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The user-defined sensitivity parameter responsible for avoiding zero division in the WENO nonlinear weights had plagued the schemes' performance in resolving smooth function with high-order critical points (CP-property) and capturing discontinuity essentially non-oscillatory (ENO-property). In this talk, a novel and simple yet effective WENO weights (Ai-weights) is devised for the (affine-invariant) Ai-WENO operator to handle the case when the function being reconstructed undergoes an affine transformation (Ai-operator) with a constant scaling and translation (Ai-coefficients) within a global stencil. The Ai-weights essentially decouple the inter-dependencies of the Ai-coefficients and sensitivity parameter effectively. For any given sensitivity parameter, the Ai-WENO operator guarantees that the WENO operator and the affine-transformation operator are commutable, as proven theoretically and validated numerically. In the presence of discontinuities, the high-order characteristic-wise (Ai-)(A-) WENO-Z finite difference scheme satisfies the ENO-property even when the classical WENO-JS and WENO-Z schemes might not. Examples in the shallow water wave equations, the Euler equations under gravitational fields solved by the characteristic-wise Ai-WENO scheme, are intrinsically well-balanced (WB-property). The two-medium γ-based model of the stiffened gas is also solved by the Ai-WENO operator, which preserves the equilibriums of velocity and pressure around the medium interface. A hybrid flux-based bound- and positivity-preserving (BP-P) limiter is also implemented to enforce the physical constraints. The theoretical analysis yields the exact CFL condition, which depends nonlinearly on the local Mach number. A variety of one-, two-, and three-dimensional benchmark two-medium shock-tube problems illustrate the high-order accuracy and enhanced robustness. In summary, any Ai-weights-based WENO reconstruction/interpolation operator enhances the robustness and reliability of the WENO scheme for solving hyperbolic conservation laws. |