## 1 Introduction  
Computational fluid dynamics (CFD) uses advanced numerical methods to study fluid flow and related phenomena. Since the mid-20 th century, the advancement of digital computers has made CFD essential in fluid mechanics research and engineering. It is widely applied in fields such as aerospace, automotive, civil engineering, energy, environmental science, and biomedical engineering for simulating complex flows, optimizing designs, and enhancing system performance [1]. Compressible flow problems involve complex characteristics such as multi-scale structures, strong shock waves, and contact discontinuities, making them extremely difficult to solve. The key to solving this challenge is to resolve small-scale vortex structures with minimal dissipation and accurately capture strong shock waves and discontinuities, thereby promoting the development of higher-precision numerical schemes [2]. In order to reduce numerical dissipation and improve the resolution of flow field results, one of the most important methods is to develop high-order numerical methods [3; 4; 5; 6]. [PERSON] and [PERSON] [3] introduced the Piecewise Parabolic Method, which uses a total variation diminishing (TVD) limiter to avoid oscillation when capturing high-gradient fluid features, improve stability, and thus achieve third-order accuracy. [PERSON] and [PERSON] [4] proposed the discontinuous Galerkin method, which uses high-order polynomials to interpolate within each cell and can accurately capture complex flow phenomena, thus achieving high-order accuracy. However, this method has a high computational cost, complex implementation, large storage requirements, and may have stability and numerical dissipation problems. [PERSON] et al. [5] proposed a high-precision shock wave capture scheme, an essentially non-oscillatory scheme (ENO). This scheme selects the reconstruction polynomial by selecting multiple candidate templates and using the difference quotient to calculate the smoothness of the algebraic polynomial defined on the candidate template. This ensures the ENO scheme remains stable without oscillations at the shock wave, maintaining precision in smooth regions. [PERSON] and [PERSON] [6; 7] proposed a high-precision ENO scheme based on the finite difference method, utilizing TVD Runge-Kutta time discretization. While the ENO scheme demonstrates robust numerical stability and superior accuracy, the rounding error near the zero point may affect the template selection, and the template change at adjacent points may cause the numerical flux to be non-smooth. In addition, when selecting the template, to achieve design accuracy, a large number of discarded templates will be generated, but only one will be chosen in the end, which will lead to a waste of computing resources and affect computing efficiency. In contrast to the ENO method, which only identifies the smoothest candidate template, it achieves high-precision spatial numerical discretization by combining all candidate templates. [PERSON] [8] proposed a Weighted Essentially Non-Oscillatory (WENO) scheme based on finite volume, which reached the third order. [PERSON] and [PERSON] [9] developed a structure for establishing a WENO scheme with \(2k+1\) order numerical accuracy from the \(k+1\) ENO scheme. The WENO-JS scheme they built is a milestone in the history of high-precision WENO schemes. On account of the high-order precision, distinguishability, and strong robustness to discontinuous, the WENO scheme has made significant progress in the finite difference method (FD-WENO) [10; 11] and the finite volume method (FV-WENO) [12; 13; 14] in recent decades. Among them, the finite difference method makes constructing a high-precision and high-resolution numerical method easier than other methods. In addition to the high-order interpolation reconstruction method, the methodology for flux evaluation is also essential for obtaining high-precision and low-dissipation simulation results. The Godunov scheme is the earliest shock wave capture scheme.