# A Two-Step High-Order Compact Corrected WENO Scheme
## 2 Numerical Methods
### Compact Corrected WENO (CCWENO)  
If we construct the corresponding compact format on each substencil of the WENO format, two third-order formulas and one fourth-order formula will be obtained as follows:  
\[ \begin{split} E_{0}:&\ 2H_{j-\frac{1}{2}}^{ \prime}+H_{j+\frac{1}{2}}^{\prime}=\frac{1}{2h_{j}}\Big{(}-H_{j-\frac{3}{2}}-4 H_{j-\frac{1}{2}}+5H_{j+\frac{1}{2}}\Big{)}\\ E_{1}:&\ H_{j-\frac{1}{2}}^{\prime}+4H_{j+\frac{1}{2}}^{ \prime}+H_{j+\frac{3}{2}}^{\prime}=\frac{3}{h_{j}}\Big{(}H_{j+\frac{3}{2}}-H_{ j-\frac{1}{2}}\Big{)}\\ E_{2}:&\ H_{j+\frac{1}{2}}^{\prime}+2H_{j+\frac{3}{2}}^{ \prime}=\frac{1}{2h_{j}}\Big{(}-5H_{j+\frac{1}{2}}+4H_{j+\frac{3}{2}}+H_{j+ \frac{5}{2}}\Big{)}.\end{split}  \tag{10}\]  
If a scheme similar to WENO is used to merge these three low-order approximations (the weights need to be recalculated), a sixth-order accurate WCS [27] can be obtained. However, WCS encounters instability in actual use and cannot be used to capture strong discontinuity problems. In addition, WCS cannot solve the derivative value at a single point alone; rather, it needs to solve a large set of linear equations to solve the values at all points at the same time, which is a problem for controlling the propagation of boundary errors and numerical stability. In order to solve the instability problem of WCS, the modified weighted compact format (MWCS) [31] combines the results of WENO and WCS with a certain weight, but there is still an instability problem when dealing with sharp shock waves. In addition, since the pentadiagonal linear system needs to be solved additionally, this improvement also doubles the computational workload. Inspired by the above two methods, we proposed a new method, which uses the compact scheme to correct the WENO flux to improve the accuracy of numerical simulation. The basic idea of this method is to enhance the weight of the central template \(E_{1}\) under certain smooth flow conditions. In this study, we further investigate this improvement and propose an alternative correction method that enhances all templates in the WENO scheme. We compare the numerical solutions of these two new methods based on flux correction using compact schemes, perform error analysis on both, compare our results with the WENO results with flux-characteristic fluxes, and propose a two-step high-order correction method applicable to more general scenarios. The compact correction method proposed in this study uses the compact schemes in Equation (10) as a correction step. By solving the flux function at \(j+\frac{1}{2}\) only, the flux functions at other points are moved to the right-hand side as follows:  
\[ \begin{split} E_{0}:&\ \hat{f}_{j+\frac{1}{2}}=\frac{1}{2h_{j}} \Big{(}-H_{j-\frac{3}{2}}-4H_{j-\frac{1}{2}}+5H_{j+\frac{1}{2}}\Big{)}-2\hat{f }_{j-\frac{1}{2}}\\ E_{1}:&\ \hat{f}_{j+\frac{1}{2}}=\frac{3}{4h_{j}} \Big{(}H_{j+\frac{3}{2}}-H_{j-\frac{1}{2}}\Big{)}-\frac{1}{4}\Big{(}\hat{f}_{j -\frac{1}{2}}+\hat{f}_{j+\frac{3}{2}}\Big{)}\\ E_{2}:&\ \hat{f}_{j+\frac{1}{2}}=\frac{1}{2h_{j}} \Big{(}-5H_{j+\frac{1}{2}}+4H_{j+\frac{3}{2}}+H_{j+\frac{5}{2}}\Big{)}-2\hat{f }_{j+\frac{3}{2}}.\end{split}  \tag{11}\]  
The primitive functions on the right-hand side of Equation (11) can be calculated by function values at the grid points, and the flux functions can be obtained by WENO. Thus, the compact correction algorithm can be succinctly summarized as follows:\[E_{0}:\;\;\hat{f}^{cc}_{j+\frac{1}{2}}=\frac{1}{2\mu_{j}}\big{(}f_{j-1}+5f_{j} \big{)}-2\hat{f}^{WENO}_{j-\frac{1}{2}} \tag{12}\] \[E_{-}1:\;\;\hat{f}^{cc}_{j+\frac{1}{2}}=\frac{3}{4\mu_{j}}\big{(}3f _{j}+f_{j+1}\big{)}-\frac{1}{4}\bigg{(}f^{WENO}_{j-\frac{1}{2}}+f^{WENO}_{j+ \frac{1}{2}}\bigg{)}\] \[E_{2}:\;\;\hat{f}^{cc}_{j+\frac{1}{2}}=\frac{1}{2\mu_{j}}\big{(}5 f_{j+1}+f_{j+2}\big{)}-2\hat{f}^{WENO}_{j+\frac{1}{2}},\]  
where \(\hat{f}^{WENO}\) denotes the flux computed using the original WENO scheme, and \(\hat{f}^{cc}\) represents the high-order corrected flux.