1. 中国空气动力研究与发展中心,绵阳,621000
2. 空天飞行空气动力科学与技术全国重点实验室,绵阳,621000
黎昊旻(1997—),男,湖北武汉人,博士研究生,研究方向:计算流体力学. E-mail:haomin.li@buaa.edu.cn
董维中*,博士,研究员,研究方向:计算流体力学. E-mail:dongwz1966@163.com
纸质出版:2026
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在数值求解双曲守恒律方程弱解时,基于多项式的高精度非线性加权格式与线性格式相比,其引入的非线性误差会对格式分辨率产生不可忽略的影响。为了降低这种影响,学者们一方面在加权模板集选取、光滑因子计算及其与非线性权函数关系等方面开展了系统性的研究工作;另一方面,也对基于非多项式的高精度非线性加权格式进行了探索性研究。本文归纳总结了三角函数、对数函数、径向基函数及双曲正切函数等非多项式的WENO格式研究成果,展示了它们的频谱分辨率和间断捕捉能力等方面的特性。三角函数和径向基函数的WENO格式在频谱分辨率上优于多项式WENO格式,双曲正切函数的WENO格式在捕捉间断时产生的数值耗散明显低于多项式的WENO格式。这些初步而积极的研究成果,反映了WENO格式研究的新动向,对推进高阶精度非线性加权格式研究具有重要指导意义。
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.
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