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.

Proper understanding and evaluation of the aerothermal environment with mass injection for high-speed aircraft are prerequisites to develop corresponding thermal protection technology. To address the prediction of aerothermal environment with mass injection, the numerical simulation method of mass injection is proposed and verified in high-temperature nonequilibrium flow. Subsequently, a modified heat flux characterization formula is theoretically derived considering comprehensive wall effects, including catalysis, ablation, pyrolysis and active injection. The new characterization method and heat reduction mechanisms are numerically investigated using a blunt wedge configuration in typical flight states. The results demonstrate the following: using the heat flux ultimately experienced by the vehicle structure as the criterion for wall heat flux assessment aligns with traditional wall heat flux formula while also providing a reasonable evaluation of heat flux with complex wall effects. Conventional heat flux formula needs to be corrected since it overestimates wall heat flux and underestimates cooling efficiency when evaluating the thermal reduction effect of mass injection. The corrected heat flux expression under mass injection conditions includes heat conduction from flow field, heat absorption/release from wall reactions, and formation enthalpy of injected media, degenerating into the traditional heat flux expression in non-ablative and non-injection conditions. For non-catalytic, non-ablative and active injection wall, heat flux consists solely of conductive heat flux. As a result, cooling effect of active injection is achieved by significantly reducing the normal temperature gradient at the wall. The cooling efficiency is further improved while incorporating the enthalpy of injected water vapor, but the dominant factor remains the reduction of conductive heat flux.