Theoretical progress on regularized lattice Boltzmann collision models on weakly incompressible and compressible thermal flows

Numerical stability has long constrained the application of lattice Boltzmann methods to high-Reynolds-number and high-Mach-number flows. Improving the stability and accuracy of collision models has therefore remained a major challenge for nearly three decades. Regularized collision models constitute an important class of lattice Boltzmann collision models and have recently undergone substantial theoretical development. They are now widely used for high-Reynolds-number turbulence, high-Mach-number compressible flow and aeroacoustics. This article systematically reviews their development and establishes a unified theoretical framework that clarifies the relationships among the principal formulations. The analysis identifies the underlying principle of the regularized multiple-relaxation collision model: higher-order kinetic moments that cannot be represented by the discrete lattice are rapidly relaxed in a coordinate-independent, self-similar moment space, thereby preventing spurious modes from contaminating the hydrodynamic equations.