# Application of a Novel High-Order WENO Scheme in LES Simulations
## 2 WENO-ZQ Scheme and Numerical Tests  
### Description of WENO-ZQ Scheme  
This section introduces the high-order WENO-ZQ scheme using the one-dimensional Navier-Stokes (N-S) equations as an example:  
\[\left\{ \begin{array}{rcl}u_{t}+f_{x}^{inv}(u)+f_{x}^{vis}(u)=0\\ u(x,0)=u_{0}(x)\end{array} \right. \tag{1}\]  
The computational domain is uniformly discretized using grid cells \(I_{i}=[x_{i-1/2},x_{i+1/2}]\), with a size set to \(h=x_{i+1/2}-x_{i-1/2}\). Upon selecting cell \(I_{i}\) as the current control volume, discretization of the spatial terms yields the following expression:  
\[ \begin{array}{rcl}\frac{d\overline{m}(x_{i},t)}{dt}&+&\frac{1}{h}\left(f^{ inv}(u(x_{i+1/2},t))-f^{inv}(u(x_{i-1/2},t))\right)\\ &+&\frac{1}{h}\left(f^{vis}(u(x_{i+1/2},t))-f^{vis}(u(x_{i-1/2},t))\right)\end{array}  =0 \tag{2}\]where the cell average of \(u(x,t)\) is \(\overline{u}(x_{i},t)=\frac{1}{h}J_{x_{i-1/2}}^{x_{i+1/2}}\,u(x,t)dx\). The approximation of Equation (2) is rendered in a conservative form as follows:  
\[\frac{d\overline{u}_{i}(t)}{dt}=L(u_{i})=-\frac{1}{h}\Big{(}\hat{f}_{i+1/2}^{ inv}-\hat{f}_{i-1/2}^{inv}\Big{)}-\frac{1}{h}\Big{(}\hat{f}_{i+1/2}^{vis}- \hat{f}_{i-1/2}^{vis}\Big{)} \tag{3}\]  
where \(\overline{u}_{i}(t)\) represents the numerical approximation of \(\overline{u}(x_{i},t)\). The viscous numerical flux \(\hat{f}_{i\pm 1/2}^{vis}\) is computed using a sixth-order scheme [32]. The inviscid numerical flux \(\hat{f}_{i+1/2}^{inv}\) can be obtained using the flow parameters \(u_{i+1/2}^{\pm}\) on either side of \(x_{i+1/2}\), i.e., \(\hat{f}_{i+1/2}^{inv}=\hat{f}^{inv}\Big{(}u_{i+1/2}^{-},u_{i+1/2}^{+}\Big{)}\). The value of \(u_{i+1/2}^{\pm}\) is reconstructed by the WENO scheme. If \(u_{i+1/2}^{\pm}=u(x_{i+1/2},t)+O(h^{\prime})\) can be reconstructed, then Equation (3) is the \(r\)-th order approximation to Equation (2). Taking \(u_{i+1/2}^{-}\) as an example, the reconstruction procedure of \(u_{i+1/2}^{-}\) using third-order and fifth-order WENO schemes (denoted as WENO-ZQ3 and WENO-ZQ5, respectively) is described in detail. \(u_{i+1/2}^{-}\) and \(u_{i+1/2}^{+}\) are reconstructed using WENO schemes with left- and right-biased stencils, respectively. Step 1. Select a larger stencil \(T_{1}=\left\{\,I_{i-\ell},\ldots,I_{i+\ell}\right\}\), with \(\ell=1\) for the WENO-ZQ3 (or \(\ell=2\) for the WENO-ZQ5). Then, a quartic polynomial \(p_{1}(x)\) can be constructed using the average values of the grid cell parameters within the stencil \(T_{1}\):  
\[\frac{1}{h}\int_{I_{i+j}}p_{1}(x)dx=\overline{u}_{i+j},j=-\ell,\ldots,\ell;\ \ \ell=1\quad\big{(}or\quad\ell=2\big{)} \tag{4}\]  
Step 2. Selection of the two smaller stencils, \(T_{2}=\left\{I_{i-1},I_{i}\right\}\) and \(T_{3}=\left\{I_{i},I_{i+1}\right\}\) is performed. Similarly, linear polynomials \(p_{2}(x)\) and \(p_{3}(x)\) are constructed using the cell averages within the stencils \(T_{2}\) and \(T_{3}\), respectively, by imposing the following conditions:  
\[\frac{1}{h}\int_{I_{i+j}}p_{2}(x)dx=\overline{u}_{i+j},j=-1,0 \tag{5}\]  
\[\frac{1}{h}\int_{I_{i+j}}p_{3}(x)dx=\overline{u}_{i+j},j=0,1 \tag{6}\]  
Step 3. Unlike the classical WENO-JS scheme, which requires computing optimal linear weights, the WENO-ZQ scheme adopts a construction approach similar to the CWENO scheme [19; 20]. It allows for the selection of arbitrary nonzero linear weights \(\gamma_{1}\) and arbitrary linear weights \(\gamma_{2}\) and \(\gamma_{3}\). Thus, \(p_{1}(x)\) can be rewritten as:  
\[p_{1}(x)=\gamma_{1}\bigg{(}\frac{1}{\gamma_{1}}p_{1}(x)-\frac{\gamma_{2}}{ \gamma_{1}}p_{2}(x)-\frac{\gamma_{3}}{\gamma_{1}}p_{3}(x)\bigg{)}+\gamma_{2}p_ {2}(x)+\gamma_{3}p_{3}(x) \tag{7}\]  
To ensure the numerical stability of spatial discretization, it is necessary for these linear weights to be positive numbers to satisfy the condition \(\gamma_{1}+\gamma_{2}+\gamma_{3}=1\), and this expression achieves third- or fifth-order accuracy. Step 4. Discontinuities may arise in the polynomial \(p_{\ell}(x)\) within the cell.