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.

翻译：Liou-Steffen分裂（AUSM）格式广泛应用于低马赫数数值模拟，然而，与许多可压缩流动数值格式类似，它们需要精细修正才能在此工况下准确解析对流特征。此类格式的既有分析通常仅关注单一离散格式处于对流极限时的表现，即便涉及声学效应，也多依赖经验性处理。在我们近期的工作（Hope-Collins & di Mare, 2023）中，推导了适用于含/不含声学效应的低马赫数格式人工耗散标度约束，并将该分析应用于Roe型有限体积格式。本文构建了Liou-Steffen分裂及其紧密相关的Zha-Bilgen与Toro-Vasquez分裂的近似耗散矩阵，并利用Hope-Collins & di Mare (2023) 所发现的约束条件，推导并分析了各分裂格式在低马赫数下的必要标度特性。通过将耗散矩阵变换至熵变量空间，我们能够识别出与Hope-Collins & di Mare (2023) 中理想形式存在差异的误差耗散项。这些误差项在Liou-Steffen分裂中渐近消失，但除非采用特定形式的界面压力，否则会在Zha-Bilgen与Toro-Vasquez分裂中导致虚假熵增。声学与对流传动的数值算例验证了分析结果，并揭示了评估此类格式时解析熵场的重要性。