# An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number
## 5 A New Low-Speed AUSM-Type Scheme  
### Construction of the Mass Flux  
It is well known that all the mass fluxes of upwind schemes can be expressed as the form of Equation (103) and are able to be the mass flux of AUSM-type schemes. Among them, the original [PERSON] scheme's mass flux wins the highest praise because it is based on the Euler equation's linearization to be with high-accuracy [32]. Therefore, we adopt it as a starting point in our new scheme's construction.  
In general, the [PERSON] scheme's mass flux can be written as follows:  
\[\hat{m}_{i+1/2}=\frac{1}{2}\Big{[}(\rho V_{n})_{i+1/2}^{+}+(\rho V_{n})_{i+1/2 }^{-}\Big{]}+D_{R\alpha e}^{\rho}\Delta\rho+D_{R\alpha e}^{V_{n}}\Delta V_{n}+ D_{R\alpha e}^{p}\Delta p \tag{105}\]  
where  
\[D_{R\alpha e}^{\rho}=-\frac{\left|\lambda\,_{1}\right|}{2} \tag{106}\]  
\[D_{R\alpha e}^{V_{n}}=\frac{(\left|\lambda\,_{3}\right|-\left|\lambda\,_{2} \right|)}{4\bar{c}} \tag{107}\]  
\[D_{R\alpha e}^{p}=\frac{2|\lambda\,_{1}|-(\left|\lambda\,_{2}\right|+\left| \lambda\,_{3}\right|)}{4\bar{c}^{2}} \tag{108}\]  
\[\begin{array}{l}\lambda\,_{1}=\widetilde{V}_{n}\\ \lambda\,_{2}=\widetilde{V}_{n}+\bar{c}\\ \lambda\,_{3}=\widetilde{V}_{n}-\bar{c}\end{array} \tag{109}\]  
It is reminded that the superscript \(\sim\) means the Roe average in this section and all the following sections.  
When the Mach number is small enough, we can obtain the following relationships:  
\[\begin{array}{l}V_{n}=O(c^{0})\ c=O(c^{1})\\ \Rightarrow M=\frac{V_{n}}{c}=O(c^{-1})\\ \Rightarrow c=O(M^{-1})\end{array} \tag{110}\]  
and the coefficient \(D_{R\alpha e}^{p}\) can be written as follows:  
\[D_{R\alpha e}^{p}=\frac{\left|\widetilde{V}_{n}\right|-\widetilde{c}}{2\bar{ c}^{2}}\approx-\frac{1}{2\bar{c}}=O(M) \tag{111}\]  
Thus, the Roe scheme's mass flux satisfies the Lemma 1 in Section 4.3.1, and we can adopt it as our new scheme's mass flux for low speeds.  
### Construction of the Mass Flux  
In general, the pressure fluxes of the AUSM-type schemes are able to be written as:  
\[\hat{p}_{i+1/2}=\frac{p_{i+1/2}^{+}+p_{i+1/2}^{-}}{2}+d_{p} \tag{112}\]  
where \(d_{p}\) is the dissipation part. For example, AUSM+'s pressure flux in Equation (19) can be written in this form as follows:  
\[\begin{array}{l}\hat{p}_{i+1/2}=\frac{p_{i+1/2}^{+}+p_{i+1/2}^{-}}{2}+\frac {\beta_{+}|_{a=3/16}-\beta_{-}|_{a=3/16}}{2}\Big{(}p_{i+1/2}^{+}-p_{i+1/2}^{-} \Big{)}\\ \hskip 142.26378 pt+(\beta_{+}|_{a=3/16}+\beta_{-}|_{a=3/16}-1)\,\frac{\big{(} p_{i+1/2}^{+}+p_{i+1/2}^{+}\big{)}}{2}\end{array} \tag{113}\]  
and  
\[\begin{array}{l}d_{p}=\frac{\beta_{+}|_{a=3/16}-\beta_{-}|_{a=3/16}}{2} \Big{(}p_{i+1/2}^{+}-p_{i+1/2}^{-}\Big{)}+\\ \hskip 142.26378 pt\beta_{+}|_{a=3/16}+\beta_{-}|_{a=3/16}-1\,\frac{\big{(}p_{ i+1/2}^{+}+p_{i+1/2}^{-}\big{)}}{2}\end{array} \tag{114}\]Combining Equation (114) with Equation (50), we can obtain  
\[d_{p}\approx\frac{15}{16}\Big{(}M_{i+1/2}^{+}p_{i+1/2}^{+}-M_{i+1/2}^{-}p_{i+1/2 }^{-}\Big{)}=O(M) \tag{115}\]  
Thus, the pressure flux does not satisfy the Lemma 2 in Section 4.3.2. To make the pressure flux applicable to low speed simulations, a function/\(f\) which satisfies Equation (116) should be used to modify the dissipation part:  
\[f=\begin{cases}\propto|M|^{k}\ (k\geq 1)&|M|<1\\ =1&|M|\geq 1\end{cases} \tag{116}\]  
In fact, many choices can be made to meet Equation (116), such as  
\[f=\min\Big{(}|M|^{k},1\Big{)}\ k\geq 1 \tag{117}\]  
Figure 2 shows that the function/\(f\) changes with different \(k\)'s definitions. However, the disparity becomes smaller with \(k\) becoming larger. In fact, our private computations show that the numerical dissipation is nearly unchangeable when \(k\geq 3\). Thus, we define the function/\(f\) and the pressure flux as follows:  
\[\hat{p}_{i+1/2}=\frac{p_{i+1/2}^{+}+p_{i+1/2}^{-}}{2}+f\cdot d_{p} \tag{118}\]  
\[f=\min\Big{(}|M|^{3},1\Big{)} \tag{119}\]  
All in all, with the mass flux defined in Equation (105) and the pressure flux defined in Equation (118), a new low-speed scheme which we call AUSMLS (AUSM for Low speed) is obtained.