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

Our past work has shown that when the critical points occur within grid intervals, the relations of accuracy of the smoothness indicators of weighted essentially non-oscillatory schemes (WENO) reported by Jiang and Shu differ from those that are obtained by assuming that the critical points occur on the grid nodes. The global smoothness indicator in the WENO-Z scheme might accordingly differ from the original one. We use this understanding to first discuss several issues regarding current improvements to third-order WENO-Z (e.g., WENO-NP3,-F3,-NN3, and-PZ3), i.e., numerical results with scale dependence, the validity of the analysis by assuming that the critical points occur on the nodes, and sensitivity in terms of the computational time step and initial conditions through the examination of the order of convergence. Numerical simulations and analyses were used to highlight defects in these improvements that occur either due to the scale dependence of the results, or the failure to recover the optimal order when the critical points occur on the half-nodes. Following this, a generic analysis that assumes that the first-order critical points occur within the grid intervals is provided. The theoretical results thus derived are used to propose two scale-independent third-order WENO-Z schemes that can be used to attain the optimal order at the critical points. The first scheme is obtained by extending a downstream smoothness indicator to derive a new global smoothness indicator and incorporating it into the mapping function. The second scheme is achieved by extending another smoothness indicator and using a different global indicator. The following val-idations are chosen and tested: the typical 1D problem of scalar advections, and 1D and 2D problems based on Euler's equations. The results verify the capability of the proposed schemes to recover the optimal order at the critical points. Moreover, the first of the above two proposed schemes outperforms the improved third-order WENO-Z scheme in terms of numerical resolution and robustness, which is usually favored by applications.