  Although there are many improvements to WENO3-Z that target the achievement
of optimal order in the occurrence of the first-order critical point (CP1),
they mainly address resolution performance, while the robustness of schemes is
of less concern and lacks understanding accordingly. In light of our analysis
considering the occurrence of critical points within grid intervals, we
theoretically prove that it is impossible for a scale-independent scheme that
has the stencil of WENO3-Z to fulfill the above order achievement, and current
scale-dependent improvements barely fulfill the job when CP1 occurs at the
middle of the grid cell. In order to achieve scale-independent improvements, we
devise new smoothness indicators that increase the error order from 2 to 4 when
CP1 occurs and perform more stably. Meanwhile, we construct a new global
smoothness indicator that increases the error order from 4 to 5 similarly,
through which new nonlinear weights with regard to WENO3-Z are derived and new
scale-independents improvements, namely WENO-ZES2 and -ZES3, are acquired.
Through 1D scalar and Euler tests, as well as 2D computations, in comparison
with typical scale-dependent improvement, the following performances of the
proposed schemes are demonstrated: The schemes can achieve third-order accuracy
at CP1 no matter its location in the stencil, indicate high resolution in
resolving flow subtleties, and manifest strong robustness in hypersonic
simulations (e.g., the accomplishment of computations on hypersonic
half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well
as essentially non-oscillatory solutions of inviscid sharp double cone flow at
M=9.59), which contrasts the comparative WENO3-Z improvement.
