# Spatiotemporal Evolution of Gas in Transmission Fluid under Acoustic Cavitation Conditions
## 2 Methods
### Numerical Solution Method  
According to the Harten-Lax theorem, the condition for quasi-linear hyperbolic conservation equations to satisfy Roe decomposition is that there is Matrix \(\mathbf{B}(\mathbf{U}_{\mathrm{L}},\mathbf{U}_{\mathrm{R}})\) such that \(\mathbf{F}(\mathbf{U}_{\mathrm{R}})-\mathbf{F}(\mathbf{U}_{\mathrm{L}})= \mathbf{B}(\mathbf{U}_{\mathrm{L}},\mathbf{U}_{\mathrm{R}})(\mathbf{U}_{ \mathrm{R}}-\mathbf{U}_{\mathrm{L}})\), and Matrix \(\mathbf{B}\) should have real eigenvalues and eigenvectors. Thus, Equation (22) is transformed from the nonlinear conserved hyperbolic Equation (1) to the linear conserved hyperbolic equation utilizing a Jacobian coefficient Matrix \(\mathbf{B}\). At the same time, the decomposed coefficient Matrix \(\mathbf{\overline{B}}\) can be obtained by a similar transformation. \[\mathbf{\overline{B}}=\mathbf{\overline{R}}\mathbf{\overline{\Lambda}}\mathbf{ \overline{L}} \tag{32}\]  
\[\mathbf{\overline{R}}= \begin{bmatrix}1&1\\ \frac{1}{\overline{u}+\overline{\gamma}}&\frac{1}{\overline{u}-\overline{ \gamma}}\end{bmatrix} ,\mathbf{\overline{\Lambda}}= \begin{bmatrix}\overline{u} +\overline{\gamma}&0\\ 0&\overline{u}-\overline{\gamma}\end{bmatrix} ,\mathbf{\overline{\Lambda}}= \frac{1}{2\overline{\gamma}} \begin{bmatrix}\overline{\gamma}-\overline{u}&1 \\ \overline{u}+\overline{\gamma}&-1\end{bmatrix}  \tag{33}\]  
where the matrices \(\mathbf{R}\) and \(\mathbf{L}\) are the right and left eigenvector matrices of the coefficient matrix \(\mathbf{B}\), respectively. \(\mathbf{\Lambda}\) is the eigenvalue diagonal matrix of the coefficient Matrix \(\mathbf{B}\). Expanding \(\mathbf{U}_{\mathrm{R}}-\mathbf{U}_{\mathrm{L}}\) by the eigenvector \(\mathbf{\overline{R}}_{k}\). \[\mathbf{U}_{R}-\mathbf{U}_{L}=\sum_{k=1}^{2}\mathbf{\overline{R}}_{k}\overline {\sigma}_{k} \tag{34}\]  
\[\overline{\sigma}_{k}=\mathbf{\overline{L}}(\mathbf{U}_{R}-\mathbf{U}_{L}) \tag{35}\]  
Noting \(\Delta\rho=\rho_{R}-\rho_{L}\), \(\Delta\rho\mathbf{U}=\rho_{R}\mathbf{U}_{R}-\rho_{L}\mathbf{U}_{L}\), and to obtain:  
\[\Delta\rho=\overline{\sigma}_{1}+\overline{\sigma}_{2} \tag{36}\]  
\[\Delta(\rho u)=\overline{u}\Delta\rho+\overline{\gamma}(\overline{\sigma}_{1}- \overline{\sigma}_{2}) \tag{37}\]  
\[\Delta(\rho u^{2}+P)=(\overline{u}^{2}+\overline{\gamma}^{2})\Delta\rho+2 \overline{u}\overline{\gamma}(\overline{\sigma}_{1}-\overline{\sigma}_{2}) \tag{38}\]  
Taking \(\overline{\sigma}_{1}=\frac{1}{2\overline{\gamma}^{2}}(\Delta P+\overline{ \rho}\overline{\gamma}\Delta u)\) and \(\overline{\sigma}_{2}=\frac{1}{2\overline{\gamma}^{2}}(\Delta P+\overline{ \rho}\overline{\gamma}\Delta u)\), then combining Equations (34)-(38) to obtain:  
\[\overline{u}=\frac{\sqrt{\rho_{R}}u_{R}+\sqrt{\rho_{L}}u_{L}}{\sqrt{\rho_{R}}+ \sqrt{\rho_{L}}} \tag{39}\]  
\[\overline{\rho}=\sqrt{\rho_{R}\rho_{L}} \tag{40}\]  
Specifically, \(\overline{\gamma}\) is obtained from Equation (31). In this study, the Roe format decomposition with the Steger-Warming flux splitting method is used to discretize the flux of Equation (22) in the space term, and the Euler forward differencing is used for the time term. Attaining the following differencing format:  
\[\mathbf{U}_{i}^{j+1}=\mathbf{U}_{i}^{j}-\frac{\Delta t}{\Delta x}\Big{[} \mathbf{F}(\mathbf{U}_{i}^{j},\mathbf{U}_{i+1}^{j})-\mathbf{\hat{F}}(\mathbf{ U}_{i-1}^{j},\mathbf{U}_{i}^{j})\Big{]}+\Delta t\mathbf{M}_{i}^{j+\frac{1}{2}} \tag{41}\]  
The numerical fluxes \(\mathbf{\hat{F}}\Big{(}\mathbf{U}_{i}^{j},\mathbf{U}_{i+  
\[\hat{\mathbf{F}}(\mathbf{U}_{i-1}^{j},\mathbf{U}_{i}^{j})=\frac{1}{2}\Bigg{[} \mathbf{F}(\mathbf{U}_{i-1}^{j})+\mathbf{F}(\mathbf{U}_{i}^{j})-\sum_{k=1}^{2} |\overline{\mathbf{\Lambda}}_{k}|\overline{\mathbf{R}}_{k}\overline{\mathbf{ \sigma}}_{k}\Bigg{]} \tag{43}\]  
The Steger-Warming flux splitting method can improve the calculation accuracy of the scheme. However, when \(|u|\)\(=0\) or on the sound velocity line (\(|u\pm\gamma|\)\(=0\)), the flow discontinuity differentiable phenomenon occurs, which can lead to numerical solution oscillation.