We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either computationally expensive or hard to converge, making these methods questionable for the simulation of highly nonequilibrium flows. By examining the convergence issue of Grad's moment methods, we propose a new moment hierarchy to bridge the hydrodynamic models and the kinetic equation, allowing nonlinear moment methods to be used as a numerical tool to discretize the velocity space for high-speed flows. For the case of one-dimensional velocity, the method is formulated for odd number of moments, and it can be extended seamlessly to the three-dimensional case. Numerical tests show that the method is capable of predicting shock structures with high Mach numbers accurately, and the results converge to the solution of the Boltzmann equation as the number of moments increases. Some applications beyond the shock structure problem are also considered, indicating that the proposed method is suitable for computation of transitional flows.

我们调查了许多矩层次结构并测试了它们在计算一维冲击结构方面的性能。发现对于高马赫数，矩层次结构要么计算成本高，要么难以收敛，这使得这些方法对于高度非平衡流动的模拟是有问题的。通过检查 Grad 矩方法的收敛问题，我们提出了一种新的矩层次来连接流体动力学模型和动力学方程，允许将非线性矩方法用作数值工具来离散化高速流动的速度空间。对于一维速度的情况，该方法是针对奇数矩制定的，并且可以无缝扩展到三维情况。数值试验表明，该方法能够准确预测高马赫数的激波结构，随着矩数的增加，结果收敛于玻尔兹曼方程的解。还考虑了冲击结构问题之外的一些应用，表明所提出的方法适用于过渡流的计算。