We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady-state computation greatly. The behavior of the solver is also numerically investigated. It is shown that the convergence rate increases, indicating the solver would be more efficient, as the total levels increases. Three order reduction strategies of the solver are considered. Numerical results show that the most efficient order reduction strategy would be $m_{l-1} = lceil m_{l} / 2 rceil$.
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