# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 4 Numerical Tests
### 2D Natural Convection in a Square Cavity
Such performance demonstrates that both USTLBM and STLBM have very good numerical stability, which makes them superior to standard LBM that can easily diverge  
Figure 3: Relative errors of volume averaged Nusselt numbers at different mesh spacing. Figure 2: Physical configuration of 2D natural convection in a square cavity. in cases with high Reynolds/Rayleigh numbers. Convergence study is also carried out with various mesh sizes adopted in these cases. It turns out that the calculated flow parameters are approaching to the reference data by refining the meshes, which verifies that both USTLBM and STLBM can converge to the correct answer. Another notable issue reflected in these tables is that at the same mesh size, the results obtained by USTLBM show smaller relative errors to the reference data. At high Rayleigh numbers, STLBM usually requires finer meshes to achieve comparable results. Here, we take the maximum vertical velocity for illustration. At \(Ra=10^{6}\), results obtained by USTLBM are within the range given by reference data at the mesh size of \(201\times 201\), while STLBM can only get comparable results at the mesh size of \(251\times 251\). In the case of \(Ra=10^{8}\), the relative error of results obtained by USTLBM at the mesh size of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **Grid** & \(\mathbf{11^{2}}\) & \(\mathbf{301^{2}}\) & \(\mathbf{351^{2}}\) & \(\mathbf{401^{2}}\) & \(\mathbf{451^{2}}\) & **Reference**[23; 24; 25] \\ \hline \multirow{2}{*}{\(Nu_{1/2}\)} & STLBM & 305.29 & 26.737 & 29.610 & 29.774 & 29.875 & \\  & USTLBM & 322.15 & 29.744 & 29.874 & 29.958 & 30.014 & 30.225–30.301 \\ \hline \multirow{2}{*}{\(u_{\max}\)} & STLBM & 1730.3 & 551.59 & 294.05 & 297.02 & 302.28 & \\  & USTLBM & 1799.6 & 312.71 & 317.23 & 319.76 & 321.27 & 321.37–323.65 \\ \hline \multirow{2}{*}{\(v_{\max}\)} & STLBM & 1919.2 & 2503.2 & 2199.3 & 2206.7 & 2212.3 & \\  & USTLBM & 2010.4 & 2219.6 & 2220.1 & 2221.6 & 2222.0 & 2222.0–2222.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparisons of flow parameters in \(Ra=10^{8}\) at different mesh sizes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & **Grid** & \(\mathbf{11^{2}}\) & \(\mathbf{151^{2}}\) & \(\mathbf{201^{2}}\) & \(\mathbf{251^{2}}\) & \(\mathbf{301^{2}}\) & **Reference**[19; 20; 21; 22] \\ \hline \multirow{2}{*}{\(Nu_{0}\)} & STLBM & 1.096 & 4.503 & 4.526 & 4.530 & 4.531 & 4.504–4.532 \\  & USTLBM & 1.365 & 4.513 & 4.527 & 4.530 & 4.530 & \\ \hline \multirow{2}{*}{\(u_{\max}\)} & STLBM & 36.900 & 34.531 & 34.566 & 34.596 & 34.624 & \\  & USTLBM & 55.550 & 34.523 & 34.577 & 34.610 & 34.635 & 34.736–34.87 \\ \hline \multirow{2}{*}{\(v_{\max}\)} & STLBM & 37.166 & 68.075 & 68.351 & 68.456 & 68.545 & \\  & USTLBM & 56.930 & 68.200 & 68.390 & 68.464 & 68.552 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparisons of flow parameters in \(Ra=10^{5}\) at different mesh sizes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & **Grid** & \(\mathbf{11^{2}}\) & \(\mathbf{201^{2}}\) & \(\mathbf{251^{2}}\) & \(\mathbf{301^{2}}\) & \(\mathbf{351^{2}}\) & **Reference**[19; 20; 21; 22] \\ \hline \multirow{2}{*}{\(Nu_{0}\)} & STLBM & 1.115 & 8.434 & 8.634 & 8.732 & 8.782 & \\  & USTLBM & 1.457 & 8.599 & 8.724 & 8.781 & 8.810 & 8.767–8.928 \\ \hline \multirow{2}{*}{\(u_{\max}\)} & STLBM & 122.78 & 63.665 & 64.005 & 64.173 & 64.364 & \\  & USTLBM & 188.10 & 64.346 & 64.441 & 64.513 & 64.577 & 64.775–65.33 \\ \hline \multirow{2}{*}{\(v_{\max}\)} & STLBM & 123.73 & 214.16 & 217.36 & 218.56 & 219.21 & \\  & USTLBM & 193.61 & 217.64 & 218.61 & 219.11 & 219.65 & 216.75–221.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons of flow parameters in \(Ra=10^{6}\) at different mesh sizes.