Abstract In this work, the characteristic-wise alternative formulation of the seventh and ninth orders conservative weighted essentially non-oscillatory (AWENO) finite difference schemes is derived. The polynomial reconstruction procedure is applied to the conservative variables rather than the flux function of the classical WENO scheme. The numerical flux contains a low order term and high order derivative terms. The low order term can use arbitrary monotone fluxes that can enhance the resolution and reduce numerical dissipation of the fine scale structures while capturing shocks essentially non-oscillatory. The high order derivative terms are approximated by the central finite difference schemes. The improved performance in terms of accuracy, essentially non-oscillatory shock capturing and resolution for the complex shocked flow with fine scale structures in the classical one- and two-dimensional problems is demonstrated. However, the inclusion of the high order derivative terms is prone to generate Gibbs oscillations around a strong discontinuity and might result in a negative density and/or pressure. Therefore, a positivity-preserving limiter [Hu et al. J. Comput. Phys. 242 (2013)] is adopted to ensure the positive density and pressure in the shocked flows with extreme conditions, such as Mach 2000 jet flow problem.

中文翻译：

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双曲守恒律的七阶和九阶特征替代WENO有限差分格式

摘要 在这项工作中，推导出了七阶和九阶保守加权基本非振荡（AWENO）有限差分格式的特征替代公式。多项式重建程序应用于保守变量而不是经典 WENO 方案的通量函数。数值通量包含低阶项和高阶导数项。低阶项可以使用任意单调通量，这可以提高分辨率并减少精细尺度结构的数值耗散，同时捕获基本上非振荡的冲击。高阶导数项由中心有限差分格式近似。在准确性方面的改进性能，证明了经典一维和二维问题中具有精细尺度结构的复杂冲击流的基本非振荡冲击捕获和解析。然而，包含高阶导数项很容易在强不连续性周围产生吉布斯振荡，并可能导致负密度和/或压力。因此，保留正性的限制器 [Hu et al. J. 计算。物理。242 (2013)] 用于确保极端条件下冲击流中的正密度和压力，例如马赫 2000 射流问题。保留正性的限制器 [Hu et al. J. 计算。物理。242 (2013)] 用于确保极端条件下冲击流中的正密度和压力，例如马赫 2000 射流问题。保留正性的限制器 [Hu et al. J. 计算。物理。242 (2013)] 用于确保极端条件下冲击流中的正密度和压力，例如马赫 2000 射流问题。