A Fifth-Order Global Smoothness Indicator for Finite Difference WENO Schemes with the UOHO Property

This work investigates the accuracy and stencil-discrimination consequences of a multiplicative global smoothness indicator (GSI) for fifth-order finite-difference WENO reconstruction, formed symmetrically from all three Jiang–Shu local indicators. Requiring the fifth-order scaling associated with WENO5-Z at regular smooth points uniquely determines the exponent 5/6 within this symmetric product family. With standard Z-type weights, the GSI yields WENO5-ZU, which satisfies the unconditionally optimal high-order (UOHO) property at critical points of arbitrary finite order but lacks algebraic troubled-substencil weight decay in the analysed nondegenerate one-troubled-substencil configuration. A modified weighting gives M-WENO5-ZU, which preserves UOHO accuracy and restores this decay under the stated nondegeneracy condition. The analysis also identifies a loss of stencil discrimination in mixed exact-zero configurations. Numerical experiments confirm fifth-order convergence in smooth advection and algebraic critical-point tests, while Euler benchmarks assess resolution and robustness. M-WENO5-ZU improves troubled-substencil discrimination relative to WENO5-ZU, whereas its performance relative to WENO5-Z remains problem-dependent.