Review on non-polynomial high-order accuracy nonlinear weighted schemes

In comparison to linear schemes, the nonlinear errors inherent in polynomial-based high-order nonlinear weighted schemes significantly affect the resolution when solving weak solutions of hyperbolic conservation laws. To mitigate this impact, researchers have systematically investigated aspects such as weighted stencils, smoothness indicators, and nonlinear weighting functions. Additionally, exploratory studies have been undertaken on high-order nonlinear weighted schemes based on non-polynomial approaches. This paper reviews the advancements in non-polynomial weighted essentially non-oscillatory (WENO) schemes, including trigonometric functions, logarithmic functions, radial basis functions (RBF), and hyperbolic tangent functions (THINC). Their performance is highlighted in terms of spectral resolution and shock-capturing capabilities. Trigonometric and RBF-based WENO schemes demonstrate superior spectral resolution compared to polynomial-based WENO schemes, while THINC-based WENO schemes exhibit significantly lower numerical dissipation when capturing discontinuities. These preliminary yet promising findings reflect a new trend in WENO scheme research and provide valuable guidance for advancing the study of high-order nonlinear weighted schemes.