## 2 Methodology
### WENO Scheme Reconstruction Interface Flow Variables
#### 2.3.1 WENO5 and WENO7 Reconstruction  
To use the WENO5 scheme to interpolate the flow variables, it is necessary to use the information of the surrounding nodes, as illustrated in Figure 5:  
The global stencil of the WENO5 scheme is \(S=\{U_{i-2},U_{i-1},U_{i},U_{i+1},U_{i+2}\}\). To construct the reconstruction polynomial, it is divided into three sub-stencils:  
\[S_{0}=\{U_{i-2},U_{i-1},U_{i}\},\ S_{1}=\{U_{i-1},U_{i},U_{i+1}\},\ S_{2}=\{U_ {i},U_{i+1},U_{i+2}\}. \tag{32}\]  
These three sub-stencils correspond to three low-order reconstruction methods, which are combined by weighted average to obtain the final WENO5 scheme [7]  
\[ \begin{array}{ll}\mathbf{U}_{i+1/2}^{L}&=w_{0}\Big{(}\frac{1}{3}\mathbf{U}_{ i-2}-\frac{7}{6}\mathbf{U}_{i-1}+\frac{11}{6}\mathbf{U}_{i}\Big{)}\\ &+w_{1}\Big{(}-\frac{1}{6}\mathbf{U}_{i-1}+\frac{5}{6}\mathbf{U}_{i}+\frac{1}{ 3}\mathbf{U}_{i+1}\Big{)}\\ &+w_{2}\Big{(}\frac{1}{3}\mathbf{U}_{i}+\frac{5}{6}\mathbf{U}_{i+1}-\frac{1}{ 6}\mathbf{U}_{i+2}\Big{)},\end{array}  \tag{33}\]  
In order to achieve the designed precision and avoid oscillation of the consequence, this paper adopts the WENO-JS scheme proposed by [PERSON] [7] to calculate the weight coefficients and smoothness indicators, as shown in the following formula:  
\[w_{k}=\frac{\alpha_{k}}{\alpha_{0}+\alpha_{1}+\alpha_{2}},\ \alpha_{k}=\frac{d_{k}}{\left(\beta_{k}+ \varepsilon\right)^{2}},\ k=0,1,2. \tag{34}\]  
Figure 5: Schematic of the global stencil and sub-stencils (\(S_{0}-S_{2}\)) for WENO5. \[ \begin{array}{l}\beta_{0}=\frac{13}{12}(\mathbf{U}_{i-2}-2\mathbf{U}_{i-1}+\mathbf{U}_{i})^{2} +\frac{1}{4}(\mathbf{U}_{i-2}-4\mathbf{U}_{i-1}+3\mathbf{U}_{i})^{2},\\ \beta_{1}=\frac{13}{12}(\mathbf{U}_{i-1}-2\mathbf{U}_{i}+\mathbf{U}_{i+1})^{2}+\frac{1}{4}( \mathbf{U}_{i-1}-\mathbf{U}_{i+1})^{2},\\ \beta_{2}=\frac{13}{12}(\mathbf{U}_{i}-2\mathbf{U}_{i+1}+\mathbf{U}_{i+2})^{2}+\frac{1}{4}( 3\mathbf{U}_{i}-4\mathbf{U}_{i+1}+\mathbf{U}_{i+2})^{2}.\end{array}  \tag{35}\]  
Following [7], the optimal linear weighting coefficients are \(d_{0}=\frac{1}{10}\), \(d_{1}=\frac{3}{5}\), \(d_{2}=\frac{3}{10}\). In order to avoid the denominator being zero, \(\varepsilon=10^{-6}\) is usually set. Through the above description, we can achieve the value of \(\mathrm{U}_{i+1/2}^{L}\), and the value of \(\mathrm{U}_{i+1/2}^{R}\) can be obtained by a symmetrical method, which will not be described here. Compared with the WENO5 scheme, WENO7 requires seven points to reconstruct the physical quantity of the interface, as illustrated in Figure 6:  
The global stencil of the WENO7 scheme is \(S=\{U_{i-3},\cdots,U_{i+3}\}\). In order to construct the reconstruction polynomial, it is divided into three sub-stencils:  
\[ \begin{array}{l}S_{0}=\{U_{i-3},U_{i-2},U_{i-1},U_{i}\},\ S_{1}=\{U_{i-2},U_{i-1},U_{i},U_{i+1}\},\\ S_{2}=\{U_{i-1},U_{i},U_{i+1},U_{i+2}\},\ S_{3}=\{U_{i},U_{i+1},U_{i+2},U_{i+3}\}, \end{array}  \tag{36}\]  
The left-state seventh-order WENO reconstruction is defined as  
\[ \begin{array}{l}\mathbf{U}_{i+1/2}^{L}\quad=w_{0}\Bigl{(}-\frac{1}{4}\mathbf{U}_{i- 3}+\frac{13}{12}\mathbf{U}_{i-2}-\frac{23}{12}\mathbf{U}_{i-1}+\frac{25}{12}\mathbf{U}_{i} \Bigr{)}\\ \quad+w_{1}\Bigl{(}\frac{1}{12}\mathbf{U}_{i-2}-\frac{5}{12}\mathbf{U}_{i-1}+\frac{13} {12}\mathbf{U}_{i}+\frac{1}{4}\mathbf{U}_{i+1}\Bigr{)}\\ \quad+w_{2}\Bigl{(}-\frac{1}{12}\mathbf{U}_{i-1}+\frac{7}{2}\mathbf{U}_{i}+\frac{7}{2} \mathbf{U}_{i+1}-\frac{1}{12}\mathbf{U}_{i+2}\Bigr{)}\\ \quad+w_{3}\Bigl{(}\frac{1}{4}\mathbf{U}_{i}+\frac{13}{12}\mathbf{U}_{i+1}-\frac{5}{12} \mathbf{U}_{i+2}+\frac{1}{12}\mathbf{U}_{i+3}\Bigr{)},\end{array}  \tag{37}\]  
The definition of weight coefficient and smoothness indicators still adopts the WENO-JS scheme proposed by SHU [7]. \[w_{k}=\frac{\alpha_{k}}{\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}},\ \alpha_{k}=\frac{d_{k}}{(\beta_{k}+\varepsilon)^{2}},\ k=0,1,2,3. \tag{38}\]  
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