In this article, we propose a modified convex combination of the polynomial reconstructions of odd-order WENO schemes to maintain the central substencil prevalence over the lateral ones in all parts of the solution. New "centered" versions of the classical WENO-Z and its less dissipative counterpart, WENO-Z+, are defined through very simple modifications of the classical nonlinear weights and show significantly superior numerical properties; for instance, a well-known dispersion error for long-term runs is fixed, along with decreased dissipation and better shock-capturing abilities. Moreover, the proposed centered version of WENO-Z+ has no ad-hoc parameters and no dependence on the powers of the grid size. All the new schemes are thoroughly analyzed concerning convergence at critical points, adding to the discussion on the relevance of such convergence to the numerical simulation of typical hyperbolic conservation laws problems. Nonlinear spectral analysis confirms the enhancement achieved by the new schemes over the standard ones.

翻译：本文提出了改进的奇数阶WENO格式多项式重构的凸组合方法，以在所有解区域内保持中心子模板相对于侧边子模板的主导地位。通过经典非线性权重的简单修改，定义了经典WENO-Z及其低耗散变体WENO-Z+的新“中心化”版本，这些版本展现出显著更优的数值特性：例如，修正了长期计算中著名的色散误差，同时降低了耗散并增强了激波捕捉能力。此外，所提出的WENO-Z+中心化版本无需引入特定参数且不依赖于网格尺寸的幂次。所有新格式在关键点处的收敛性均得到严格分析，补充了此类收敛性对典型双曲守恒律数值模拟重要性的讨论。非线性谱分析证实了新格式相较标准格式的改进效果。