# An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number
## 4 Theoretical Analyses on Low Speed Issues
### The AUSM-Type Schemes
#### 4.2.3. Slau  
As can be seen in Equations (29), (31) and (32), the following equations can be obtained with the Mach number approaching 0:  
\[f_{p}\approx 1-2\hat{M} \tag{80}\]  
\[g=0 \tag{81}\]  
\[\left|\overline{V}_{n}\right|\approx\frac{\left|\overline{V}_{n}\right|_{i+1/2}^ {+}+\left|\overline{V}_{n}\right|_{i+1/2}^{-}}{2} \tag{82}\]  
Combine Equations (80)-(82) with Equations (49) and (50) and applying SLAU in a finite volume context of first order, we can obtain set of semi-discrete equations as follows:  
\[\delta\frac{\partial}{\partial t}\rho_{i}-\frac{1}{2}\underset{l\in\wp(i)}{ \sum}[\hat{m}_{il}+\left|\hat{m}_{il}\right|]-\frac{1}{2}\underset{l\in\wp(i)} {\sum}[\hat{m}_{il}-\left|\hat{m}_{il}\right|]=0 \tag{83}\]  
\[\delta\frac{\partial}{\partial t}\rho_{i}u_{i}-\underset{l\in\wp(i)}{\sum} \frac{u_{i}}{2}[\hat{m}_{il}+\left|\hat{m}_{il}\right|]-\underset{l\in\wp(i)} {\sum}\frac{u_{i}}{2}[\hat{m}_{il}-\left|\hat{m}_{il}\right|] \tag{84}\]  
\[\delta\frac{\partial}{\partial t}\rho_{i}u_{i}-\underset{l\in\wp(i)}{\sum} \frac{v_{i}}{2}[\hat{m}_{il}+\left|\hat{m}_{il}\right|]-\underset{l\in\wp(i)} {\sum}\frac{v_{i}}{2}[\hat{m}_{il}-\left|\hat{m}_{il}\right|]\]  
\[-\frac{1}{2}\underset{l\in\wp(i)}{\sum}\left[p_{l}\left(n_{y}\right)_{il} \right]-\frac{15}{32}\underset{l\in\wp(i)}{\sum}\left[\left(\frac{\left(\frac {u_{i}\cdot\cdot\cdot\cdot\cdot_{il}}{c_{il}}\right)}{c_{il}}+\frac{u_{i}\cdot \cdot\cdot_{il}}{c_{il}}\right)\Delta\rho_{il}\left(n_{y}\right)_{il}\right] \tag{85}\]  
\[-\frac{15}{16}\underset{l\in\wp(i)}{\sum}\left[\frac{\left(p_{i}+p_{l}\right) }{\sqrt{2}}\frac{\sqrt{u_{i}^{2}+v_{i}^{2}+u_{i}^{2}+v_{i}^{2}}}{c_{il}} \left(\frac{\left(\frac{u_{i}\cdot\cdot\cdot_{il}}{c_{il}}\right)}{c_{il}}- \frac{u_{i}\cdot\cdot\cdot_{il}}{c_{il}}\right)\left(n_{y}\right)_{il}\right]=0\]  
\[\delta\frac{\partial}{\partial t}\rho_{i}u_{i}-\underset{l\in\wp(i)}{\sum} \frac{v_{i}}{2}[\hat{m}_{il}+\left|\hat{m}_{il}\right|]-\underset{l\in\wp(i)} {\sum}\frac{h_{l}}{2}[\hat{m}_{il}-\left|\hat{m}_{il}\right|]=0 \tag{86}\]  
where  
\[\hat{m}_{il}=\frac{\mathbf{u}_{i}\cdot\cdot\cdot_{il}}{2}\rho_{i}+\frac{ \mathbf{u}_{i}\cdot\cdot\cdot_{il}}{2}\rho_{l}-\frac{\left|\mathbf{u}_{i}\cdot \cdot\cdot_{il}\right|+\left|\mathbf{u}_{i}\cdot\cdot\cdot_{il}\right|}{2}( \rho_{l}-\rho_{i})\]  
Similar as the analyses above, the systems in Equations (83)-(87) can be non-dimensionalized as follows:  
\[g=0 \tag{81}\]0  
\[g=0 \tag{81}\]1  
\[g=0 \tag{81}\]2  
\[g=0 \tag{81}\]3\[ \begin{array}{l}\widetilde{\delta}\frac{\partial}{\partial t}\rho_{i}v_{i}-\sum \limits_{l\in\pi(i)}\frac{v_{l}}{2}[\widetilde{m}_{il}+|\widetilde{m}_{il}|]- \sum\limits_{l\in\pi(i)}\frac{v_{l}}{2}[\widetilde{m}_{il}-|\widetilde{m}_{il} |]\\ -\frac{1}{2M_{*}^{2}}\sum\limits_{l\in\pi(i)}[p_{l}\big{(}n_{y}\big{)}_{il}]- \frac{15}{32M_{*}}\sum\limits_{l\in\pi(i)}\left[\left(\frac{\mathbf{u}_{i} \cdot\overrightarrow{n_{il}}}{c_{il}}+\frac{\mathbf{u}_{i}\cdot\overrightarrow{ n_{il}}}{c_{il}}\right)\Delta p_{il}\big{(}n_{y}\big{)}_{il}\right]\\ -\frac{15}{16}\sum\limits_{l\in\pi(i)}\left[\frac{(p_{i}+p_{j})}{\sqrt{2}} \frac{\sqrt{l_{i}^{2}+p_{j}^{2}+p_{l}^{2}+p_{l}^{2}}}{c_{il}}\left(\frac{ \mathbf{u}_{i}\cdot\overrightarrow{n_{il}}}{c_{il}}-\frac{\mathbf{u}_{i} \cdot\overrightarrow{n_{il}}}{c_{il}}\right)\big{(}n_{y}\big{)}_{il}\right]=0 \end{array}  \tag{90}\]  
\[ \begin{array}{l}\widetilde{\delta}\frac{\partial}{\partial t}\rho_{i}h_{i}- \sum\limits_{l\in\pi(i)}\frac{h_{i}}{2}[\widetilde{m}_{il}+|\widetilde{m}_{il }|]-\sum\limits_{l\in\pi(i)}\frac{h_{l}}{2}[\widetilde{m}_{il}-|\widetilde{m}_ {il}|]=0\end{array}  \tag{91}\]  
where  
\[