Precision Analysis of the 3rd-Order WENO-Z Scheme and Its Improved Scheme
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摘要: 首先通过理论推导给出了三阶WENO格式(WENO-JS3格式)满足收敛精度的充分条件.采用Taylor(泰勒)级数展开的方法,分析发现传统的三阶WENO-Z格式(WENO-Z3格式)在光滑流场极值点处精度降低.为了提高WENO-Z3格式在极值点处的计算精度,根据收敛精度的充分条件构造一种改进的三阶WENO-Z格式(WENO-NZ3格式),并综合权衡计算精度和计算稳定性确定所构造格式的参数.通过两个典型的精度测试,验证了WENO-NZ3格式在光滑流场极值点区域逼近三阶精度.选用Sod激波管、激波与熵波相互作用、Rayleigh-Taylor不稳定性、二维Riemann(黎曼)问题经典算例,进一步证实了本文提出的WENO-NZ3格式相较其他格式(WENO-JS3、WENO-Z3、WENO-N3),不仅提高了计算精度,而且提高了对复杂流场结构的分辨率.Abstract: Firstly, the sufficient conditions for the 3rd-order WENO scheme satisfying the convergence precision were deduced. Based on the Taylor series method, the precision of the conventional 3rd-order WENO-Z scheme in the smooth flow field was analyzed. It was found that at the critical points, the 3rd-order WENO-Z scheme fails to achieve the convergence precision. In order to improve the precision near the critical points for the 3rd-order WENO-Z scheme, an improved 3rd-order WENO-Z scheme (WENO-NZ3) was constructed in view of the balance between precision and stability to finally determine the parameters. The improvement of the precision was verified through 2 typical numerical tests. What is more, the Sod shock wave tube, the shock-entropy wave interaction, the Rayleigh-Taylor instability and the 2D Riemann problem were calculated to confirm that the WENO-NZ3 scheme performs better than the conventional WENO schemes like WENO-JS3, WENO-Z3 and WENO-N3.
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