  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
non-equilibrium 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 the computation of transitional flows.
