# A Two-Step High-Order Compact Corrected WENO Scheme
## 1 Introduction  
With the advancement of computing, the demand for high-order numerical methods in numerical simulations has been steadily increasing. These methods serve as essential tools for simulating complex physical phenomena and are widely applied across various scientific and engineering fields. Over the past two decades, significant progress has been made in this domain, resulting in the development of various high-order numerical methods. Based on finite element principles, discontinuous Galerkin (DG) methods [1, 2, 3] have emerged as a flexible framework for high-order discretization. These methods divide the computational domain into discrete elements and use polynomial approximations within each element to effectively represent solution gradients and discontinuities. By introducing discontinuities at element interfaces and employing suitable numerical fluxes, DG methods excel in handling shocks, making them ideal for simulating compressible flows, turbulence modeling, and multi-phase flows. Spectral element (SE) methods [4] represent another class of high-order numerical techniques that achieve exceptional solution accuracy by utilizing the spectral precision of polynomial approximations within elements. By combining spectral accuracy with the flexibility of element-based discretization, SE methods excel in applications such as turbulence simulations, aerodynamics, and environmental fluid dynamics. Spectral volume methods and spectral difference methods also belong to the category of spectral-based high-order numerical techniques. Similarly, spectral volume methods (SVMs) [5, 6] combine spectral accuracy with control volume principles to precisely represent fluxes and gradients within computational cells. Other spectral difference methods (SDMs) [7, 8], on the other hand, integrate spectral approximations with finite-difference methods, achieving a balancebetween computational efficiency and accuracy. Both approaches are effective in simulating phenomena such as turbulent flows, shock waves, and combustion processes. Group velocity control methods [9] offer a unique approach, focusing on regulating the propagation characteristics of numerical methods to enhance stability and accuracy. By adjusting the group velocity within the discretization framework, these methods reduce dispersion and dissipation errors, thereby improving the fidelity of wave propagation simulations. These techniques are applicable in fields such as acoustics, electromagnetics, and solid mechanics. While the aforementioned methods excel in achieving extreme accuracy and handling discontinuity problems, simulating high-speed, complex flows requires numerical methods that not only attain high precision to capture fine structures in flow fields but are also robust enough to handle strong discontinuities, such as shock waves. Essentially, non-oscillatory (ENO) methods and their weighted variants (WENO) [10; 11; 12; 13; 14; 15], based on traditional finite difference, are still the practical way for most high-speed complex flow simulations. These high-order methods provide precise techniques for the discretization of partial differential equations, enabling accurate analysis of complex physical mechanisms, the capture of intricate wave propagation, and the handling of phenomena characterized by steep gradients and discontinuities. WENO methods, based on ENO principles, stand out for their stability and ability to suppress oscillations when handling discontinuities. By employing multiple stencils and smoothness-based weighted reconstructions, WENO methods achieve a balance between accuracy and stability, proving effective in scenarios involving shock waves and other discontinuities. Additionally, [PERSON] et al. [16; 17] proposed a fifth-order shock-capturing scheme with a two-stage boundary variation diminishing (BVD) algorithm with an improved approach combining upwind-biased interpolations to further enhance the capability of handling discontinuities, offering robust performance for high-speed flow simulations. [PERSON] et al.