arXiv:2610.33537v1 Announce Type: new Abstract: The fast iterative moment (FIM) method developed in [G. Li, C. Wang, and Z. Hu, Comput. Fluids, 311 (2026), 107031] achieves fast convergence for near-continuum gas flows modeled by the Boltzmann-BGK equation through alternating iterations between the high-order moment system and the consistent hydrodynamic equations. In this paper, we extend the FIM solver to the Shakhov model and improve it in the following aspects. A finite volume method with linear reconstruction is employed for high-resolution spatial discretization of both systems. Following the original FIM method, an implicit-explicit scheme treating the Shakhov collision term implicitly is adopted for the high-order moment system. A local Fourier analysis of a linearized one-dimensional BGK model reveals the need for more hydrodynamic iterations at smaller Knudsen numbers and their slower convergence on refined grids. Motivated by these findings, a Newton-multigrid method with a new regularization strategy is introduced for the hydrodynamic equations, generally providing an effective hydrodynamic correction with one step per alternating iteration, thereby reducing the need for empirical tuning. Numerical experiments for planar Couette flow, lid-driven cavity flow, and heat transfer in a bottom-heated cavity are carried out to investigate the performance of the improved FIM solver. The results demonstrate improved efficiency and robustness over the original FIM method, particularly at small Knudsen numbers, and favorable performance with both first- and second-order spatial discretizations.