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

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