# A Two-Step High-Order Compact Corrected WENO Scheme
## 3 Numerical Results  
In the following section, we analyze the three benchmark cases mentioned above using several numerical schemes, including the fifth-order WENO scheme, the MWCS, and the two proposed schemes, CCWENO and CCWENO2, introduced in this paper. All of these schemes utilize the same five-point stencil and three substencils, ensuring a consistent basis for comparison. Additionally, we include the numerical results of the WENO scheme incorporating characteristic-wise reconstruction, as discussed in [34]. By reconstructing the variables along characteristic directions, this approach significantly reduces spurious oscillations and numerical dissipation, which are commonly encountered in high-order schemes. However, this enhanced accuracy comes at the cost of a considerable increase in computational effort due to the added complexity of characteristic-wise operations. In contrast, the proposed CCWENO and CCWENO2 schemes achieve comparable accuracy without the need for characteristic-wise reconstruction. This innovative feature simplifies their implementation and reduces computational overhead while maintaining robust performance in capturing complex flow features. Figure 2 shows the numerical results for the Sod shock tube problem, a classic test case in computational fluid dynamics used to evaluate the ability of numerical schemes to resolve discontinuities and shock waves. Specifically, Figure 2b provides a magnified view of the shock discontinuity region, allowing for a closer inspection of the schemes' accuracy and resolution at the discontinuity. For all numerical schemes analyzed, a uniform grid size of 200 points was employed, alongside a consistent time step of \(dt=10^{-4}\), ensuring equitable comparisons across methods. To provide a benchmark reference for comparison, the WENO scheme was utilized with a significantly finer grid resolution of 1600 points, coupled with a reduced time step of \(dt=10^{-5}\), delivering enhanced spatial and temporal accuracy in the refined solution. The results indicate that the proposed CCWENO and CCWENO2 schemes exhibit highly comparable numerical behavior, particularly in their ability to accurately resolve the shock wave discontinuity. Both schemes demonstrate a notable advantage over other methods, achieving closer alignment to the true shock wave position while maintaining high-resolution detail. These findings highlight the strengths of compact corrected WENO schemes in applications involving sharp gradients and shock waves. Figure 3 presents the numerical results for the Shu-Osher problem, a benchmark test case designed to assess the accuracy and resolution of numerical schemes in capturing fine-scale structures and shock interactions. Figure 3b provides a magnified view of the region exhibiting significant fluctuations and the primary shock wave discontinuity, allowing for a closer examination of the schemes' performance in resolving intricate flow features. For all numerical schemes under consideration, we used the same grid size (200 grid points) and time accuracy (\(10^{-4}\)), ensuring consistency across the simulations. The standard WENO scheme with a much higher grid density of 1600 points and smaller a time step of \(dt=10^{-5}\) was used to obtain the refined solution. The results reveal that  
Figure 2: Numerical results of the Sod problem: (**a**) whole domain; (**b**) enlarged view. both the CCWENO and CCWENO2 schemes excel in capturing the fine vortex structures generated in this problem. Among the two, CCWENO2 achieves a closer match to the amplitude of the refined solution, demonstrating its capability to approximate the true solution with remarkable fidelity. However, it introduces a more pronounced phase error (dispersion) compared to the CCWENO scheme, which maintains greater alignment with the reference solution in terms of phase accuracy. Figure 4 illustrates the numerical results obtained for the challenging two interacting blast waves problem, which involves both complex wave systems and strong discontinuities.