/3$. Shaded area represents the stability domain $u\in [-0.1835, 0.1835]$.
• Figure 4: Attenuation rates of normal modes \ref{['eq:attenuation_rates_1D']}, $\mathcal{R}^+$ in red and $\mathcal{R}^-$ in blue, for van der Waals equation of state \ref{['eq:vdw']} for $T_r=0.8$, $a=1/49$, $b=2/21$, $\rho_r=0.24$. Shaded area represents stability domain $u\in [-0.1605, 0.1605]$.
• Figure 5: Condition \ref{['eq:cond_3']} in the $(c^-,c^+,k)$ parameter space. By periodicity of the function $\cos^2(k/2)$, only the interval $0\le k\le \pi/2$ is shown. The interior of the iso-surface represents the inequality \ref{['eq:cond_3']}.