## 2 WENO-ZQ Schemes  
Before applying the finite-volume form of the WENO-ZQ scheme, let us first provide a brief introduction. Consider the one-dimensional hyperbolic conservation law  
\[ \begin{cases}u_{t}+f(u)_{x}=0,\\ u(x,0)=u_{0}(x),\end{cases}  \tag{1}\]  
which is discretized in the computational domain using a uniform cell size \(h\). Integrating Equation (1) in a control volume \(I_{i}=[x_{i-1/2},x_{i+1/2}]\) cell yields  
\[\frac{d\overline{u}(x_{i},t)}{dt}+\frac{1}{h}[f(u(x_{i+1/2},t))-f(u(x_{i-1/2}, t))]=0, \tag{2}\]  
where \(\overline{u}(x_{i},t)\) is the average value of \(u\) over the cell \(I_{i}\) at time. Equation (2) can be rewritten as  
\[\frac{d\overline{u}(x_{i},t)}{dt}=L(u_{i})=-\frac{1}{h}\Big{(}\hat{f}_{i+1/2}- \hat{f}_{i-1/2}\Big{)}.\]The Roe-difference splitting scheme [23] and AUSM-up flux splitting scheme [24] are employed in this paper to calculate the numerical flux using the equation  
\[\left\{ \begin{aligned} & ROE:\quad\quad\quad\quad\quad\quad \hat{f}_{i+1/2}=\frac{1}{2}\big{(}f\big{(}u^{L}\big{)}+f\big{(}u^{R}\big{)} \big{)}-\frac{1}{2}\Big{|}\overline{A}\Big{|}\big{(}u^{R}-u^{L}\big{)},\\ &\text{AUSM }-up:\quad\hat{f}_{i+1/2}=F_{i+1/2}^{(c)}+P_{i+1/2}=m_{i+1/2} \psi_{i+1/2}+P_{i+1/2},\end{aligned} \right. \tag{3}\]  
where \(u^{L}\) and \(u^{R}\) are the reconstructed left and right valuables on the face of the control volume \(l_{i}\). The Roe scheme is used in transonic flow, and AUSM-up is used in low-speed flow, supersonic flow, and LES simulation. Based on this approach [13], a new fifth-order WENO-ZQ scheme (referred to as WENO-ZQS) was proposed by [PERSON] and [PERSON] [11; 12]. They innovatively constructed an adaptive formula, enabling the use of a large stencil in smooth regions and two small stencils in capturing discontinuities. This new fifth-order WENO-ZQ scheme features positive linear weights and high accuracy. The procedure of WENO-ZQS is summarized as follows:  
1. Choose the big central spatial stencil \(T_{1}=\{I_{i-2},I_{i-1},I_{i},I_{i+1},I_{i+2}\}\) and the other two smaller stencils \(T_{2}=\{I_{i-1},I_{i}\}\) and \(T_{3}=\{I_{i},I_{i+1}\}\) to reconstruct the polynomials \(p_{1}(x)\), \(p_{2}(x)\), and \(p_{3}(x)\). Additionally, the generalized expression for the reconstructed polynomial on nonuniform meshes provided by [PERSON] [25] is adopted in this paper. 2. Compute the smoothness indicators \(\beta_{l}\), \(l=1,2,3\), which are obtained through a multiple of the local grid spacing and the difference in polynomial values at adjacent points: \[\beta_{l}=\sum_{k=1}^{r}\int_{I_{i}}h^{2k-1}\bigg{(}\frac{d^{k}p_{l}(x)}{dx^{ k}}\bigg{)}^{2}dx,l=1,2,3\] (4)
3. Calculate the nonlinear weights based on the linear weights and the smoothness indicators. An adaptive formula for \(\tau\) is written based on the difference between \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) as follows: \[\tau=\bigg{(}\frac{|\beta_{1}-\beta_{2}|+|\beta_{1}-\beta_{3}|}{2}\bigg{)}^{2}\] The nonlinear weights \(\omega_{l}\), \(l=1\), \(2\), \(3\) in the expression are then expressed as \[\omega_{l}=\bigg{(}\frac{\overline{\omega}_{l}}{\overline{\omega}_{1}+ \overline{\omega}_{2}+\overline{\omega}_{3}}\bigg{)},\ \overline{\omega}_{l}=\gamma_{l}\bigg{(}1+\frac{\tau}{\beta_{l}+\varepsilon} \bigg{)},\ l=1,2,3\] where \(\gamma_{l}\) represents the positive linear weights, with the only requirement being that \(\gamma_{1}+\gamma_{2}+\gamma_{3}=1\) (\(\gamma_{1}\
eq 0\)). Here, \(\varepsilon\) is a small positive number to avoid the denominator becoming zero. 4. The final reconstruction formulation of conservative values \(u(x,\ t)\) at the point \(x_{i+1/2}\) of the target cell \(l_{i}\) is given by \(\beta_{l},l=1,2,3\). \[u_{l+\frac{1}{2}}^{L}=\omega_{1}\bigg{(}\frac{1}{\gamma_{1}}p_{1}(x)-\frac{ \gamma_{2}}{\gamma_{1}}p_{2}(x)-\frac{\gamma_{3}}{\gamma_{1}}p_{3}(x)\bigg{)}+ \omega_{2}p_{2}(x)+\omega_{3}p_{3}(x)\] (5) The procedure of WENO-JS5 is summarized as follows: 1.