On the idea of mapped WENO scheme, properties of mapping methods are analyzed, uncertainties in mapping development are investigated, and new piecewise rational mappings are proposed. Based on our former understandings, i.e. the mapping at endpoints {0, 1} tending to identity mapping, a so-called C_n,m condition is summarized for function development. Uncertainties, i.e., whether the pattern at endpoints of mapping would make mapped scheme behave like WENO or ENO, whether piecewise implementation of mapping would entail numerical instability, and whether WENO3 could preserve the third-order at first-order critical points by mapping, are analyzed and clarified. A new piecewise rational mapping with sufficient regulation capability is developed afterwards, where the flatness of mapping around the linear weights and the profile at endpoint tending toward identity mapping can be coordinated explicitly and simultaneously. Hence, the increase of resolution and preservation of stability can be balanced. Especially, concrete mappings are determined for {WENO3, 5, 7}. Numerical examples are tested for the new mapped WENO, which regard preservation and convergence rate of accuracy, numerical stability including that in the long-time computation, resolution and robustness. For comparison, some recent mappings such as IM by [App. Math. Comput. 232, 2014:453–468], RM by [J. Sci. Comput. 67, 2016:540–580] and AIM by [J. Comput. Phys. 381, 2019:162–188] are tested; in addition, some recent WENO-Z type schemes are chosen as well. The results manifest that new schemes can preserve optimal orders at corresponding critical points, achieve numerical stability, and indicate overall comparative advantages regarding accuracy, resolution and robustness.

基于映射WENO方案的思想，分析了映射方法的性质，研究了映射开发中的不确定性，提出了新的分段有理映射。根据我们之前的理解，即端点 {0, 1} 处的映射倾向于恒等映射，即所谓的C _{n , m}为功能开发总结条件。不确定性，即映射端点处的模式是否会使映射方案表现得像 WENO 或 ENO，映射的分段实现是否会导致数值不稳定性，以及 WENO3 是否可以通过映射保持一阶临界点的三阶，是进行了分析和澄清。之后开发了一种新的具有足够调节能力的分段有理映射，其中线性权重周围的映射平坦度和趋向身份映射的端点的轮廓可以明确且同时地进行协调。因此，可以平衡分辨率的增加和稳定性的保持。特别地，为{WENO3, 5, 7}确定具体映射。为新映射的 WENO 测试了数值示例，包括精度的保持和收敛速度，包括长时间计算在内的数值稳定性，分辨率和鲁棒性。为了比较，一些最近的映射，例如 [App. 数学。计算。232, 2014:453–468]，RM [J. 科学。计算。67, 2016:540–580] 和 AIM [J. 计算。物理。381, 2019:162–188] 进行了测试；此外，还选择了一些最近的 WENO-Z 类型方案。结果表明，新方案可以在相应的关键点保持最优阶数，实现数值稳定性，并表明在精度、分辨率和鲁棒性方面的整体比较优势。计算。67, 2016:540–580] 和 AIM [J. 计算。物理。381, 2019:162–188] 进行了测试；此外，还选择了一些最近的 WENO-Z 类型方案。结果表明，新方案可以在相应的关键点保持最优阶数，实现数值稳定性，并表明在精度、分辨率和鲁棒性方面的整体比较优势。计算。67, 2016:540–580] 和 AIM [J. 计算。物理。381, 2019:162–188] 进行了测试；此外，还选择了一些最近的 WENO-Z 类型方案。结果表明，新方案可以在相应的关键点保持最优阶数，实现数值稳定性，并表明在精度、分辨率和鲁棒性方面的整体比较优势。