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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.

Liou-Steffen splitting (AUSM) schemes are popular for low Mach number simulations, however, like many numerical schemes for compressible flow they require careful modification to accurately resolve convective features in this regime. Previous analyses of these schemes usually focus only on a single discrete scheme at the convective limit, only considering flow with acoustic effects empirically, if at all. In our recent paper Hope-Collins & di Mare, 2023 we derived constraints on the artificial diffusion scaling of low Mach number schemes for flows both with and without acoustic effects, and applied this analysis to Roe-type finite-volume schemes. In this paper we form approximate diffusion matrices for the Liou-Steffen splitting, as well as the closely related Zha-Bilgen and Toro-Vasquez splittings. We use the constraints found in Hope-Collins & di Mare, 2023 to derive and analyse the required scaling of each splitting at low Mach number. By transforming the diffusion matrices to the entropy variables we can identify erroneous diffusion terms compared to the ideal form used in Hope-Collins & di Mare, 2023. These terms vanish asymptotically for the Liou-Steffen splitting, but result in spurious entropy generation for the Zha-Bilgen and Toro-Vasquez splittings unless a particular form of the interface pressure is used. Numerical examples for acoustic and convective flow verify the results of the analysis, and show the importance of considering the resolution of the entropy field when assessing schemes of this type., Comment: 34 pages, 17 figures, submitted to journal