  This paper develops a numerical procedure to accelerate the convergence of
the Favre-averaged Non-Linear Harmonic (FNLH) method. The scheme provides a
unified mathematical framework for solving the sparse linear systems formed by
the mean flow and the time-linearized harmonic flows of FNLH in an explicit or
implicit fashion. The approach explores the similarity of the sparse linear
systems of FNLH and leads to a memory efficient procedure, so that its memory
consumption does not depend on the number of harmonics to compute. The proposed
method has been implemented in the industrial CFD solver HYDRA. Two test cases
are used to conduct a comparative study of explicit and implicit schemes in
terms of convergence, computational efficiency, and memory consumption.
Comparisons show that the implicit scheme yields better convergence than the
explicit scheme and is also roughly 7 to 10 times more computationally
efficient than the explicit scheme with 4 levels of multigrid. Furthermore, the
implicit scheme consumes only approximately $50\%$ of the explicit scheme with
four levels of multigrid. Compared with the full annulus unsteady Reynolds
averaged Navier-Stokes (URANS) simulations, the implicit scheme produces
comparable results to URANS with computational time and memory consumption that
are two orders of magnitude smaller.
