# Application of a Novel High-Order WENO Scheme in LES Simulations
## 2 WENO-ZQ Scheme and Numerical Tests
### Description of WENO-ZQ Scheme
Therefore, the smoothness of reconstruction polynomials within the cell \(I_{i}\) is denoted by the smoothness coefficients \(\beta_{\ell}\):  
\[\beta_{\ell}=\sum_{\kappa=1}^{r}\int_{I_{i}}h^{2\kappa-1}\bigg{(}\frac{d^{ \kappa}p_{\ell}(x)}{dx^{\kappa}}\bigg{)}^{2}dx,\ell=1,2,3 \tag{8}\]  
where \(r=2\) (or \(r=4\)) for \(\ell=1\) and \(r=1\) for \(\ell=2,3\). Step 5. The nonlinear weights \(\omega_{\ell}\) are calculated using a combination of linear weights \(\gamma_{\ell}\) and smoothing coefficients \(\beta_{\ell}\). The adaptive formula of \(\tau\) can be defined in various ways [21; 25]; in this study, it is defined based on the absolute differences between \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\):  
\[\tau=\left(\frac{|\beta_{1}-\beta_{2}|+|\beta_{1}-\beta_{3}|}{2}\right)^{2} \tag{9}\]  
Notably, this definition differs from the formula outlined in [33; 34; 35] due to variations in the reconstruction stencils. Equation (10) defines the nonlinear weights, with a small positive value \(\epsilon=10^{-6}\) to avert division by zero in the denominators [22; 24; 25]:  
\[\omega_{\ell}=\frac{\overline{\omega}_{\ell}}{\sum_{\ell\ell=1}^{3}\overline{ \omega}_{\ell\ell}},\quad\overline{\omega}_{\ell}=\gamma_{\ell}\Big{(}1+ \tfrac{\tau}{\epsilon+\overline{\rho}_{\ell}}\Big{)},\qquad\ell=1,2,3 \tag{10}\]  
Step 6. Substitute the nonlinear weights \(\omega_{\ell}\) in Equation (10) for the linear weights \(\gamma_{\ell}\) in Equation (7), yielding the final reconstruction formula for the conserved variable \(u(x,t)\) at the point \(x_{i+1/2}\) within the target cell \(I_{i}\):  
\[u_{i+1/2}^{-}=\omega_{1}\bigg{(}\frac{1}{\gamma_{1}}p_{1}(x_{i+1/2})-\frac{ \gamma_{2}}{\gamma_{1}}p_{2}(x_{i+1/2})-\frac{\gamma_{3}}{\gamma_{1}}p_{3}(x_ {i+1/2})\bigg{)}+\omega_{2}p_{2}(x_{i+1/2})+\omega_{3}p_{3}(x_{i+1/2}) \tag{11}\]  
Step 7. The semi-discretization scheme presented in Equation (3) undergoes temporal discretization using a third-order TVD Runge-Kutta (R-K) method [36]:  
\[ \begin{cases}u_{i}^{(1)}&=\quad\quad\quad u_{i}^{n}+\Delta tL\big{(}u_{i}^{n} \big{)}\\ u_{i}^{(2)}&=\quad\tfrac{3}{4}u_{i}^{n}+\tfrac{1}{4}u_{i}^{(1)}+\tfrac{1}{4} \Delta tL\big{(}u_{i}^{(1)}\big{)}\\ u_{i}^{n+1}&=\quad\tfrac{1}{3}u_{i}^{n}+\tfrac{2}{3}u_{i}^{(2)}+\tfrac{2}{3} \Delta tL\big{(}u_{i}^{(2)}\big{)}\end{cases}  \tag{12}\]  
The WENO-ZQ schemes are readily adaptable to two-dimensional or three-dimensional scenarios through a dimension-by-dimension strategy. Unlike the classical WENO-JS schemes [37], the finite-volume WENO-ZQ schemes can employ identical linear weights at various quadrature points along the boundaries of the target cell. This approach eliminates the need for real-time calculation of linear weights and avoids concerns regarding negative linear weights. Consequently, this enhancement bolsters the computational process's stability and diminishes the computational overheads associated with the reconstruction procedure [21; 22; 23; 24]. Since the linear weights no longer depend on the mesh topology, this scheme is convenient for applications with a non-uniform and adaptive mesh [24; 38]. Furthermore, it can be observed that constructing the third-order or fifth-order accuracy WENO-ZQ model requires the replacement of polynomials and smoothness coefficients of the large spatial stencil.