Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical finite difference WENO (FD-WENO) method (Shu and Osher, J. Comput. Phys., 83 (1989) 32-78) relies two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative finite difference WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing AFD-WENO up to ninth order. The third reason is that AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution ...

翻译：守恒律的高阶有限差分加权本质无振荡（WENO）格式因其在多维问题中以远低于有限体积WENO或DG格式的成本实现高阶精度而广受欢迎。此类格式存在两种构造方式。经典的有限差分WENO（FD-WENO）方法（Shu and Osher, J. Comput. Phys., 83 (1989) 32-78）通过对两个分裂通量应用两步重构步骤实现。然而，该方法无法兼容不同类型的黎曼求解器，也无法在曲线网格上保持自由来流边界条件，这限制了其实用性。交替有限差分WENO（AFD-WENO）方法能够克服这些缺陷，但针对此方法的研究工作相对较少。其原因有三：首先，普通读者难以理解需要在网格边界处评估通量高阶导数的复杂逻辑，推导AFD-WENO格式更新方程的解析方法较为深奥。为克服这一困难，我们在本文附录A中提供了一个基于计算机代数系统的易用脚本。其次，该方法依赖插值而非重构，而文献中对WENO插值公式的记载远不如WENO重构公式详尽。本文明确给出了实现九阶精度AFD-WENO所需的所有必要WENO插值公式。第三个原因在于，AFD-WENO要求网格边界处存在通量高阶导数。由于这些导数通常通过网格中心通量的有限差分获得，当解出现不连续性时，它们易受吉布斯现象影响……