# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 3 Unconditionally Stable Thermal Lattice Boltzmann Method (USTLBM)
### Unconditionally Stable Thermal Lattice Boltzmann Method (USTLBM)  
With the modifications made on previous simplified thermal lattice Boltzmann method (STLBM), formulations of the present unconditionally stable thermal lattice Boltzmann method (USTLBM) are shown as follows:  
**Predictor step:**  
\[\rho^{*}=\sum_{\alpha}f_{\alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha}\delta_{ t},t-\delta_{t}) \tag{39}\]  
\[\rho^{*}\mathbf{u}^{*}=\sum_{\alpha}\mathbf{e}_{\alpha}f_{\alpha}^{eq}( \mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t}) \tag{40}\]  
\[\rho^{*}e^{*}=\sum_{\alpha}g_{\alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha} \delta_{t},t-\delta_{t}) \tag{41}\]  
**Corrector step:**  
\[\rho(\mathbf{r},t)=\rho^{*} \tag{42}\]  
\[\rho(\mathbf{r},t)\mathbf{u}(\mathbf{r},t)=\rho^{*}\mathbf{u}^{*}-\left(1- \frac{1}{\tau_{c}}\right)\sum_{\alpha}\mathbf{e}_{\alpha}\Big{[}f_{\alpha}^{neq }\Big{(}\mathbf{r}+\frac{1}{2}\mathbf{e}_{\alpha}\delta_{t},t-\frac{1}{2} \delta_{t}\Big{)}-f_{\alpha}^{neq}\Big{(}\mathbf{r}-\frac{1}{2}\mathbf{e}_{ \alpha}\delta_{t},t-\frac{1}{2}\delta_{t}\Big{)}\Big{]}+\mathbf{F}_{E}\delta_{t} \tag{43}\]  
\[\rho(\mathbf{r},t)e(\mathbf{r},t)=\rho^{*}e^{*}-\left(1-\frac{1}{\tau_{c}} \right)\sum_{\alpha}\bigg{[}g_{\alpha}^{neq}\Big{(}\mathbf{r}+\frac{1}{2} \mathbf{e}_{\alpha}\delta_{t},t-\frac{1}{2}\delta_{t}\Big{)}-g_{\alpha}^{neq }\bigg{(}\mathbf{r}-\frac{1}{2}\mathbf{e}_{\alpha}\delta_{t},t-\frac{1}{2} \delta_{t}\bigg{)}\bigg{]} \tag{44}\]  
where the non-equilibrium terms are evaluated by  
\[f_{\alpha}^{neq}\bigg{(}\mathbf{r}-\frac{1}{2}\mathbf{e}_{\alpha}\delta_{t}, t-\frac{1}{2}\delta_{t}\bigg{)}=-\tau_{v}\Big{[}f_{\alpha}^{eq}(\mathbf{r},t)-f_{ \alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t})\Big{]} \tag{45}\]  
\[g_{\alpha}^{neq}\bigg{(}\mathbf{r}-\frac{1}{2}\mathbf{e}_{\alpha}\delta_{t}, t-\frac{1}{2}\delta_{t}\bigg{)}=-\tau_{c}\Big{[}g_{\alpha}^{eq}(\mathbf{r},t)-g_{ \alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t})\Big{]} \tag{46}\]  
To ensure the second-order of accuracy in time, the external force term \(\mathbf{F}_{E}\) should be defined on \((\mathbf{r},t-0.5\delta_{t})\). According to Boussinesq approximation shown in Equation (23), this term is approximated by  
\[F_{Ex}=0,\ F_{Ey}=0,\ F_{Ez}=-0.5\rho\beta[T(\mathbf{r},t)+T(\mathbf{r},t- \delta_{t})-2T_{m}] \tag{47}\]  
Since \(T(\mathbf{r},t)\) can be solved firstly and then substituted into Equation (47) in the same time step, the equations are decoupled and the explicitness of the scheme is ensured.  
As discussed above, USTLBM is an improved version of STLBM and is a truly second-order method. Nevertheless, the basic idea of USTLBM is similar to that of STLBM, which is to reconstruct solutions to macroscopic governing equations within LBM framework. Such characteristics grant USTLBM advantages from both STLBM and standard LBM. On one hand, USTLBM maintains the simplicity and explicitness, and ensures the second-order of accuracy in time and space as standard LBM does. On the other hand, USTLBM directly updates macroscopic variables rather than distribution functions, which greatly lowers virtual memories and facilitates implementation of physical boundary conditions. Moreover, as its name indicates, USTLBM can be theoretically proven to be unconditionally stable, which will be discussed in detail in following sections.