## 2 Methodology
### WENO Scheme Reconstruction Interface Flow Variables
#### 2.3.2 Characteristic Space Transform  
Typically, conservative variables are reconstructed in physical space, followed by the calculation of the corresponding flow variables. However, when the reconstruction in the physical space cannot solve the shock wave problem well, there will still be problems such as low resolution and numerical oscillation. Therefore, this paper adopts the reconstruction in the characteristic space proposed by [PERSON] [8].  
We already have the conservative variables for all grid points. The eigenvectors of the eigenmatrix \(\left.\frac{\partial\mathbf{F}}{\partial\mathbf{U}}\right|_{x_{i+1/2}y_{i}}\) in the x direction are  
\[R^{F}=\left(\begin{array}{cccc}1&1&1&1\\ u_{1}-c&u_{1}&u_{1}&u_{1}+c\\ u_{2}&0&1&u_{2}\\ \gamma e+\frac{u_{1}^{2}+u_{2}^{2}}{2}-u_{1}c&\frac{u_{1}^{2}-u_{2}^{2}}{2}&u_ {2}+\frac{u_{1}^{2}-u_{2}^{2}}{2}&\gamma e+\frac{u_{1}^{2}+u_{2}^{2}}{2}+u_{1} c\\ \end{array}\right), \tag{40}\] \[L^{F}=\left(\begin{array}{cccc}\frac{u_{1}^{2}+u_{2}^{2}}{4 \gamma e}+\frac{u_{1}}{2c}&-\frac{1}{2c}-\frac{u_{1}}{2\gamma e}&-\frac{u_{2}}{ 2\gamma e}&\frac{1}{2\gamma e}\\ 1-\frac{u_{1}+u_{2}^{2}}{2\gamma e}+\frac{u_{2}(u_{1}^{2}+u_{2}^{2})}{2\gamma e }&\frac{u_{1}-u_{1}u_{2}}{\gamma e}&\frac{u_{2}-u_{2}^{2}}{\gamma e}-1&\frac{u _{2}-1}{\gamma e}\\ \frac{u_{2}(u_{1}^{2}+u_{2}^{2})}{2\gamma e}&\frac{u_{1}u_{2}}{\gamma e}&1+ \frac{u_{2}^{2}}{\gamma e}&-\frac{u_{2}}{\gamma e}\\ \frac{u_{1}^{2}+u_{2}^{2}}{4\gamma e}-\frac{u_{1}}{2c}&\frac{1}{2c}-\frac{u_{1 }}{2\gamma e}&-\frac{u_{2}}{2\gamma e}&\frac{1}{2\gamma e}\\ \end{array}\right).\]  
The conservative variables \(\mathbf{U}_{i+1/2,j}\) at \(x_{i+1/2,j}\) is  
\[\mathbf{U}_{i+1/2,j}=\frac{1}{2}\big{(}\mathbf{U}_{i,j}+\mathbf{U}_{i+1,j} \big{)}, \tag{41}\]  
Through the left characteristic variable \(L^{F}\), the conservation variables \(\mathbf{U}_{k,j}\) can be converted to the local characteristic space to obtain the local characteristic variable \(\mathbf{V}_{k,j}\)  
\[\mathbf{V}_{k,j}=L^{F}\mathbf{U}_{k,j}, \tag{42}\]  
where \(k\) denotes the data points to construct the \(\mathbf{V}_{i+1/2,j}^{\pm}\) value on both sides of the interface. When reconstructed using the WENO5 scheme, the value range of \(k\) is \((i-2,\cdots,i+3)\), and when reconstructed using the WENO7 scheme, the value range of \(k\) is \((i-3,\cdots,i+4)\). After obtaining the local characteristic variable \(\mathbf{V}_{i+1/2,j}^{\pm}\), we can convert it to the characteristic space through the right characteristic matrix  
\[\mathbf{U}_{i+1/2,j}^{\pm}=R^{F}\mathbf{V}_{i+1/2,j}^{\pm}. \tag{43}\]  
Finally, the corresponding flow variable \(\rho_{i+1/2}^{\pm}\), \(u_{i+1/2}^{\pm}\), \(v_{i+1/2}^{\pm}\), \(p_{i+1/2}^{\pm}\) can be obtained through the conservation variables.  
The y direction is a similar process, except that the expression of the eigenvector matrix is different.  
\[R^{G}=\left(\begin{array}{cccc}1&1&1&1\\ u_{1}&0&1&u_{1}\\ u_{2}-c&u_{2}&u_{2}&u_{2}+c\\ \gamma e+\frac{u_{1}^{2}+u_{2}^{2}}{2}-u_{2}c&\frac{u_{2}^{2}-u_{1}^{2}}{2}&u+ \frac{u_{2}^{2}-u_{1}^{2}}{2}&\gamma c+\frac{u_{1}^{2}+u_{2}^{2}}{2}+u_{2}c \end{array}\right), \tag{44}\] \[L^{G}=\left(\begin{array}{cccc}\frac{u_{2}^{2}+u_{2}^{2}}{4 \gamma e}+\frac{u_{2}}{2c}&-\frac{u_{1}}{2\gamma e}&-\frac{1}{2c}-\frac{u_{2}}{ 2\gamma e}&\frac{1}{2\gamma e}\\ 1-\frac{u_{1}^{2}+u_{2}^{2}}{2\gamma e}&\frac{u_{1}(u_{1}^{2}+u_{2}^{2})}{2 \gamma e}&\frac{u_{1}-u_{1}^{2}}{2\gamma e}&\frac{u_{1}-u_{1}\gamma}{\gamma e} \\ -\frac{u_{1}(u_{1}^{2}+u_{2}^{2})}{2\gamma e}&1+\frac{u_{1}^{2}}{\gamma e}& \frac{u_{1}u_{2}}{\gamma e}&-\frac{u_{1}}{\gamma e}\\ \frac{u_{2}^{2}+u_{2}^{2}}{4\gamma e}-\frac{u_{2}}{2\gamma e}&-\frac{u_{1}}{2 \gamma e}&\frac{1}{2c}-\frac{u_{2}}{2\gamma e}&\frac{1}{2\gamma e}\end{array} \right).\]  
Unless otherwise specified, all reconstruction processes are performed in the characteristic space in this paper.