  We study the iterative methods for large moment systems derived from the
linearized Boltzmann equation. By Fourier analysis, it is shown that the direct
application of the block symmetric Gauss-Seidel (BSGS) method has slower
convergence for smaller Knudsen numbers. Better convergence rates for dense
flows are then achieved by coupling the BSGS method with the micro-macro
decomposition, which treats the moment equations as a coupled system with a
microscopic part and a macroscopic part. Since the macroscopic part contains
only a small number of equations, it can be solved accurately during the
iteration with a relatively small computational cost, which accelerates the
overall iteration. The method is further generalized to the multiscale
decomposition which splits the moment system into many subsystems with
different orders of magnitude. Both one- and two-dimensional numerical tests
are carried out to examine the performances of these methods. Possible issues
regarding the efficiency and convergence are discussed in the conclusion.
