On Developing Piecewise Rational Mapping with Fine Regulation Capability for WENO Schemes

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 Cn,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.