Numerical computation of shock wave using wavelet methods
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摘要: 基于自适应小波配点法和人工黏性技术，构造出一种简单稳定的冲击波数值计算方法。采用小波阈值滤波，生成适应流场分布的多尺度自适应网格，并利用密度场最细尺度的小波系数构造幂函数形式的冲击波定位函数，用以判断冲击波位置。联合人工黏性与冲击波定位函数，自动根据流场梯度严格控制人工黏性的大小和分布。对强/弱冲击波管问题进行计算，结果表明，该方法能够准确捕捉冲击波和有效抑制数值振荡，并且使用简单、分辨率高、计算量小。Abstract: A simple and stable wavelet method, which is based on adaptive wavelet collocation methods and artificial viscosity techniques, was proposed to compute shock waves. Dynamic multiscale grids generated by wavelet threshold filtering adaptive to the flow field were used. The shock waves can be checked out by the shock locator functions with power formula, which are constructed through using the magnitudes of the wavelet coefficients on the finest level in the density fields. Then, the artificial viscous terms including viscosity and shock locator functions strictly control the magnitudes and distributions of the artificial viscosity according to the gradients in the flow field. A strong and a weak shock tubes were computed, which shows that the method can accurately capture shock fronts and effectively restrain numerical oscillations. By the way, it is easy to manipulate, high of resolution and small of computational costs.
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图 1 冲击波与对应的小波系数和定位函数
Figure 1. Shock and the corresponding wavelet coefficients and shock locator functions
图 6 自适应小波配点法相对迎风、ENO、WENO格式的计算时间
Figure 6. Relative computational time costs of AWCM to Up-wind, ENO and WENO schemes
图 7 J=12时三种格式密度分布对比
Figure 7. Spatial distributions of density through three schemes with J=12
表 1 弱冲击波管计算参数
Table 1. Computational parameters for weak shock tube
J N α ε/10−4 9 513 1.5 1.0 10 1 025 1.5 1.0 11 2 049 1.5 1.0 
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表 2 强冲击波管计算参数
Table 2. Computational parameters for strong shock tube
J N α ε/10−5 10 1 025 0.3 1.0 11 2 049 0.3 1.0 12 4 097 0.3 1.0
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