## 5 Lattice Boltzmann Methods (LBMs)
In another comparative study [134], the efficiency of hybrid LBM was compared to FDM by considering the problem of transient conduction radiation in a 1D planar geometry. It has been shown that the LBM requires more iterations and longer computation time than the FDM, but that it has computational advantages in the case of a multidimensional medium. [PERSON] and [PERSON] [135] conducted a comparative study of the discrete ordinate (DOM)-LBM and DOM-FDM methods by dealing with transient conduction-radiation heat transfer problems in 1D and 2D square cavities containing an absorbing, emitting and isotropically diffusing medium. They stated that the CPU times for the LBM-DOM were slightly faster than those of the FDM-DOM with almost the same number of iterations. [PERSON] and [PERSON] [136] used the LBM and SFV (_Stream Function-Vorticity_) schemes to simulate a flow over a flat plate. They were interested in the CPU/GPU solver execution time and acceleration and the type of GPU arrays used. They found that although the execution time of SFV is lower than that of LBM, the potential for accelerating LBM using NVIDIA GTX 480 is better. [PERSON] et al. [137] addressed numerically the internal convective heat transfer phenomena through cavities and enclosures using both the FVM (_finite volume method_) and LBM with various discretization schemes.They concluded that at the corners, the FVM can exhibit more accurate results compared to those of the LBM. By simulating turbulent flows, [PERSON] and [PERSON] [138] applied the gas kinetic method (GKM) while assessing its capacity by comparison with LBM and Navier-Stokes solvers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **Method** & **Ra** & **Re** & **Grid Number** & **Iterations Number** & **CPU Time (s)** \\ \hline \multirow{4}{*}{**Buoyancy-driven flow**} & LBM & \multirow{2}{*}{710} & \multirow{2}{*}{-} & \multirow{2}{*}{50 \(\times\) 50} & 11,146 & 282 \\  & FDM & & & & 8501 & 542 \\ \cline{2-6}  & LBM & & & & 27,352 & 705 \\  & FDM & & & & 30,148 & 1904 \\ \cline{2-6}  & LBM & & & 100 \(\times\) 100 & 58,101 & 4802 \\  & FDM & & & & 32,029 & 8594 \\ \hline \multirow{4}{*}{**Lid-driven flow**} & LBM & \multirow{2}{*}{-} & \multirow{2}{*}{100} & \multirow{2}{*}{100 \(\times\) 100} & 10,963 & 886 \\  & FDM & & & & 5149 & 1131 \\ \cline{1-1} \cline{2-6}  & LBM & & & & 30,557 & 5249 \\ \cline{1-1}  & FDM & & & 10,848 & 5360 \\ \cline{1-1} \cline{2-6}  & LBM & & & & 50,853 & 15,902 \\ \cline{1-1}  & FDM & & & & 13,964 & 12,214 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison between LBM and FDM in cases of buoyancy-driven cavityand lid-driven cavity flows [133]. They stipulated that LBM simulations are performed with the same physical time as GKM and NS but with fewer iterations. Nonetheless, the LBM also has some disadvantages.Among these, we can cite the need for a large memory for the simulations, which can pose problems for 3D problems, some ambiguity concerning the values of non-hydrodynamic moments, the inadequacy of dealing with highly compressible flows and flows with large temperature fluctuations, which remain tough tasks and, thereby, are still under investigation [139]. Still, the LBM is now admitted as a powerful and effective approach for the flows in its ranges of application. Here, a suitable choice of a lattice model (_arrangement of nodes in space_) is an important first step. It is basically a Cartesian mesh lattice structure (_grid_) grid where particles propagate from one node to the next. Indeed, in general, the LBM is based on a uniform or regular grid for an easy implementation. However, there are scientific issues where non-uniform domain mesh should be implemented (e.g., _complex porous media_) without increasing the computation cost.