  On the idea of mapped WENO-JS scheme, properties of mapping methods are
analyzed, uncertainties in mapping development are investigated, and new
rational mappings are proposed. Based on our former understandings, i.e.
mapping at endpoints {0, 1} tending to identity mapping, an integrated Cm,n
condition is summarized for function development. Uncertainties, i.e., whether
the mapping at endpoints would make mapped scheme behave like WENO or ENO,
whether piecewise implementation would entail numerical instability, and
whether WENO3-JS 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 its endpoint convergence 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-JS. Numerical cases are tested
for the new mapped WENO-JS, which regards numerical stability including that in
long time computation, resolution and robustness. In purpose of 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 chosen; in addition, some recent WENO-Z type scheme are
selected also. Proposed new schemes can preserve optimal orders at
corresponding critical points, achieve numerical stability and indicate overall
comparative advantages regarding accuracy, resolution and robustness.
