Not yet reviewed by Pith; the record is open.

To address the order degradation at critical points in the WENO3-Z scheme, some improvements have been proposed , but these approaches generally fail to consider the occurrence of critical points at arbitrary positions within grid intervals, resulting in their inability to maintain third-order accuracy when a first-order critical point (CP1) occurs. Also, most previous improved schemes suffer from a relatively large exponent p of the ratio of global to local smoothness indicators, which adversely affects the numerical resolution. Concerning these limitations, introduced here is an accuracy-optimization lemma demonstrating that the accuracy of nonlinear weights can be enhanced providing that smoothness indicators satisfy specific conditions, thereby establishing a methodology for elevating the accuracy of nonlinear weights. Leveraging this lemma, a local smoothness indicator is constructed with error terms achieving second-order in smooth regions and fourth-order at CP1, alongside a global smoothness indicator yielding fourth-order accuracy in smooth regions and fifth-order at CP1, enabling the derivation of new nonlinear weights that meet accuracy requirements even when employing p=1. Furthermore, a resolution-optimization lemma is proposed to analyze the relationship between parameters in local smoothness indicators and resolution. By integrating theoretical analysis with numerical practices, free parameters in non-normalized weights and local smoothness indicators are determined under the balance of numerical resolution and robustness, which leads to the development of WENO3-ZES4, a new WENO3-Z improvement that preserves the optimal order at CP1 especially with p=1. 1D and 2D validating tests show that the new scheme consistently achieves third-order in the case of CP1 regardless of its position and exhibits good resolution as well as preferable robustness.