# Application of a Novel High-Order WENO Scheme in LES Simulations
## 1 Introduction  
The accurate computation of turbulent separation [1; 2], transition [3; 4], and shockwave-turbulent boundary layer interactions (SWTBI) [5; 6] is pivotal for the design and development of aero-engines [7; 8]. The limitations of traditional Reynolds-averaged Navier-Stokes (RANS) approaches in accurately capturing these complex flow phenomena have been well-documented [9; 10]. Advanced high-fidelity numerical strategies, particularly large eddy simulation (LES), are instrumental in providing a detailed understanding of these flows and their transient characteristics [11; 12]. Therefore, to achieve high-fidelity LES predictions of such complex flows while maintaining manageable computational costs, it is essential to employ high-order numerical schemes that offer high resolution, low dissipation, and are suitable for compressible flows. High-order numerical schemes have been the subject of intense investigation over recent decades. [PERSON] [13] pioneered the notion of total variation diminishing (TVD) byanalyzing the causes of non-physical oscillations near shockwaves and devised a TVD finite-difference scheme that maintains second-order accuracy in smooth regions. To rectify the degradation of the TVD scheme at smooth extrema, [PERSON] and colleagues [14; 15] subsequently developed the high-precision, essentially non-oscillatory (ENO) scheme. The ENO scheme maintains non-oscillatory properties in discontinuous regions by selecting the \"smoothest\" template among all candidates for function reconstruction, thereby avoiding accuracy degradation in smooth regions. However, the ENO scheme relies on information from only one template, resulting in some loss of flow field information while increasing computational cost and storage requirements. Moreover, the template selection process in the ENO scheme involves substantial logical judgments, which impairs the parallelization of the algorithm [16] and decreases the computational efficiency of multidimensional ENO schemes. To overcome these limitations, [PERSON] et al. [17] proposed the weighted essentially non-oscillatory (WENO) scheme, which performs reconstruction using a weighted average of all templates. [PERSON] and [PERSON] [16] further established a framework for developing finite-difference WENO schemes with arbitrary accuracy orders and introduced the fifth-order WENO (WENO-JS) scheme. They provided the essential methodology for creating smoothness indicators and nonlinear weights. A significant innovation of the WENO-JS scheme is its method of combining multiple local reconstructions using different-sized spatial stencils to achieve the final spatial reconstruction. However, the classical WENO schemes still present significant challenges [18], including high computational costs and process complexity, as well as the dependence of optimal (linear) weights on mesh topology, which can lead to negative weights and compromise robustness. These issues are particularly pronounced in three-dimensional contexts. To enhance the effectiveness of WENO schemes, [PERSON] and colleagues investigated the issue of non-existent linear weights when performing third-order WENO reconstruction at the target cell center, as discussed in their paper [19]. They observed that WENO schemes using linear weights exhibit slightly varying accuracy depending on the grid resolution. In contrast, schemes utilizing nonlinear weights show reduced errors and more stable accuracy under sparse grid conditions. They proposed a technique called the central WENO (CWENO) scheme [19; 20] to deal with this difficulty. Based on the classical WENO scheme, [PERSON] and [PERSON] [21; 22] introduced the core concept of constructing non-equidistant spatial templates and convex combinations of different-order polynomials. They designed a simple and practical high-precision finite-difference and finite-volume WENO scheme with unequal stencil sizes, referred to as the WENO-ZQ scheme. Recently, the WENO-ZQ scheme has seen advancements and applications across various engineering fields. [PERSON] et al.