# A Two-Step High-Order Compact Corrected WENO Scheme
## 1 Introduction
[18] also proposed a new family of WENO schemes which satisfy the scale-invariant property. To tackle greater challenges in numerical simulations of complex high-speed fluid dynamics, combustion, and other intricate flow phenomena, researchers have further enhanced WENO by developing various variants. For instance, WENO-JS [14] introduced a novel smoothness indicator based on Lagrange interpolation polynomials. WENO-Z [19] refined this methodology by employing a global smoothness indicator derived from WENO-JS stencils, reducing numerical dissipation near shocks while preserving high-order accuracy. [PERSON] et al. [20] extended the framework of smoothness indicators, enabling WENO-Z to achieve arbitrary odd-order precision. [PERSON] et al. [21] proposed a new smoothness indicator to develop a fifth-order WENO-Z method tailored specifically for first-order critical points. [PERSON] et al. [22] presented WENO-Z+, which improves numerical resolution by optimizing less smooth substencils. [PERSON] et al. [23] identified limitations in the accuracy of fifth-order WENO-Z methods at high-order critical points and proposed WENO-D--a fifth-order variant that incorporates a corrective function to enhance convergence accuracy relative to the weight functions used in WENO-Z. The Central WENO (CWENO) method [24; 25] prioritizes the central stencil as the most stable option under smooth flow conditions, consistently showcasing its reliability. For non-smooth flows, smoothness indicators guide the reconstruction process to select the optimal one-sided stencil. Additionally, the adaptive order WENO (WENO-AO) method [26] integrates a fourth-degree polynomial and three quadratic polynomials into a new adaptive fifth-order approach, utilizing Legendre polynomials to calculate smoothness indicators. However, most of these improvements are based on extra stencils, leading to an increase in calculation. Meanwhile, using more points to calculate the derivatives near the strong discontinuities may also lead to inaccurate results. The Weighted Compact Scheme (WCS), proposed by [PERSON], [PERSON], and [PERSON] [27], integrates the principles of WENO and compact schemes [28; 29; 30]. By employing Hermite polynomials instead of Lagrange polynomials within each stencil, WCS achieves enhanced accuracy. In shock regions, WCS effectively adjusts stencil contributions to limit the influence of those containing discontinuities. However, the original formulation of WCS faces instability issues when addressing strong discontinuities and fails to sufficiently resolve global dependency problems associated with derivatives in compact schemes. To address these challenges, [PERSON] et al. [31] introduced the Modified Weighted Compact Scheme (MWCS). This method combines results from WENO and WCS, assigning appropriate weights to both. The MWCS eliminates oscillations while enabling local computation of weight coefficients for pressure and density at each time step. However, compared to traditional WENO methods, the MWCS incurs additional computational costs due to it solving a five-diagonal matrix system, effectively doubling the workload. Furthermore, stability issues arising from compact schemes remain unresolved. In addition, this leads to the rapid propagation of boundary errors throughout the flow field. In our previous study [32], we modified and improved the linear scheme used in the central stencil in WENO by incorporating a high-order compact scheme inspired by the MWCS, enhancing computational accuracy with the same points as used in the fifth-order WENO scheme. In this study, we further investigate this improvement and propose another correction method which enhances all stencils in the WENO scheme. We compare the numerical solutions obtained from these two new schemes, perform error analysis for both, compare the results with the results from WENO with characteristic flux, and propose a two-step high-order correction method applicable to more general scenarios.