An Improved Third-Order WENO-Z Scheme for Achieving Optimal Order near Critical Points and Its Application
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摘要: 高精度、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。为了提高三阶WENO-Z格式在极值点处的计算精度，首先通过理论推导给出三阶WENO格式满足收敛精度的充分条件。采用泰勒级数展开的方式，推导确定所构造格式的参数。通过精度测试证明改进格式在光滑流场区域能收敛到三阶精度。选用Sod激波管、Rayleigh-Taylor不稳定性等经典算例证实了提出的改进格式WENO-NN3相较其他格式（WENO-SJ3、WENO-Z3和WENO-N3）具有精度高、耗散低、对流场结构分辨率高的特性。Abstract: A high-precision and resolution shock capturing scheme is of great significance for numerical simulation of the complex flow field containing shock waves.In this study, to improve the convergence accuracy of the conventional third-order WENO-Z scheme at the critical points, we firstly derived the sufficient conditions for satisfying the convergence precision of the third-order WENO scheme from the theoretical derivation, then determined the parameters of the constructed scheme using the Taylor series expansion for satisfying the sufficient conditions, and proved using the accuracy test that the proposed scheme converges to the third order precision in smooth flow field including the critical points.Furthermore, we selected the Sod shock tube, the Rayleigh-Taylor instability and some other classic examples, verifying that the improved scheme WENO-NN3 was capable of giving more precision and high resolution results of the complex flow field structures compared with other WENO schemes such as the WENO-JS3, WENO-Z3, and WENO-N3.
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图 1 Sod激波管计算结束后密度曲线及其局部放大图
Figure 1. Density curve and partially enlarged detail at the final time for the Sod problem
图 2 激波与熵波相互作用计算结束后密度曲线及其局部放大图
Figure 2. Density curve and partially enlarged detail at the final time for the Shu-Osher problem
图 3 Rayleigh-Taylor不稳定性问题不同格式(WENO-JS3、WENO-Z3、WENO-N3、WENO-NN3)密度曲线图
Figure 3. Density contours of the Rayleigh-Taylor instability computed using WENO-JS3, WENO-Z3, WENO-N3, and WENO-NN3 schemes
图 4 三模Richtmyer-Meshkov不稳定性初始条件设置
Figure 4. Initial condition of the treble-mode Richtmyer-Meshkov instability
图 5 三模Richtmyer-Meshkov不稳定性问题不同格式(WENO-JS3、WENO-Z3、WENO-N3、WENO-NN3)密度曲线图
Figure 5. Density contours of the treble-mode Richtmyer-Meshkov instability computed using WENO-JS3, WENO-Z3, WENO-N3, and WENO-NN3 schemes
表 1 针对初始条件(37)式在计算时间t=2时不同数值计算格式L1误差和精度比较
Table 1. A comparative study of L1 (error and order) for different schemes with initial condition Eq.(37) at t=2
N WENO-JS3 WENO-Z3 WENO-N3 WENO-NN3 L1 (Error) L1 (Order) L1 (Error) L1 (Order) L1 (Error) L1 (Order) L1 (Error) L1 (Order) 25 1.242 9×10-1 5.727 6×10-2 4.825 4×10-2 2.595 4×10-2 50 4.605 0×10-2 1.432 4 1.511 6×10-2 1.921 9 1.131 5×10-2 2.092 4 3.796 1×10-3 2.773 4 100 1.301 0×10-2 1.823 6 3.464 5×10-3 2.125 4 2.554 9×10-3 2.146 9 4.660 5×10-4 3.026 0 200 1.513 2×10-3 3.103 9 8.372 1×10-4 2.049 0 4.980 9×10-4 2.358 8 4.865 7×10-5 3.259 8 400 1.461 8×10-4 3.371 8 1.682 3×10-4 2.315 2 1.006 7×10-4 2.306 8 6.294 1×10-6 2.950 6 
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