# An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number
## 6 The AUSMLS Scheme's Improvements for All Speeds
### 2D Inviscid NCA0012 Airfoil  
The flow conditions and the calculating settings are the same as those in Section 5.3. Furthermore, the computations within the two different grids are conducted to check the grid dependencies of the schemes, in which the grid with 120 (circumferential) \(\times\) 50 (wall normal) is called the coarse grid (Figure 3) and another with 240 (circumferential) \(\times\) 100 (wall normal) is called the dense grid. In addition, we adopt three different reconstruction methods to analyze the schemes' disparities. These reconstruction technologies including the reconstruction of first order, MUSCL reconstruction of second order combined with van Leer limiter [28], third order reconstruction combined with van Leer limiter [35]. In terms of the time integration, the CFL number is equal to 10 with the implicit LU-SGS approach adopted. To achieve at least ten orders reduction of the density residual, all the computations are conducted for 100,000 steps. The error histories of these schemes combined with first reconstruction method in different grids are shown in Figure 10. In them we can see that all the computations are convergent after iterating 100,000 steps and AUSMAS converges much faster than the others. Figure 11 indicates that AUSMAS and SLAU are capable of obtaining accurate flow fields at low speeds while the others obtain inaccurate ones. Furthermore, the AUSMAS scheme avoids the weak oscillations in the vicinity of the wall while SLAU scheme is inclined to show this unphysical phenomenon.  
Figure 12a depicts the pressure fluctuations of the schemes vs. the Mach number with first order reconstruction method. In this paper, the pressure fluctuation is defined as \(\text{Ind}(p)=(P_{\text{max}}-P_{\text{min}})/P_{\text{max}}\).  
Figure 9: Pressure and density distributions of the Sod shock tube II test (\(t=1.5\) s).  
Figure 11: Schemes’ pressure contours of inviscid flows around NACA0012 airfoil with the first order reconstruction (Ma = 0.1, coarse grid); (**a**) AUSM+; (**b**) AUSPW+; (**c**) SLAU; (**d**) AUSMAS.  
Figure 10: Error histories of inviscid flow over NACA0012 airfoil (\(M_{\infty}\) = 0.1, first order reconstruction, (a) coarse grid, (**b**) dense grid).  
According to the theoretical asymptotic prediction, in the continuous case, the pressure fluctuation is proportional to the square of the _Mach_ number. Therefore, none of these schemes are able to agree with the theoretical predictions except for the AUSMAS scheme.  
Figure 12b,c display the pressure fluctuations obtained by different schemes coupled with reconstruction methods of higher order. In them, we can find that only AUSMAS is able to obtain the physical solutions no matter which reconstruction method it is combined with at low speeds. Furthermore, the result of the AUSMAS scheme with the reconstruction method of first order is even better than the results of the other schemes combined with higher order reconstruction methods.  
Similarly, Figure 13 indicates that AUSMAS performs the best among all these schemes in the dense grid.  
In theory, the drag coefficient of the airfoil should be 0 in this case. Therefore, the drag coefficient is also a good measurement of the schemes' numerical errors [36].  
Figures 14 and 15 display the drag coefficients obtained by different schemes with different reconstruction methods or different computational grids. As can be seen in the results, the numerical error of the AUSMAS is much less than others in all these cases.  
Figure 12: Pressure fluctuations vs inflow Mach number (coarse grid, (**a**) first order reconstruction, (**b**) second order reconstruction, (**c**) third order reconstruction).  
Figure 14: Drag coefficients vs. inflow Mach number (coarse grid, (**a**) first order reconstruction, (**b**) second order reconstruction, (**c**) third order reconstruction).  
Figure 13: Resolutions of the pressure fluctuations with different schemes (first order reconstruction, dense grid).