三阶WENO格式精度分析与改进

高分辨率激波捕捉格式对含激波流场的数值模拟具有重要意义。通过理论推导分析了WENO-JS3格式和WENO-Z3格式的精度，发现两种格式在光滑流场区域（包含极值点处）具有相同的理论精度且均低于三阶设计精度，WENO-Z3格式由于增大了非光滑模板的非线性权重使其计算精度有所提高。在理论推导的基础上，提出了WENO-Z+3格式及其改进格式（WENO-Z+3P1和WENO-Z+3P32），且改进格式在光滑流场区域能满足所设计的三阶精度要求。选用一维平面黎曼问题及双马赫反射等经典算例，验证了本文提出的WENO-Z+3格式及其改进格式相较其他格式具有耗散低和对流场结构分辨率高的特性。

Precision analysis and improvement of third-order WENO scheme

High resolution shock capturing schemes are of great significance for numerical simulation of flow fields containing shock waves.Precision of both WENO-JS3 scheme and WENO-Z3 scheme are firstly analyzed theoretically.It is indicated that the two schemes share the same accuracy in the smooth flow field including critical points and actually fail to achieve designed third-order convergence.However,WENO-Z3 scheme is better than WENO-JS3 scheme because of larger weight assignment to the discontinuous stencils.WENO-Z+3 scheme and its improved high-resolution schemes (WENO-Z+3P1,WENO-Z+3P32) are then proposed in present paper.And it is proved that the improved schemes can recover the designed-order convergence at critical points as well as other smooth flow fields theoretically.Several classical one-dimensional Riemann problems and double Mach reflection case are simulated based on the schemes discussed in this paper.The proposed WENO-Z+3 scheme and its improved schemes are proved to have better characteristics with low numerical dissipation and high resolution to the flow features.