# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 2 Simplified Thermal Lattice Boltzmann Method (STLBM) without Evolution of Distribution Functions
### Simplified Thermal Lattice Boltzmann Method (STLBM)  
The simplified thermal lattice Boltzmann method (STLBM) is a novel method that is recently developed within LBM framework [12]. Its derivation starts from the macroscopic governing equations recovered from TLBM, i.e., Equations (8)-(10). Assisted by fractional step technique, these equations can be resolved in a predictor-corrector scheme  
**Predictor step:**  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\sum_{\alpha}\mathbf{e}_{ \alpha}f_{\alpha}^{eq}\right)=0 \tag{17}\]  
\[\frac{\partial\rho\mathbf{u}}{\partial t}+\
abla\cdot\left[\sum_{\alpha} \left(\mathbf{e}_{\alpha}\right)_{\beta}\left(\mathbf{e}_{\alpha}\right)_{ \gamma}f_{\alpha}^{eq}+\frac{1}{2\tau_{\upsilon}}\sum_{\alpha}\left(\mathbf{e }_{\alpha}\right)_{\beta}\left(\mathbf{e}_{\alpha}\right)_{\gamma}f_{\alpha}^ {neq}\right]=0 \tag{18}\]  
\[\frac{\partial\rho e}{\partial t}+\
abla\cdot\left[\sum_{\alpha}\mathbf{e}_{ \alpha}g_{\alpha}^{eq}+\frac{1}{2\tau_{\varsigma}}\sum_{\alpha}\mathbf{e}_{ \alpha}g_{\alpha}^{neq}\right]=0 \tag{19}\]  
**Corrector step:**  
\[\frac{\partial\rho}{\partial t}=0 \tag{20}\]  
\[\frac{\partial\rho\mathbf{u}}{\partial t}+\
abla\cdot\left[\left(1-\frac{1}{ \tau_{\upsilon}}\right)\sum_{\alpha}\left(\mathbf{e}_{\alpha}\right)_{\beta} \left(\mathbf{e}_{\alpha}\right)_{\gamma}f_{\alpha}^{neq}\right]-\mathbf{F}_{E}=0 \tag{21}\]  
\[\frac{\partial\rho e}{\partial t}+\
abla\cdot\left[\left(1-\frac{1}{\tau_{ \varsigma}}\right)\sum_{\alpha}\mathbf{e}_{\alpha}g_{\alpha}^{neq}\right]=0 \tag{22}\]According to the Boussinesq approximation [16], the external force term can be regarded as a buoyance force related to the temperature profile, i.e.,  
\[\mathbf{F}_{E}=\left( \begin{array}{c}0\\ 0\\ -\rho g\beta(T-T_{m})\end{array} \right) \tag{23}\]  
where \(g\) is the gravitational acceleration; \(\beta\) is the thermal expansion coefficient; and \(T_{m}\) is the average temperature. By using lattice properties and the relationships given by C-E analysis, the above Equations (17)-(21) can be reconstructed within the LBM framework to obtain the formulations of simplified thermal lattice Boltzmann method (STLBM)  
**Predictor step:**  
\[\rho^{*}=\sum_{\alpha}f_{\alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha}\delta_{ t},t-\delta_{t}) \tag{24}\]  
\[\rho^{*}\mathbf{u}^{*}=\sum_{\alpha}\mathbf{e}_{\alpha}f_{\alpha}^{eq}( \mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t}) \tag{25}\]  
\[\rho^{*}e^{*}=\sum_{\alpha}g_{\alpha}^{eq}(\mathbf{r}-\mathbf{e}_{\alpha} \delta_{t},t-\delta_{t}) \tag{26}\]  
**Corrector step:**  
\[\rho(\mathbf{r},t)=\rho^{*} \tag{27}\]  
\[\rho(\mathbf{r},t)\mathbf{u}(\mathbf{r},t)=\rho^{*}\mathbf{u}^{*}+\bigg{(}1- \frac{1}{\tau_{v}}\bigg{)}\sum_{\alpha}\mathbf{e}_{\alpha}f_{\alpha}^{neq}( \mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t)+\mathbf{F}_{E}\delta_{t} \tag{28}\]  
\[\rho(\mathbf{r},t)e(\mathbf{r},t)=\rho^{*}e^{*}+\bigg{(}1-\frac{1}{\tau_{c}} \bigg{)}\sum_{\alpha}g_{\alpha}^{neq}(\mathbf{r}-\mathbf{e}_{\alpha}\delta_{t },t) \tag{29}\]  
Inspired by the treatment in lattice Boltzmann flux solver (LBFS), the non-equilibrium terms involved in Equations (28) and (29) are approximated by the difference between two equilibrium distribution functions at different locations and time levels, i.e.,  
\[f_{\alpha}^{neq}(\mathbf{r},t)=-\tau_{v}\delta_{t}\mathrm{D}f_{\alpha}^{eq}( \mathbf{r},t)=-\tau_{v}\Big{[}f_{\alpha}^{eq}(\mathbf{r},t)-f_{\alpha}^{eq}( \mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t})\Big{]} \tag{30}\]  
\[g_{\alpha}^{neq}(\mathbf{r},t)=-\tau_{c}\delta_{t}\mathrm{D}g_{\alpha}^{eq}( \mathbf{r},t)=-\tau_{c}\Big{[}g_{\alpha}^{eq}(\mathbf{r},t)-g_{\alpha}^{eq}( \mathbf{r}-\mathbf{e}_{\alpha}\delta_{t},t-\delta_{t})\Big{]} \tag{31}\]  
According to [PERSON] et al. [17] and [PERSON] et al. [18], the equilibrium distribution functions at time level of t is evaluated from the intermediate macroscopic variables obtained in the predictor step. Compared to the standard TLBM, simplified thermal lattice Boltzmann method (STLBM) possesses many advantages.