## 2 Mathematical and Physical Formulation
### Cascaded Lattice Boltzmann Method (CLBM)
#### 2.2.2 CLBM for the Temperature Field  
In this subsection, a D2Q5 (the five discrete velocity set, \(\{e_{i}=\left[|e_{ix}>,|e_{iy}>|i=0,1,\ldots 4\}\right)\), CLBM is proposed to solve the convection-diffusion equation for the temperature field. Similarly, the raw moments and central moments of the temperature distribution functions \(g_{i}\) can be defined by  
\[K_{mn}^{T}=\left\langle g_{i}\middle|\rho_{ix}^{m}e_{iy}^{n}\right\rangle \tag{20}\]  
\[\overline{K}_{mn}^{T}=\left\langle g_{i}\middle|\left(e_{ix}-u_{x}\right)^{m} \left(e_{iu}-u_{y}\right)^{n}\right\rangle \tag{21}\]  
The simplified raw-moment set (as opposed to the recombined raw-moment set) is also used in the D2Q5 lattice [31]:  
Recombined Raw-Moment Set:  
\[\left|I_{i}^{T}>=\left[k_{00}^{T},K_{10}^{T},K_{01}^{T},K_{20+02}^{T},K_{02-2 0}^{T}\right]^{T}\right. \tag{22}\]  
Simplified Raw-Moment Set:  
\[\left|I_{i}^{T}>=\left[k_{00}^{T},K_{10}^{T},K_{01}^{T},K_{20}^{T},K_{02}^{T} \right]^{T}\right. \tag{23}\]  
And so do the recombined central moments \(\left|\widetilde{I_{i}^{T}}>\). Analogously, the raw moments and central moments can be calculated through a transformation matrix \(M_{T}\) and a shift matrix \(N_{T}\), respectively.  
\[\Big{|}I_{i}^{T}>=M_{T}|g_{i}>\text{and}\ \Big{|}\widetilde{I_{i}^{\widetilde{T} _{i}}}>=N_{T}\Big{|}I_{i}^{T}> \tag{24}\]  
Explicitly, \(M_{T}\) and \(N_{T}\) are expressed as,  
\[M_{T}=\begin{bmatrix}1&1&1&1&1\\ 0&1&0&-1&0\\ 0&0&1&0&-1\\ 0&1&0&1&0\\ 0&0&1&0&1\end{bmatrix} \tag{25}\]  
\[N_{T}=\begin{bmatrix}1&0&0&0&0\\ -u_{x}&1&0&0&0\\ -u_{y}&0&1&0&0\\ u_{x}^{2}&-2u_{x}&0&1&0\\ u_{y}^{2}&0&-2u_{y}&0&1\end{bmatrix} \tag{26}\]  
The collision in central moments can also be written as  
\[\Big{|}\widetilde{I_{i}^{T,*}}>=(I-S_{T})\Big{|}\widetilde{I_{i}^{T}}>+S_{T} \Bigg{|}\widetilde{I_{i}^{T,eq}}> \tag{27}\]  
where \(S_{T}=diag(\lambda_{0},\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2})\) is the matrix of diagonal relaxation. A continuous 'temperature' equilibrium distribution is developed in the continuous velocity space that is comparable to the Maxwell-Boltzmann distribution (\(\varepsilon_{x},\varepsilon_{y}\)).  
\[g^{eq}=\frac{T}{2\pi c_{T}^{2}}exp\left[-\frac{\big{(}\varepsilon-\mathbf{u}\big{)} ^{2}}{2c_{T}^{2}}\right] \tag{28}\]  
where \"Sound speed \(C_{T}\)\" is defined. Next, a crucial step is taken by equating the discrete central moments \(g_{i}^{eq}\) to the continuous central moments \(g_{eq}\):  
\[\Big{\langle}g_{i}^{eq}\Big{|}\big{(}\varepsilon_{ix}-u_{x}\big{)}^{m}\big{(} \varepsilon_{iy}-u_{y}\big{)}^{n}\Big{\rangle}=\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}g^{eq}(\varepsilon_{x}-u_{x})^{m}\big{(}\varepsilon_{y}-u_{y} \big{)}^{n}d\varepsilon_{x}d\varepsilon_{y} \tag{29}\]  
Consequently, the values \(\widetilde{I_{i}}\) in equilibrium may be expressed as  
\[\Bigg{|}\widetilde{I_{i}^{T,eq}}>=\Big{[}T,0,0,Tc_{T}^{2},Tc_{T}^{2}\Big{]}^{T} \tag{30}\]  
The post-collision temperature distribution functions \(g_{i}^{*}\) can be obtained by  
\[g_{i}^{*}=M_{T}^{-1}N_{T}^{-1}\Big{|}\widetilde{I_{i}^{T,*}}> \tag{31}\]  
The streaming step for \(g_{i}\) also takes the form:  
\[g_{i}=(\mathbf{X}+\mathbf{e_{i}}\Delta t,t+\Delta t)=g_{i}^{*}(\mathbf{X},t) \tag{32}\]  
The temperature \(T\) is calculated as follows:  
\[\theta=\sum_{i=0}^{4}g_{i} \tag{33}\]