# An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number
## 4 Theoretical Analyses on Low Speed Issues  
### The Continuous Case  
By ignoring the influence of the viscous terms, the compressible NS equations become the compressible Euler equations. In this section, we will discuss the singular limit for the compressible Euler equations when the Mach number is approaching zero. More details can be obtained in Refs. [29; 30].  
Let \(\rho^{*}=\max(\rho),u^{*}=\max(\sqrt{u^{2}+v^{2}})\), and the sound speed scale \(c^{*}=\sqrt{\gamma\max(p)/\rho^{*}}\). The flow variables can be normalized in the following forms:  
\[\begin{array}{l}\widetilde{\rho}=\frac{\rho}{p^{*}},\ \ \widetilde{u}=\frac{u}{u^{*}},\ \ \widetilde{v}=\frac{v}{v^{*}},\ \ \widetilde{x}=\frac{x}{x^{*}},\ \ \widetilde{y}=\frac{y}{y^{*}},\\ \widetilde{p}=\frac{p}{\rho^{*}(c^{*})^{2}},\ \ \widetilde{E}=\frac{E}{(c^{*})^{2}},\ \ \widetilde{t}=\frac{tu^{*}}{x^{*}}\end{array} \tag{36}\]  
where \(x^{*}\) is arbitrary length scale.  
The normalized Euler equations can be written as follows by adopting these non-dimensional variables into the compressible Euler equations.  
**Continuity Equation:**  
\[\frac{\partial}{\partial\widetilde{t}}\widetilde{\rho}+\mathrm{div}( \widetilde{\rho}\widetilde{\mathbf{u}})=0 \tag{37}\]  
**Momentum Equation:**  
\[\frac{\partial}{\partial\widetilde{t}}\widetilde{\rho}\widetilde{\mathbf{u}} +\mathrm{div}(\widetilde{\rho}\widetilde{\mathbf{u}}\otimes\widetilde{ \mathbf{u}})+\frac{1}{M_{*}^{2}}\
abla\widetilde{p}=0 \tag{38}\]  
**Energy Equation:**  
\[\frac{\partial}{\partial\widetilde{t}}\widetilde{\rho}\widetilde{e}+\mathrm{ div}(\widetilde{\rho}\widetilde{e}\widetilde{\mathbf{u}}+\widetilde{p} \widetilde{\mathbf{u}})=0 \tag{39}\]  
where \(M_{*}=\frac{\sqrt{u^{*}-u^{*}+v^{*}\rho^{*}}}{c^{*}}\) is the reference Mach number and \(\widetilde{\mathbf{u}}=\left(u,v\right)^{\mathrm{T}}\) is the velocity vector.  
By using powers of the reference Mach number to expand the dimensionless variables asymptotically, the flow variables non-dimensioned can be written as follows:  
\[\widetilde{u}=\widetilde{u}_{0}+M_{*}\widetilde{u}_{1}+M_{*}^{2} \widetilde{u}_{2}+\ldots \tag{40}\]  
\[\widetilde{v}=\widetilde{v}_{0}+M_{*}\widetilde{v}_{1}+M_{*}^{2} \widetilde{v}_{2}+\ldots \tag{41}\]  
\[\widetilde{p}=\widetilde{p}_{0}+M_{*}\widetilde{p}_{1}+M_{*}^{2} \widetilde{p}_{2}+\ldots \tag{42}\]  
\[\widetilde{\rho}=\widetilde{\rho}_{0}+M_{*}\widetilde{\rho}_{1}+M_{*}^{2} \widetilde{\rho}_{2}+\ldots \tag{43}\]  
\[\widetilde{e}=\widetilde{e}_{0}+M_{*}\widetilde{e}_{1}+M_{*}^{2} \widetilde{e}_{2}+\ldots \tag{44}\]  
If we combine Equations (40)-(44) with Equations (37)-(39) and merge the same terms based on power of \(M_{*}\), the following conclusions can be obtained (the superscript \(\sim\) is dropped):  
1. Order \(1/M_{*}^{2}\) \[\
abla p_{0}=0\] (45)
2. Order \(1/M_{*}\) \[\
abla p_{1}=0\] (46)
3. Order \(1\) \[\begin{array}{l}\frac{\partial}{\partial t}\rho_{0}+\mathrm{div}(\rho_{0} \mathbf{u}_{0})=0\\ \frac{\partial}{\partial t}\rho_{0}\mathbf{u}_{0}+\mathrm{div}(\rho_{0} \mathbf{u}_{0}\otimes\mathbf{u}_{0})+\
abla p_{2}=0\\ \frac{\partial}{\partial t}\rho_{0}\epsilon_{0}+\mathrm{div}(\rho_{0} \epsilon_{0}\mathbf{u}_{0}+p_{0}\mathbf{u}_{0})=0\end{array}\] (47)In other words, if in the low Mach limit, the pressure varies with the square of the Mach number in space as follows:  
\[p(x,t)=P_{0}(t)+M_{*}^{2}p_{2}(x,t) \tag{48}\]