A linear stability analysis of compressible hybrid lattice Boltzmann methods

TL;DR

This paper conducts a spectral stability analysis of compressible hybrid lattice Boltzmann methods, identifying instability mechanisms and proposing stabilization strategies, ultimately validating the approach for practical compressible flow simulations.

Contribution

It provides the first detailed spectral stability analysis of compressible HLBM, introduces stabilization techniques, and explores the macroscopic modal behavior of entropy-based classes.

Findings

All continuous HLBM classes recover Navier-Stokes Fourier behavior in linear approximation.

Instabilities occur at high Mach numbers, but can be mitigated by specific stabilization strategies.

Entropy-based HLBM classes are suitable for compressible applications, revealing novel shear-to-entropy and entropy-to-shear transfer phenomena.

Abstract

An original spectral study of the compressible hybrid lattice Boltzmann method (HLBM) on standard lattice is proposed. In this framework, the mass and momentum equations are addressed using the lattice Boltzmann method (LBM), while finite difference (FD) schemes solve an energy equation. Both systems are coupled with each other thanks to an ideal gas equation of state. This work aims at answering some questions regarding the numerical stability of such models, which strongly depends on the choice of numerical parameters. To this extent, several one- and two-dimensional HLBM classes based on different energy variables, formulation (primitive or conservative), collision terms and numerical schemes are scrutinized. Once appropriate corrective terms introduced, it is shown that all continuous HLBM classes recover the Navier-Stokes Fourier behavior in the linear approximation. However, striking differences arise between HLBM classes when their discrete counterparts are analysed. Multiple instability mechanisms arising at relatively high Mach number are pointed out and two exhaustive stabilization strategies are introduced: (1) decreasing the time step by changing the reference temperature $T_{ref}$ and (2) introducing a controllable numerical dissipation $\sigma$ via the collision operator. A complete parametric study reveals that only HLBM classes based on the primitive and conservative entropy equations are found usable for compressible applications. Finally, an innovative study of the macroscopic modal composition of the entropy classes is conducted. Through this study, two original phenomena, referred to as shear-to-entropy and entropy-to-shear transfers, are highlighted and confirmed on standard two-dimensional test cases.