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As we found previously, when critical points occur within grid intervals, theaccuracy relations of smoothness indicators of WENO-JS would differ from thatassuming critical points occurring on grid nodes, and accordingly the globalsmoothness indicator in WENO-Z scheme will differ from the original one. Basedon above understandings, we first discuss several issues regarding currentthird-order WENO-Z improvements (e.g. WENO-NP3, -F3, -NN3, -PZ3 and -P+3), i.e.the numerical results with scale dependency, the validity of analysis assumingcritical points occurring on nodes, and the sensitivity regarding computationaltime step and initial condition in order convergence studies. By analyses andnumerical validations, the defections of present improvements are demonstrated,either scale-dependency of results or failure to recover optimal order whencritical points occurring at half nodes, then a generic analysis is providedwhich considers the first-order critical point occurring within grid intervals.Based on achieved theoretical outcomes, two scale-independent, third-orderWENO-Z schemes are proposed which could truly recover the optimal order atcritical points: the first one is acquired by limited expansion of gridstencil, deriving new global smoothness indicator and incorporating with themapping function; the second one is achieved by further expanding grid stenciland employing a different global smoothness indicator. For validating, typical1-D scalar advection problems, 1-D and 2-D problems by Euler equations arechosen and tested. The consequences verify the optimal order recovery atcritical points by proposed schemes and show that: the first scheme outperformsaforementioned third-order WENO-Z improvements in terms of numericalresolution, while the second scheme indicates weak numerical robustness inspite of improved resolution and is mainly of theoretical significance

Third-order scale-independent WENO-Z scheme achieving optimal order at critical points