# An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number
## 1 Introduction  
Since being proposed by [PERSON] [1], AUSM-type schemes have been widely used to simulate the inviscid fluxes of the Navier-Stokes equations in aerospace engineering areas [2, 3]. Among them, AUSM+ [4], AUSMPW+ [5, 6], AUSM+UP [7], and SLAU [8] have garnered the highest praises due to their outstanding performances. However, all these schemes suffer various problems in both cases of low speed and high speed in simulations.  
For example, AUSM+ is inclined to show the expansion shock unphysically [5]. Furthermore, it is similar to AUSMPW+ in that both of them are unable to obtain physical solutions at low speeds, which is illustrated in Ref. [9] and Section 4.3 in this paper. To make them applicable to the simulations of low speeds, preconditioning matrix methods should be used [10, 11, 12, 13, 14, 15]. However, these methods need one tuning coefficient at least and encounter the global cut-off problem where high speeds and low speeds are coexisting. In addition, they inherit shortcomings in high Mach number flows from the original ones [16, 17]. For example, the unphysical expansion shock still exists in hypersonic flows for the AUSM+ scheme coupled with preconditioned matrix. For these, the applicability of the preconditioning matrix methods is limited. AUSM+UP has been proposed, which is able to ensure a high level of accuracy for all speeds. However, it still encounters the global cut-off problem which is the same as the preconditioning matrix methods [18]. SLAU avoids this problem and can have a high level of accuracy for low speeds. However, it is similar to the others in that all these schemes are not robust against the shock anomaly [19, 20, 21, 22, 23]. Moreover, it still shows unphysical solutions when the Mach number approaches 0 [8].  
In this paper, a new scheme called AUSMAS is proposed. In terms of high speeds, it avoids the expansion shock's appearance by modifying the maximum and minimumspeeds of the wave. Furthermore, it has high robustness against the abnormal shock by damping the damping rates of the density perturbation based on theoretical liner analyses. In terms of low speeds, it is capable of avoiding the checkerboard problem, non-physical behavior and the problem of the global cut-off, based on theoretical analyses. All in all, the numerical tests in the following sections indicate that the AUSMAS scheme proposed has the following attractive properties:  
1. Robustness against the abnormal shock in capturing strong shock;
2. High discontinuity resolution;
3. Robustness in simulating expansion regions and being free from the appearance of the entropy-violating expansion shock;
4. High resolution at low speeds;
5. Independent of any tunable parameter.  
This paper is organized as follows. In Section 2, a brief overview will be given of the governing equations in 2D forms. Introductions to the reconstruction methods and the parameter free AUSM-type schemes are given in Section 3. We conduct a theoretical study of the AUSM-type schemes on low speed issues in Sections 4 and 5. AUSMAS's construction for high speeds is shown in the Section 6. To ensure a thorough analysis, we conduct several numerical tests in Section 7. The last section contains concluding remarks.