## 2 Methodology
### Evaluating Numerical Flux for Compressible Flow
#### 2.2.2 The URLBFS Developed Based on RLBFS  
From the above description, we know that in the process of solving the flux using LBFS, due to the use of the 1D D1Q4 model, it can be seen from Equations (10)-(19) that the model only considers the interface normal method. When applying the 1D LBM to calculate the inviscid flux, the contribution of the tangential velocity is not considered. A common approach for calculating tangential velocity is to use Equation (21) to split it into values on the interface, substitute these into the respective equilibrium distribution functions, and obtain the complete numerical flux through the interface. In this study, we have redefined the value of \(U_{\tau}^{*}\) using the upwind scheme. By Equation (20), we can calculate the density and the momentum at the interface. \[ \begin{array}{l}\rho=\sum\limits_{i=1,3}g_{i}^{L}+\sum\limits_{i=2,4}g_{i }^{R},\\ \rho U_{n}=\sum\limits_{i=1,3}\xi_{i}g_{i}^{L}+\sum\limits_{i=2,4}\xi_{i}g_{i }^{R}.\end{array}  \tag{26}\]From the above equation, we can obtain the \(U_{n}^{*}\) at the interface. Let \(U_{n}^{upwind}=U_{n}^{*}\) and determine the value of \(U_{\tau}^{*}\) according to the windward direction of \(U_{n}^{upwind}\) are given as  
\[U_{\tau}^{*}=\left\{ \begin{array}{l}U_{\tau}^{L},\text{ if }U_{n}^{upwind} \geq 0,\\ U_{\tau}^{R},\text{ if }U_{n}^{upwind}<0.\end{array} \right. \tag{27}\]  
Then, Equation (21) can be rewritten as  
\[\left\{ \begin{array}{l}\left(\rho U_{\tau}\right)^{*}=\sum\limits_{i=1}^{4}g _{i}U_{\tau}^{*}=\sum\limits_{i=1,3}g_{i}^{L}\cdot U_{\tau}^{L}+\sum\limits_{i =2\mathcal{A}}g_{i}^{R}\cdot U_{\tau}^{L},\\ \left(\rho U_{n}U_{\tau}\right)^{*}=\sum\limits_{i=1}^{4}\xi_{i}g_{i}U_{\tau}^ {*}=\sum\limits_{i=1,3}\xi_{i}g_{i}^{L}\cdot U_{\tau}^{L}+\sum\limits_{i=2 \mathcal{A}}\xi_{i}g_{i}^{R}\cdot U_{\tau}^{L},\\ \left(\rho U_{n}U_{\tau}^{2}\right)^{*}=\sum\limits_{i=1}^{4}\xi_{i}g_{i}(U_{ \tau}^{*})^{2}=\sum\limits_{i=1,3}\xi_{i}g_{i}^{L}\cdot\left(U_{\tau}^{L} \right)^{2}+\sum\limits_{i=2\mathcal{A}}\xi_{i}g_{i}^{R}\cdot\left(U_{\tau}^{ L}\right)^{2},\\ \left(\rho U_{\tau})^{*}=\sum\limits_{i=1}^{4}g_{i}U_{\tau}^{*}=\sum\limits_{i =1,3}g_{i}^{L}\cdot U_{\tau}^{R}+\sum\limits_{i=2\mathcal{A}}g_{i}^{R}\cdot U _{\tau}^{R},\\ \left(\rho U_{n}U_{\tau}\right)^{*}=\sum\limits_{i=1}^{4}\xi_{i}g_{i}U_{\tau}^ {*}=\sum\limits_{i=1,3}\xi_{i}g_{i}^{L}\cdot U_{\tau}^{R}+\sum\limits_{i=2 \mathcal{A}}\xi_{i}g_{i}^{R}\cdot U_{\tau}^{R},\\ \left(\rho U_{n}U_{\tau}^{2}\right)^{*}=\sum\limits_{i=1}^{4}\xi_{i}g_{i}(U_{ \tau}^{*})^{2}=\sum\limits_{i=1,3}\xi_{i}g_{i}^{L}\cdot\left(U_{\tau}^{R} \right)^{2}+\sum\limits_{i=2\mathcal{A}}\xi_{i}g_{i}^{R}\cdot\left(U_{\tau}^{ R}\right)^{2},\end{array} \right. \tag{28}\]  
Using this method to determine the tangential velocity can effectively suppress high-frequency oscillations and improve numerical stability. Combining it with the rotating lattice Boltzmann method will result in a new numerical scheme, the upwind rotating lattice Boltzmann flux solver. Unlike previous LBFS methods, the RLBFS by [PERSON] [41] breaks down normal vectors \(\mathbf{n}\) into two orthogonal vectors \(\mathbf{n_{1}}\) and \(\mathbf{n_{2}}\), as shown in Figure 4.  
The relationship between the vectors is given as  
\[ \begin{array}{l}\mathbf{n_{1}}\cdot\mathbf{n_{2}}=0,\\ \mathbf{n}=\alpha_{1}\mathbf{n_{1}}+\alpha_{2}\mathbf{n_{2}},\end{array}  \tag{29}\]  
where \(|\mathbf{n_{1}}|\mathbf{=}|\mathbf{n_{2}}|=1\), \(\alpha_{1}=\mathbf{n}\cdot\mathbf{n_{1}}\) and \(\alpha_{2}=\mathbf{n}\cdot\mathbf{n_{2}}\). To ensure that \(\mathbf{n_{1}}\) and \(\mathbf{n_{2}}\) are on the same side of the interface is necessary, then \(\alpha_{1}\) and \(\alpha_{2}\) are greater than or equal to 0. The final flux is  
\[\mathbf{F}_{RLBFS}=\mathbf{F}_{RLBFS}(n)=\alpha_{1}\mathbf{F}_{LBFS}(n_{1})+ \alpha_{2}\mathbf{F}_{LBFS}(n_{2}). \tag{30}\]  
where \(\mathbf{F}_{LBFS}(n_{1})\) and \(\mathbf{F}_{LBFS}(n_{2})\) can be determined by Equation (24).