  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.
