The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not achieve the desired order of convergence in smooth monotone regions and at critical points. In this article, this drawback is achieved with the WENO-LOC smoothness indicators by constructing a WENO-Z type nonlinear weights which contains a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil, as a result, the proposed numerical scheme could decrease the dissipation near the discontinuous regions. The theoretical and numerical experiments to achieve the required order of convergence in smooth monotone regions, at critical points, the essentially non-oscillatory (ENO), the analysis of parameters involved in the nonlinear weights like $\epsilon$ and $p$ are studied. From this study, we conclude that the imposition of certain conditions on $\epsilon$ and $p$, the proposed scheme achieves the global order of accuracy in the presence of an arbitrary number of critical points. Numerical tests for scalar, one and two-dimensional system of Euler equations are presented to show the effective performance of the proposed numerical scheme.

中文翻译：

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双曲线守恒定律的简单平滑度指标 WENO-Z 方案

WENO-JS5方案的优势[J. Comput. 物理。1996] 关于 WENO-LOC 方案 [J. 计算。Phys.1994] 是 WENO-LOC 非线性权重在平滑单调区域和临界点未达到所需的收敛顺序。在本文中，通过构建包含新型全局平滑度指标的 WENO-Z 型非线性权重，使用 WENO-LOC 平滑度指标解决了这一缺点。这种新颖的平滑度指标测量全局模板中重建通量的导数，因此，所提出的数值方案可以减少不连续区域附近的耗散。理论和数值实验在平滑单调区域中达到所需的收敛顺序，在临界点，基本上非振荡 (ENO)，研究了 $\epsilon$ 和 $p$ 等非线性权重中涉及的参数的分析。从这项研究中，我们得出结论，在 $\epsilon$ 和 $p$ 上施加某些条件，所提出的方案在存在任意数量的临界点的情况下实现了全局精度顺序。提出了对欧拉方程的标量、一维和二维系统的数值测试，以显示所提出的数值方案的有效性能。