# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 1 Introduction  
Incompressible thermal flows are frequently encountered in engineering applications [1]. Therefore, much research has been conducted on this topic. Numerical simulation is an emerging and popular approach in studying this topic [2; 3; 4]. Among various numerical methods, lattice Boltzmann method (LBM) is attracting more and more interest because of its simplicity and explicitness [5; 6; 7; 8; 9]. However, with its broader applications, some drawbacks of standard LBM are revealed and reported, which include high cost in virtual memories, inconvenient implementation of physical boundary conditions, lattice uniformity and poor numerical stability in high Reynolds/Rayleigh numbers [10].  
To overcome or alleviate these drawbacks, a simplified lattice Boltzmann method (SLBM) was recently developed and extended to simulations of thermal flows [11; 12]. Simplified thermal lattice Boltzmann method (STLBM) is derived from reconstructing solutions to the macroscopic governing equations recovered from lattice Boltzmann equation and resolved in a predictor-corrector scheme. Final formulations of STLBM only involve the equilibrium and the non-equilibrium distribution functions. The equilibrium distribution functions are evaluated from the macroscopic variables, while the non-equilibrium distribution function is approximated from the difference of two equilibrium distribution functions at different locations and time levels. Consequently, the macroscopic variables are directly updated in STLBM, which brings various merits to this method. On the one hand, virtualmemories are reduced dramatically because the distribution functions no longer need to be saved. On the other hand, physical boundary conditions can be directly implemented. However, one flaw in STLBM constrains its robustness in numerical perspective, which is its first-order of accuracy in time in the corrector step.  
Under the motivation of eliminating this flaw, we present a truly second-order unconditionally stable thermal lattice Boltzmann method (USTLBM) in this paper. Assisted by Chapman-Enskog expansion analysis and Taylor series expansion formulas, higher order of accuracy can be achieved in approximating the non-equilibrium terms. Meanwhile, slight shifting of half time step can be obtained in the time level, which enables us to construct a typical central difference scheme that ensures the second-order of accuracy in time.  
While eliminating the flaws, USTLBM maintains all merits of STLBM. The final formulations of USTLBM are very similar to those of STLBM, with slight changes made in the corrector step. Therefore, the simplicity and explicitness are well maintained. At the same time, USTLBM also directly updates the macroscopic variables. Thus, virtual memories are greatly lowered and physical boundary conditions can be directly implemented. Most importantly, through von Neumann stability analysis, USTLBM can be theoretically proven unconditionally stable, which indicates that it inherits the good numerical stability of STLBM and has great advantages over the standard LBM.  
The remaining parts of this paper are laid out as follows. Section 0 gives brief introduction to the conventional thermal lattice Boltzmann method and the recently developed simplified thermal lattice Boltzmann method (STLBM). Derivations of the unconditionally stable thermal lattice Boltzmann method (USTLBM), as well as its boundary treatments and stability analysis, are discussed in detail in Section 3. Four typical numerical examples are presented in Section 4 for comprehensive evaluation of the robustness and flexibility of USTLBM. Conclusions are finally summarized in Section 5.