## 4 Conclusions  
In this paper, a new high-precision and low-dissipation numerical method, WENO-URLBFS, has been proposed based on the finite difference framework. In order to enhance the accuracy and reduce the numerical dissipation, the URLBFS is introduced, which uses the theoretical solution of the Euler equation and adopts the upwind direction of the normal velocity at the cell interface to redefine the tangential velocity. Different from the traditional FD-WENO scheme that directly reconstructs the numerical flux on the interface, this paper adopts two high-order methods, WENO5 and WENO7, to reconstruct the flow variables on both sides of the interface and then brings the reconstructed flow variables into the URLBFS scheme to calculate the numerical flux. To avoid unphysical oscillations around the shock waves that would lead to divergent results, the eigenvector of the Jacobian matrix is introduced to transform the reconstruction process into the characteristic space, which finally produces a stable, high-precision, and high-resolution numerical scheme.  
The order of accuracy of the present WENO-URLBFS scheme has been successfully validated by simulating the density perturbation advection example. Both WENO5-URLBFS and WENO7-URLBFS have reached the fifth and seventh-order accuracy, respectively. In addition, many challenging benchmark examples are used to verify the scheme's resolution and shock wave capture capabilities. The one-dimensional examples include shock tube problems, Shu-Osher problems, and blast wave problems. The two-dimensional examples include double Mach reflections, several configurations of two-dimensional Riemann problems, KH instability problems, and High Mach number astrophysical jets.  
The numerical results of many challenging one-dimensional and two-dimensional examples mentioned above show that the WENO-URLBFS scheme has very low numerical dissipation and is robust when capturing small-scale flow structures and resolving strong shock waves and contact waves. This indicates that the WENO-URLBFS scheme is advantageous in solving complex flow problems with strong shock waves and contact discontinuities in compressible flows, which may promote its wide applications in engineering and scientific research. It is worth noting that this paper only focuses on inviscid compressible flows, which can be further expanded to the simulation of incompressible and viscous flows in the future. In terms of numerical methods, this paper uses the most basic WENO-JS scheme. In the future, we can try to make various improvements to the WENO scheme, such as introducing weighted restrictive correction, convection restrictive enhancement, etc., to improve the calculation accuracy and stability. These improvements will provide stronger numerical solution capabilities for more complex flow problems and further broaden this method's application scope.  
Conceptualization, Y.Z.; Methodology, [PERSON] ([PERSON]), J.C., [PERSON] ([PERSON]) and Y.Z.; Validation, Y.W. ([PERSON]), Y.W. ([PERSON]) and S.K.; Formal analysis, J.C., Y.W. ([PERSON]) and S.K.; Investigation, J.C.; Resources, S.K.; Writing--original draft, [PERSON] ([PERSON]): Writing--review & editing, [PERSON] ([PERSON]); Supervision, [PERSON] ([PERSON]); Funding acquisition, Y.Z. and S.K. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China (Grant No. 12272178 and 52321165649), Aeronautical Science Foundation of China (Grant No. 20220012052004) and the Spring Sunshine Program of the Ministry of Education of China and the Funding for Outstanding Doctoral Dissertation in NUAA (Grant No. BCXJ24-01). This work is also partially supported by High Performance Computing Platform of Nanjing University of Aeronautics and Astro-nautics.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.