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, formulations (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 analyzed. 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 $T_r$ and (2) introducing a controllable numerical dissipation σ 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.

提出了标准晶格上可压缩混合晶格玻尔兹曼方法（HLBM）的原始光谱研究。在此框架中，质量和动量方程使用晶格玻尔兹曼方法 (LBM) 解决，而有限差分 (FD) 方案求解能量方程。由于理想气体状态方程，两个系统相互耦合。这项工作旨在回答有关此类模型的数值稳定性的一些问题，这在很大程度上取决于数值参数的选择。在这方面，基于不同的能量变量、公式（原始或保守）、碰撞项和数值方案的几个一维和二维 HLBM 类被仔细检查。一旦引入了适当的纠正条款，结果表明，所有连续的 HLBM 类都在线性近似中恢复了 Navier-Stokes-Fourier 行为。然而，当分析它们的离散对应物时，HLBM 类之间会出现显着差异。指出了在相对较高马赫数下产生的多种不稳定机制，并引入了两种详尽的稳定策略：（1）通过改变参考温度来减少时间步长${吨}_r$(2)通过碰撞算子引入可控的数值耗散σ。一项完整的参数研究表明，只有基于原始和保守熵方程的 HLBM 类可用于可压缩应用。最后，对熵类的宏观模态组成进行了创新研究。通过这项研究，在标准二维测试案例中突出并证实了两种原始现象，称为剪切到熵和熵到剪切转移。