A Framework Unifying Three‐Cell‐Based Scale‐Invariant Exponential and Trigonometric WENO Weighting Functions With Optimal Shape Parameters

ABSTRACT Exponential and trigonometric functions have been extensively employed as the kernel of reconstruction operators within numerous WENO (Weighted Essentially Non‐oscillatory) schemes to accelerate the convergence rate. However, most of them are scale‐dependent, compromising the robustness required for multi‐scale flow simulations. Thus, this work aims to develop novel three‐cell‐based scale‐invariant WENO schemes that use exponential and trigonometric functions as the kernel of non‐linear weights. First, to achieve the scale‐invariant property, this work reformulates the newly proposed scale‐invariant ROUND (Reconstruction Operators in Unified Normalized‐variable Diagram) schemes into the form of WENO weighting functions, thereby facilitating the design of scale‐invariant WENO schemes. Then, this work proposes new WENO non‐linear weights using exponential and trigonometric functions—such as Gaussian, hyperbolic, and cosine functions—to enhance the accuracy of the three‐cell‐based WENO scheme. The proposed WENO weights contain a shape parameter that controls the errors between the non‐linear weight and the ideal weight. As the value of the shape parameter increases, the non‐linear weight converges towards the ideal weight but also becomes more likely to produce numerical oscillations. To approximate the optimal value of the shape parameter, the WENO reconstruction operator is projected into normalized variable space, and the shape parameters are fine‐tuned to ensure the normalized reconstruction operator falls into the CBC (Convection Bounded Criterion) region of UND (Unified Normalized‐variable Diagram). The accuracy analysis reveals that the proposed weighting functions outperform classical WENO schemes, particularly when the smooth function contains a first‐order critical point. The accuracy and shock‐capturing properties of the proposed schemes are further validated through benchmark tests. Thus, this work demonstrates using the ROUND framework to design scale‐invariant three‐cell‐based WENO schemes with exponential and trigonometric functions and optimal shape parameters.