Mechanisms for Stabilizing Thermal Lattice Boltzmann Equation and Their Applications for Convection-Diffusion Problem on Nonuniform Meshes (Advances in Applied Mathematics and Mechanics, 2025)
The thermal lattice Boltzmann equation (TLBE) has superior numerical stability as an explicit algorithm. However, its applications on nonuniform meshes are complicated. This paper clarifies the intrinsic mechanism for stabilizing computations in TLBE and proposes two solvers that combine the numerical stability of TLBE and flexible finite difference/volume schemes for nonuniform meshes. Through a brief review of the lattice Boltzmann method, it is concluded that the entropy increase of the collision operator is essential for numerical stability. This paper first proposes a macroscopic entropy-increasing (MEI) model for convection-diffusion problems by combining the MEI process and TLBE. The von Neumann stability analysis proves that the MEI model has no upper limit for mesh Fourier number. However, the accuracy of the MEI model is found to be sensitive to higher-order deviation terms. Therefore, a hybrid model that combines the MEI and the equilibrium-moment-based models is proposed to solve the problem. The von Neumann stability analysis demonstrates that the hybrid model can completely recover the numerical stability of TLBE. Numerical investigations validate the good stability and accuracy of the hybrid model. Most importantly, it can be easily applied to nonuniform meshes, whereas implementing TLBE on nonuniform meshes is relatively complicated.