## 2 Methodology
### Evaluating Numerical Flux for Compressible Flow
#### 2.2.1 The RLBFS Scheme
The expression is  
\[f_{i}(0,t)=g_{i}(0,t)-\tau_{0}[g_{i}(0,t)-g_{i}(-\xi_{i}\delta t,t-\delta t)]+ \mathcal{O}(\delta t), \tag{14}\]  
where \(g_{i}(0,t)\) and \(g_{i}(-\xi_{i}\delta t,t-\delta t)\) represent the equilibrium distribution functions at the interface and the neighboring points, respectively. \(\tau_{0}=\tau/\delta t\), \(\delta t\) is the streaming time step, \(\tau_{0}\) is the dimensionless collision time. Then, the inviscid flux in Equation (13) can be expressed as  
\[ \begin{split}(\mathbf{F}_{c})_{i+1/2}^{*}&=(1-\tau _{0})\sum\limits_{i=1}^{4}\xi_{i}\varphi_{\alpha}g_{i}(0,t)+\tau_{0}\sum \limits_{i=1}^{4}\xi_{i}\varphi_{\alpha}g_{i}(-\xi_{i}\delta t,t-\delta t)\\ &=(1-\tau_{0})\mathbf{F}_{c,i+1/2}^{(Is)}+\tau_{0}\mathbf{F}_{c,i+ 1/2}^{(II*)}.\end{split}  \tag{15}\]  
Therefore, to accurately calculate the total inviscid flux, obtaining the values of the two equilibrium distribution functions \(g_{i}(-\xi_{i}\delta t,t-\delta t)\) and \(g_{i}(0,t)\) is necessary. Figure 2: Application of 1D LB model at cell interface in 2D cases. among them, \(U_{\tau}^{*}\) is the tangential velocities. The tangential velocity on the left side of the interface is \(U_{\tau}^{L}\), and the velocity on the right side is \(U_{\tau}^{R}\). The primitive variables \(\rho^{*},\;U_{n}^{*},\;U_{\tau}^{*},\;p^{*}\) were obtained in this way. Afterward, the inviscid flux at \(i+1/2\) is  
\[\mathbf{F}_{c,i+1/2}^{(I)}= \begin{bmatrix}\rho U_{n}\\ (\rho U_{n}U_{n}+p)n_{x}-\rho U_{n}U_{t}n_{y}\\ (\rho U_{n}U_{n}+p)n_{y}+\rho U_{n}U_{\tau}n_{x}\\ (\rho E+p)U_{n}\end{bmatrix} _{i+1/2}^{*}, \tag{22}\]  
Combining Equations (18) and (21), we can achieve the convective flux as  
\[\mathbf{F}_{c,i+1/2}^{(II)}= \begin{bmatrix}\sum\limits_{i=1}^{4}\xi_{i}g_{i} \\ \sum\limits_{i=1}^{4}\xi_{i}\xi_{i}g_{i}n_{x}-\sum\limits_{i=1}^{4}\xi_{i}g_{i }\cdot U_{\tau}^{*}n_{y}\\ \sum\limits_{i=1}^{4}\xi_{i}\xi_{i}g_{i}n_{y}+\sum\limits_{i=1}^{4}\xi_{i}g_{i }\cdot U_{\tau}^{*}n_{x}\\ \sum\limits_{i=1}^{4}\xi_{i}\left(\frac{1}{2}\xi_{i}\xi_{i}+e_{p}\right)g_{i} +\frac{1}{2}\sum\limits_{i=1}^{4}\xi_{i}g_{i}\cdot(U_{\tau}^{*})^{2}\end{bmatrix} , \tag{23}\]  
Finally, the total numerical flux is  
\[\mathbf{F}_{c,i+1/2}=\big{(}1-\tau_{0}\big{)}\mathbf{F}_{c,i+1/2}^{(I)}+\tau_ {0}\mathbf{F}_{c,i+1/2}^{(II)}. \tag{24}\]  
The \(\tau_{0}\) in above equation has been mentioned before and is expressed as dimensionless collision time. Since this research focuses solely on inviscid compressible flow, \(\tau_{0}\) can be considered as a coefficient controlling numerical dissipation. When \(\tau_{0}\) approaches zero, it indicates minimal dissipation while approaching one suggests significant dissipation. This article adopts a switching function proposed by [PERSON] [32] to control \(\tau_{0}\). When approaching the boundary layer, \(\tau_{0}\) can be controlled to be close to zero, and near a strong shock wave, \(\tau_{0}\) can be controlled to be close to one. The function is  
\[\tau_{0}=\tanh\bigg{(}C\frac{|p_{L}-p_{R}|}{p_{L}+p_{R}}\bigg{)}. \tag{25}\]  
where \(C\) is a magnifying factor, the value is 1-100. \(p_{L}\) and \(p_{R}\) are the pressures at \(i+1/2\).